1 ALGEBRA READINESS DIAGNOSTIC TEST PRACTICE

Directions: Study the examples, work the problems, then check your answers at the end of each topic. If you don’t get the answer given, check your work and look for mistakes. If you have trouble, ask a math teacher or someone else who understands the topic.

TOPIC 1: INTEGERS A. What is an integer? Any natural number (1, 2, 3, 4, 5,…), its opposite (-1,-2, -3, -4, -5,…), or zero (0). (Integers are useful for problems involving “below normal,” debts, “below sea level,” etc.)

Problems 1-10: Identify each number as an integer (I) or not an integer (NI):

1. 367 6. 0 2. !4.4 7.

!

" 23

3.

!

21

2 8. 0.027

4. –1010 9.

!

1

2

5.

!

100 10. 23

Problems 11-14: Write the opposite of each integer:

11. 42 13. 0 12. !3 14. !43

Problems 15-19: Choose the greater:

15. 5, –10 18. –5, 0 16. 5,–5 19. –5,–10 17. 5, 0

20. What is the result of adding an integer and its opposite?

21. What number is its own opposite?

B. Absolute Value: Absolute value is used for finding distance, explaining addition of integers, etc. The absolute value of a positive number or zero is itself. The absolute value of a negative number is its opposite.

Problems 22-26: Choose the integer with the greater absolute value:

22. 4 or –3 25. 3 or 0 23. –4 or 3 26. – 3 or 0 24. 3 or –3

C. Adding, subtracting, multiplying and dividing integers:

To add two integers:

Both positive: add as natural numbers: example: Add 4 and 3: 4 + 3 = 7

Both negative: add as though positive; make the result negative:

example: Add –4 and –3: Treat as positive and add: 4 + 3 = 7 . The answer is –7 because it must be negative.

One positive, one negative: treat each as positive, subtract, make the answer sign of the one with the greater absolute value: example: Add –4 and 3: 4 ! 3 = 1 ; the answer is

–1 because –4 has the greater absolute value. example: Add 4 and –3: 4 ! 3 = 1 ; the answer is

1 because 4 has the greater absolute value.

Problems 27-33: Add the two integers:

27. 4 and –3 (This means 4( ) + !3( ) )

28. 4 and 3 31. –4 and 3 29. –4 and –3 32. 16 and –7 30. 4 and 0 33. –3 and 0

To subtract two integers: add the opposite of the one to be subtracted: example: 3 subtract –4, or 3( ) ! !4( ) : The opposite

of –4 is 4, so we add 4 (rather than subtract –4). We change the problem from 3( ) ! !4( ) to 3( ) + 4( ) , which we know how to do: 3( ) + 4( ) = 3 + 4 = 7

example: !4( ) ! 3( ) : Add the opposite of 3, namely –3: !4( ) ! 3( ) = !4( ) + !3( ) = !7

example: 4( ) ! 3( ) = 4( ) + !3( ) = 1 example: !5 ! 8 = !5( ) ! 8( ) = !5( ) + !8( ) = !13

Problems 34-43: Calculate:

34. 12( ) ! 3( ) =

35. !12 !3 = (Hint: this means

!

"12+ "3( ))

36. !12 ! !3( ) = 40. 0 ! 3 = 37. 3 !12 = 41. 0 + 4 = 38. !3 !12 = 42. !12 +3 = 39. !7( ) ! !7( ) = 43. !3( ) + !12( ) =

To multiply two integers: 1stinteger! 2ndinteger= Answer

+ + + – + – + – – – – +

2 Both positive: multiply as two natural numbers. example: 3( ) ! 4( ) = 3 ! 4 =12

Both negative: multiply as if positive; and make the answer positive. and remember, two negatives make a positive. When multiplying two negative numbers, you always get a positive answer. example: !3( ) !4( ) so 3 ! 4 = 12 ; make it

positive, and the answer is 12.

One positive, one negative: When multiplying a negative number and a positive number, the answer is always negative. example: 3( ) !4( )so 3 ! 4 = 12 ; make the

answer negative; answer –12.

Problems 44-55: Multiply:

44. 3 ! "4( ) = 50.

!

"4( ) •0 = 45.

!

3( ) • "4( ) = 51. 02 = 46. 3( ) !4( ) = 52. !3( )

2

= 47. 3 !4( ) = 53. 4( )

2

= 48. !3( ) !4( ) = 54.

!

"3( ) • 4 = 49. !3 !4( ) = 55.

!

3• 4 =

Reciprocals are used for dividing. Every integer except zero has a reciprocal. The reciprocal is the number that multiplies the integer to give 1.

example:

!

6• 1

6=1, so the reciprocal of 6 is

!

1

6.

(And the reciprocal of

!

1

6 is 6.)

example:

!

"4( ) " 1

4( ) =1, so the reciprocal of –4

is

!

" 1

4.

Problems 56-59: Find the reciprocal:

56. –5 57. 1 58. 10 59. –1

60. What number is its own reciprocal? (Can you find “more than one”?)

61. Using the reciprocal definition, explain why there is no reciprocal of zero.

To divide two integers: multiply by the reciprocal of the one to be divided by: example: 20 divided by !5 = 20 ÷ !5( ) .

The reciprocal of –5 is

!

" 15

so we multiply by

!

" 15

: 20 ÷ !5( ) = 20 " ! 1

5( ) =

!

20

1" # 1

5( ) = # 20•11•5

= # 205

= #4

example:

!

"520

= "5 ÷ 20 = "5• 1

20= " 1

4

example:

!

"3"6

= "3÷ "6( ) = "3• " 16( )

!

= 3

6= 1

2

(Note negative times negative is positive.)

example:

!

0

3= 0 ÷ 3 = 0• 1

3= 0

Problems 62-67: Calculate:

62. !14( ) ÷ !2( ) = 65.

!

"153

= 63. 2 ÷ 3 = 66.

!

"50

=(careful)* 64. 3 ÷ 2 = 67.

!

0

7=

* Problem 61 says

!

1

0 has no value (you cannot

divide by zero).

68. From the rule for division, why is it impossible to divide by zero?

To “sum” it all up: Positive+positive= larger positive Negative+negative=more negative Positive+negative= in between both

Positive! positive= positive Negative! negative= positive Positive! negative= negative

To subtract add the opposite. To divide, multiply by the reciprocal.

69. Given the statement “Two negatives make a positive.” Provide an example of a situation where the statement would be true and another when it would be false.

70. Write “18 divided by 30” in three ways: using

!

÷, ) , and using a fraction bar ––.

Problems 71- 80: Calculate:

71. 4 !10 + 3 ! 2 = 76. !2 !6( ) 8( ) + 9[ ] = 72. 4 ! 10 +3 ! 2( ) = 77. 5 + 3 ! 7( ) = 73. 4 + 3 ! 10 ! 2( ) = 78. 5 ! 3 ! 7( ) = 74. 6 8 ! 3( ) = 79. 5 ! 3 + 7 = 75. !6( ) 8( ) + 9 = 80. !1 + 2 ! 3 + 4 =

81. What is the meaning of “sum”, “product”, quotient”, and “difference”?

D. Factoring: If a number is the product of two (or more) integers, then the integers are factors of the number.

example: 40 = 4 !10, 2 ! 20, 1 ! 40 , and 8 ! 5 . So 1, 2, 4, 5, 8, 10, 20, 40 are all factors of 40. (So are all their negatives.) Problems 82-86: Find all positive factors of:

82. 10 83. 7 84. 24 85. 9 86. 1 If a positive integer has exactly two positive factors, it is a prime number. Prime numbers are

3 used to find the greatest common factor (GCF) and least common multiple (LCM), which are used to reduce fractions and find common denominators, which in turn are often needed for adding and subtracting fractions.

example: The only positive factors of 7 are 1 and 7, so 7 is a prime number.

example: 6 is not prime, as it has 4 positive factors: 1, 2, 3, 6.

87. From the prime number definition, why is 1 not a prime?

88. Write the 25 prime numbers from 1 to 100.

Every positive integer has one way it can be factored into primes, called its prime factorization.

example: Find the prime factorization (PF) of 30: 30 = 3! 10 = 3 ! 2 ! 5, so the PF of 30 is

!

2• 3•5. example:

!

72 = 2•2•2• 3• 3 = 23• 3

2 , the PF. (The PF can be found by making a “factor tree.”)

Problems 89-91: Find the PF:

89. 36 90. 10 91. 7

Greatest common factor (GCF) and least common multiple (LCM). If you need to review GCF or LCM, see the worksheet in this series: “Topic 2: Fractions”.

Problems 92-95: Find the GCF and the LCM of:

92. 4 and 6 93. 4 and 7

94. 4 and 8 95. 3 and 5

E. Word problems:

96. The temperature goes from !14° to 28°C. How many degrees Celsius does it change?

97. 28 ! !14( ) = 98. Derek owes $43, has $95, so “is worth”…? 99. If you hike in Death Valley from 282 feet

below sea level to 1000 feet above sea level, how many feet of elevation have you gained?

100. 1000 ! !282( ) = 101. A hike from 243 feet below sea level

(FBSL) to 85 FBSL means a gain in elevation of how many feet?

102. !85 ! !243( ) = 103. What number added to –14 gives –24? 104. What does “an integral number” mean? 105. Jim wrote a check for $318. His balance is

then $2126. What was the balance before he wrote the check?

106. What number multiplied by 6 gives –18? 107. If you hike downhill and lose 1700 feet of

elevation and end at 3985 feet above sea level (FASL), what was your starting elevation?

108. Anne was 38 miles south of her home. She drove 56 miles north. How far from home was she at that time and in what direction?

109. 5 subtracted from what number gives –12? 110. What number minus negative four gives ten?

Answers: 1. I 2. NI 3. NI 4. I 5. I 6. I 7. NI 8. NI 9. NI 10. I 11. –42 12. 3 13. 0 14. 64 15. 5 16. 5 17. 5 18. 0 19. –5 20. zero 21. zero 22. 4

23. –4 24. both same 25. 3 26. –3 27. 1 28. 7 29. –7 30. 4 31. –1 32. 9 33. –3 34. 9 35. –15 36. –9 37. –9 38. –15 39. 0 40. –3 41. 4 42. –9 43. –15 44. –12

45. –12 46. –12 47. –12 48. 12 49. 12 50. 0 51. 0 52. 9 53. 16 54. –12 55. 12 56.

!

" 1

5

57. 1 58.

!

1

10

59. –1 60. 1; also –1 61. no number times 0 = 1 62. 7 63.

!

2

3

64.

!

3

2

4 65. –5 66. no value (not defined) 67. 0 68. zero has no reciprocal 69. true if ! , false if

!

+. 70.

!

18 ÷ 30,

!

30 18) ,

!

18 30 71. –5 72. –7 73. –1 74. 30 75. –39 76. 78 77. 1 78. 9 79. 9 80. 2

81. +, !, ÷, " 82. 1, 2, 5, 10 83. 1, 7 84. 1, 2, 3, 4, 6, 8, 12, 24 85. 1, 3, 9 86. 1 87. 1 has one factor 88. 2, 3, 5, 7, 11, 13, 17, 19,

23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

89.

!

22•3

2 90.

!

2•5 91. 7 92. 2, 12 93. 1, 28 94. 4, 8

95. 1, 15 96. 42 97. 42 98. $52 99. 1282

100. 1282 101. 158 102. 158 103. –10 104. an integer 105. $2444 106. –3 107. 5685 FASL 108. 18 mi. N 109. –7 110. 6

TOPIC 2: FRACTIONS A. Greatest Common Factor (GCF): The GCF of two integers is used to simplify (reduce, rename) a fraction to an equivalent fraction. A factor is an integer multiplier. A prime number is a positive whole number with exactly two positive factors.

example: the prime factorization of 18 is

!

2• 3• 3, or

!

2• 32.

Problems 1-2: Find the prime factorization:

1. 24 2. 42

example: Find the factors of 42. Factor into primes:

!

42 = 2• 3• 7 1 is always a factor 2 is a prime factor 3 is a prime factor 4 is a prime factor 7 is a prime factor

!

2• 3 = 6 is a factor

!

2• 7 =14 is a factor

!

3• 7 = 21is a factor

!

2• 3• 7 = 42 is a factor Thus 42 has 8 factors.

Problems 3-4: Find all positive factors:

3. 18 4. 24

To find the GCF: example: Looking at the factors of 42 and 24,

we see that the common factors of both are 1, 2, 3, and 6, of which the greatest it 6; so: the GCF of 42 and 24 is 6. (Notice that “common factor” means “shared factor.”)

Problems 5-7: Find the GCF of:

5. 18 and 36 6. 27 and 36 7. 8 and 15

B. Simplifying fractions:

example: Reduce

!

27

36:

!

27

36= 9•39•4

= 9

9• 3

4=1• 3

4= 3

4

(Note that you must be able to find a common factor, in this case 9, in both the top and bottom in order to reduce.)

Problems 8-13: Reduce:

8.

!

13

52= 11.

!

16

64=

9.

!

26

65= 12.

!

24

42=

10.

!

3+63+9

= 13.

!

24

18=

C. Equivalent Fractions:

example:

!

3

4 is equivalent to how many eighths?

!

3

4=8( )

!

3

4=1• 3

4= 2

2• 3

4= 2•32•4

= 6

8

Problems 14-17: Complete:

14.

!

4

9=72

15.

!

3

5 is how many twentieths?

16.

!

56

100=50

17. How many halves are in 3? (Hint: think

!

3 = 3

1=2

)

D. Ratio: If the ratio of boys to girls in a class is 2 to 3, it means that for every 2 boys, there are 3 girls. A

5 ratio is like a fraction: think of the ratio 2 to 3 as the fraction

!

2

3.

example: If the class had 12 boys, how many girls are there? Write the fraction ratio:

!

number of boys

number of girls= 2

3= 12

Complete the equivalent fraction:

!

2

3= 2•63•6

= 12

18

So there are 18 girls.

18. If the class had 21 girls and the ratio of boys to girls was 2 to 3, how many boys would be in the class?

19. If the ratio of X to Y is 4 to 3, and there are 462 Y’s, how many X’s are there?

20. If the ratio of games won to games played is 6 to 7 and 18 games were won, how many games were played?

E. Least common multiple (LCM): The LCM of two or more integers is used to find the lowest common denominator of fractions in order to add or subtract them.

To find the LCM: example: Find the LCM of 27 and 36.

First factor into primes: 27 = 3

3

!

36 = 22

• 32

Make the LCM by taking each prime factor to its greatest power: LCM

!

= 22

• 33

= 4 •27 =108

Problems 21-25: Find the LCM:

21. 6 and 15 24. 8 and 12 22. 4 and 8 25. 8, 12, and 15 23. 3 and 5

F. Lowest common denominator (LCD):

To find LCD fractions for two or more given fractions: example: Given

!

5

6 and

!

8

15

First find LCM of 6 and 15:

!

6 = 2• 3

!

15 = 3•5 LCM

!

= 2• 3•5 = 30 =LCD So

!

5

6= 25

30 and

!

8

15= 16

30

Problems 26-32: Find equivalent fractions with the LCD:

26.

!

2

3and

!

2

9 29.

!

1

2,

!

2

3, and

!

3

4

27.

!

3

8and

!

7

12 30.

!

7

8 and

!

5

8

28.

!

4

5and

!

2

3

31. Which is larger,

!

5

7 or

!

3

4? (Hint: find and

compare LCD fractions) 32. Which is larger,

!

3

8 or

!

1

3?

G. Adding and subtracting fractions:

If denominators are the same, combine the numerators: example:

!

7

10" 1

10= 7"1

10= 6

10= 3

5

Problems 33-37: Find the sum or difference (reduce if possible):

33.

!

4

7+ 2

7= 36.

!

3+ 1

2=

34.

!

5

6+ 1

6= 37.

!

1" 2

3=

35.

!

7

8" 5

8=

If the denominators are different, first find equivalent fractions with common denominators (preferably the LCD): example:

!

4

5+ 2

3= 12

15+ 10

15= 22

15=1 7

15

example:

!

1

2" 2

3= 3

6" 4

6= 3"4

6= "1

6

Problems 38-43: Calculate:

38.

!

3

5" 2

3= 41.

!

23

4+ 5 7

8=

39.

!

5

8+ 1

4= 42.

!

31

4" 3

4( ) + 1

2=

40.

!

5

2+ 5

4= 43.

!

41

3" 3

1

2" 3( ) =

H. Multiplying and dividing fractions:

To multiply fractions, multiply the tops, multiply the bottoms, and reduce if possible: example:

!

3

4• 2

5= 3•24•5

= 6

20= 3

10

Problems 44-52: Calculate:

44.

!

2

3• 3

8= 49.

!

21

2( )2

= 45.

!

1

2• 2

3= 50.

!

4

5• 30 =

46.

!

4

5" 5 = 51.

!

8• 3

4=

47.

!

3

4( )2

= 52.

!

15

21" 14

25=

48.

!

3

2( )2

=

Divide fractions by making a compound fraction and then multiply the top and bottom (of the larger fraction) by the lowest common denominator (LCD) of both.

example:

!

3

4÷ 2

3=

3

4

2

3

The LCD is 12, so multiply by 12:

!

3

4•12

2

3•12

=9

8

6

example:

!

7

2

3" 1

2

=7•6

2

3" 1

2( ) •6

!

=42

2

3•6 " 1

2•6

=42

4 " 3=42

1= 42

Problems 53-62: Calculate:

53.

!

3

2÷ 1

4= 58.

!

2

3

4

=

54.

!

113

8÷ 3

4= 59.

!

2

3

4=

55.

!

3

4÷ 2 = 60.

!

4

5÷ 5 =

56.

!

3

4

2

3

= 61.

!

3

8÷ 3 =

57.

!

1+ 1

2

1" 3

4

= 62.

!

21

3" 1

3

32

3+11

2

=

I. Comparing fractions:

example: Arrange small to large:

!

7

9, 5

7, and

!

3

4

LCD is

!

22

• 32

• 7 = 252

!

7

9= 7•289•28

= 196

252

!

5

7= 5•367•36

= 180

252

!

3

4= 3•634•63

= 189

252

So the order is

!

5

7, 3

4, 7

9

Fractions can also be compared by writing in decimal from and comparing the decimals.

Problems 63-65: Arrange small to large:

63.

!

15

8, 11

6 65.

!

2

3, 7

12, 5

6, 25

36

64.

!

7

8, 5

6, 11

12

Word Problems: 66. How many 2’s are in 8? 67. How many

!

1

2’s are in 8?

68. Three fourths is equal to how many twelfths? 69. What is

!

3

4 of a dozen?

70. Joe and Mae are decorating the gym for a dance. Joe has done

!

1

3 of the work and Mae has done

!

2

5.

What fraction of the work still must be done? 71. The ratio of winning tickets to tickets sold is 2

to 5. If 3,500,000 are sold, how many tickets are winners?

72. An

!

113

8-inch wide board can be cut into how

many strips of width

!

5

8 inch, if each cut takes

!

1

8

inch of the width? (Must the answer be a whole number?)

Problems 73-76: Inga and Lee each work for $4.60 per hour:

73. If Inga works

!

31

2 hours, what will her pay be?

74. If Lee works

!

23

4 hours, what will he be paid?

75. Together, what is the total time they work? 76. What is their total pay?

Visual Problems: Problems 77-80: What fraction of the figure is shaded?

77. 79.

78. 80.

Problems 81-83: What letter best locates the given number?

81.

!

5

9 82.

!

3

4 83.

!

2

3

Answers:

1.

!

23

•3 2.

!

2•3•7 3. 1, 2, 3, 6. 9, 18 4. 1, 2, 3, 4, 6, 8, 12, 24 5. 18 6. 9 7. 1 8.

!

1

4

9.

!

2

5

10.

!

3

4

11.

!

1

4

12.

!

4

7

13.

!

4

3

14. 32 15. 12 16. 28 17. 6 18. 14 19. 616 20. 21 21. 30 22. 8

23. 15 24. 24 25. 120 26.

!

6

9, 2

9

27.

!

9

24, 14

24

28.

!

12

15, 10

15

29.

!

6

12, 8

12, 9

12

30.

!

7

8, 5

8

31.

!

3

4 (because

!

20

28< 21

28)

32.

!

3

8 (because

!

9

24> 8

24)

0 1

P Q R S T

!

1

7

!

2

7

!

3

7

!

4

7

!

5

7

!

6

7

7 33.

!

6

7

34. 1 35.

!

1

4

36.

!

3 12

37.

!

1

3

38.

!

" 1

15

39.

!

7

8

40.

!

15

4

41.

!

8 58

42. 3 43.

!

3 56

44.

!

1

4

45.

!

1

3

46. 4 47.

!

9

16

48.

!

9

4

49.

!

6 14

or

!

25

4

50. 24 51. 6 52.

!

2

5

53. 6 54.

!

15 16

55.

!

3

8

56.

!

9

8

57. 6 58.

!

8

3

59.

!

1

6

60.

!

4

25

61.

!

1

8

62.

!

12

31

63.

!

11

6, 15

8

64.

!

5

6, 7

8, 11

12

65.

!

7

12, 2

3, 25

36, 5

6

66. 4

67. 16 68. 9 69. 9 70.

!

4

15

71. 1,400,000 72. 18; yes 73. $16.10 74. $12.65 75.

!

6 14

76. $28.75 77.

!

2

3

78.

!

2

5

79.

!

11

35

80.

!

5

12

81. Q 82. T 83. S

TOPIC 3: DECIMALS A. Meaning of Places:

Each digit position has a value ten times the place to its right. The part to the left of the point is the whole number part.

example:

!

324.519

!

= 3"100( ) + 2 "10( ) + (4 "1)

+ 5 " 110( ) + 1" 1

100( ) + 9 " 11000( )

Problems 1-5: Which is larger?

1. .59 or .7 4. 1.9 or 1.09 2. .02 or .03 5. .5 or .49 3. .2 or .03

Problems 6-8: Arrange in order of size from smallest to largest:

6. .02, .2, .19, .0085 8. 4.5, 5.4, 4.49, 5.41 7. .45, .449, .451, .5

Repeating decimals are shown with a bar over the repeating block of digits: example:

!

.3 means .333333333… example:

!

.43 means .4343434343… example:

!

.43 means .4333333333…

Problems 9-10: Arrange in order, large to small:

9.

!

.3 , .3, .34 10. .6,

!

.67 , .67,

!

.67 ,

!

.6

B. Fraction-decimal conversion:

Fraction to decimal: divide the top by the bottom: example:

!

3

4= 3÷ 4 = 0.75

example:

!

20

3= 20 ÷ 3 = 6.6

example:

!

32

5= 3+ 2

5= 3+ 2 ÷ 5( )

!

= 3+ .4 = 3.4

Problems 11-14: Write each as a decimal. If the decimal repeats, show the repeating block:

11.

!

5

8= 13.

!

41

3=

12.

!

3

7= 14.

!

3

100=

Non-repeating decimals to fractions: say the number as a fraction, write the fraction you say; reduce if possible: example:

!

.4 = four tenths

!

= 4

10= 2

5

example: 3.76 = three and seventy six hundredths

!

= 3 76

100= 3 19

25

Problems 15-18: Write as a fraction:

15. .01 = 17. 4.9 = 16. .38 = 18. 1.25 =

Comparison of fractions and decimals: usually it is easiest to convert fractions to decimals, then compare: example: Arrange from small to large: .3,

!

2

5, .3 ,

!

2

7

8 As decimals these are: .3, .4, .33333…,

!

.285714… So the order is:

!

.285714 , .3. .3 , .4, or

!

2

7, .3, .3 , 2

5

Problems 19-21: Arrange in order, small to large:

19.

!

2

3, .6, .67, .67 21.

!

1

100, .01, .009 , 5

500

20.

!

7

8, 0.87, 13

16, 0.88

Adding and subtracting decimals: like places must be combined (line up the points): example: 4 + .3 = 4.3

example: 3.43 + .791+12 :

!

3.430

.791

12.000

16.221

example: 8 ! 4.96 :

!

8.00

"4.96

3.04

example: 6.04 ! 2 !1.4( ) = 6.04 ! .6 = 5.44

Problems 22-30: Calculate:

22. 5.4 + .78 = 23. 1.36 ! 0.63 = 24. 4 ! .3 + .001 ! .01 + .1 = 25. $3.54 !$1.68 = 26. $17 ! $10.50 = 27. 17.5 !10 = 28. 4 + .3 + .02 + .001 = 29. 8.3 ! 0.92 = 30. 4.7 + 47 + 0.47 =

C. Multiplying and dividing decimals:

Multiplying decimals example: .3 ! .5 = .15 example: .3 ! .2 = .06 example: .03( )

2

= .0009

31. 3.24 ! 10 = 34. 5 ! 0.4 = 32. .01! .2 = 35. .51( )

2

= 33. .04( )

2

=

Dividing decimals: Change the problem to an equivalent whole number problem by multiplying both numbers by the same power of 10: example: .3 ÷ .03 Multiply both by 100 to get 30 ÷ 3 = 10 example:

!

.014

.07

Multiply both by 1000 to get 14 ÷ 70 = .2

36. .013 ÷100 = 40.

!

7.20

2.4=

37. .053 ÷ .2 = 41. 1.44 ÷ 2.4 = 38.

!

340

3.4= 42.

!

36.8

10=

39.

!

8.4

0.07=

D. Percent:

Meaning: translate percent as hundredths: example: 8% means 8 hundredths or .08 or

!

8

100= 2

25

Percent-decimal conversion: To change a decimal to percent form, multiply by 100 (move the point 2 places right), write the percent symbol (%): example: .075 = 7.5% example:

!

11

4=1.25 =125%

Problems 43-45: Write as a percent:

43. .3 = 44. 4 = 45. .085 =

To change a percent to decimal form, move the point 2 places left (divide by 100) and drop the % symbol: example: 8.76% = .0876 example: 67% = .67

Problems 46-49: Write as a decimal:

46. 10% = 48. .03% = 47. 136% = 49. 4% =

Solving percent problems: Step 1: Without changing the meaning, write the problems so it says “__ of __ is __”, and from this, identify a, b, and c: a % of b is c

Problems 50-52: Write in the form a% of b is c, and tell the values of a, b, and c:

50. 3% of 40 is 1.2 51. 600 is 150% of 400 52. 3 out of 12 is 25%

Step 2: Given a and b, change a% to a decimal and multiply (“of” can be translated “multiply”). Or, given c and one of the others, divide c by the other (first change percent to decimal); if answer is a, write it as a percent:

example: What is 9.4% of $5000? Compare a% of b is c: 9.4% of $5000 is __? Given a and b: multiply: .094 ! $5000 = $470

example: 56 problems correct out of 80 is what percent? Compare a% of b is c: __% of 80 is 56? Given c and other (b): 56 ÷80 = .7 = 70%

example: 5610 people, which is 60% of the registered voters, vote in an election. How many are registered? Compare a% of b is c: 60% of __ is 5610? Given c and other (a): 5610 ÷ .6 = 9350

9 53. 4% of 9 is what? 54. What percent of 70 is 56? 55. 15% of what is 60? 56. What is 43% of 500? 57. 10 is what percent of 40? E. Estimation and approximation:

Rounding to one significant digit: example: 3.67 rounds to 4 example: .0449 rounds to .04 example: 850 rounds to either 800 or 900 example: .4 = .44444... rounds to .4

Problems 58-61: Round to one significant digit:

58. 45.01 60. .00083 59. 1.09 61. 0.5

To estimate an answer, it is often sufficient to round each given number to one significant digit, then compute: example: .0298 ! .000513 Round and compute: .03 ! .0005 = .000015 .000015 is the estimate

Problems 62-66: Select the best approximation of the answer:

62. 1.2346825 ! 367.003246 = (4, 40, 400, 4000, 40000)

63. .0042210398 ÷ .01904982 = (.02, .2, .5, 5, 20, 50)

64. 101.7283507 + 3.14159265 = (2, 4, 98, 105, 400)

65. 4.36285903( )3

= (12, 64, 640, 5000, 12000)

66. 1.147 ! 114.7 = (–100, –10, 0, 10, 100)

Word Problems:

Problems 67-69: A cassette which cost $9.50 last year costs $11 now.

67. What is the amount of the increase? 68. What percent of the original price is the increase? 69. What is the percent increase? Problems 70-71: Jodi’s weekly pay is $89.20. She gets a 5% raise.

70. What will be her new weekly pay? 71. How much more will she get? Problems 72-74: Sixty percent of those registered voted in the last election.

72. What fraction voted? 73. If there was 45,000 registered, how many voted? 74. If 33,000 voted, how many were registered?

75. A person weighs 125 pounds. Their ideal weight is 130 pounds. Their actual weight is what percent of their ideal weight?

Answers: 1. .7 2. .03 3. .2 4. 1.9 5. .5 6. .0085, .02, .19, .2 7. .449, .45, .451, .5 8. 4.49, 4.5, 5.4, 5.41 9. .34,

!

.3 ,.3 10.

!

.67 , .67, .67,

!

.6 , .6 11. .625 12.

!

.428571 13.

!

4.3 14. .03 15.

!

1

100

16.

!

19

50

17.

!

4 9

10= 49

50

18.

!

1 14

= 5

4

19. .6,

!

2

3, .67,

!

.67 20.

!

13

16, .87,

!

7

8, .88

21. all equal

!

1

100

22. 6.18 23. .73 24. 3.791 25. $1.86 26. $6.50 27. 7.5 28. 4.321 29. 7.38 30. 52.17 31. 32.4 32. .002 33. .0016 34. 2 35. .2601 36. .00013 37. .265 38. 100 39. 120 40. 3 41. .6 42. 3.68

43. 30% 44. 400% 45. 8.5% 46. .1 47. 1.36 48. .0003 49. .04 50. 3% of 40 is 1.2;

a = 3%, b = 40,c = 1.2 51. 150% of 400 is 600;

a = 150% , b = 400,c = 600 52. 25% of 12 is 3;

a = 25% , b = 12 , c = 3 53. .36 54. 80% 55. 400 56. 215 57. 25% 58. 50 59. 1 60. .0008 61. .6

10 62. 400 63. .2 64. 105 65. 64 66. –100

67. $1.50 68.

!

"15.8% 69.

!

"15.8% 70. $93.66 71. $4.46

72.

!

3

5

73. 27,000 74. 55,000 75.

!

" 96%

TOPIC 4: EXPONENTS A. Positive integer exponents:

Meaning of exponents: example: 34 = 3 ! 3 ! 3 ! 3 = 3• 3 • 3 • 3 = 81 example: 43 = 4 • 4 • 4 = 64

Problems 1-12: Find the value:

1. 32 = 7. !2( )3

= 2. 23 = 8. 1002 = 3. !3( )

2

= 9. 2.1( )2

= 4. ! 3( )2 = 10.

!

".1( )3

=

5.

!

"32 = " 32( ) = 11.

!

2

3( )3

=

6. !23 = 12.

!

" 23( )3

=

ab means use a as a factor b times. (b is the

exponent or power of a) example: 25 means 2 • 2 • 2 • 2 • 2 2

5 has a value 32 5 is the exponent or power 2 is the factor example: 5 • 5 can be written 52 . Its value is 25. example: 41 = 4

Problems 13-24: Write the meaning and find the value:

13. 63 = 19. 0.1( )4

=

14. !4( )2

= 20.

!

2

3( )4

=

15. 04 = 21.

!

11

2( )2

=

16. 71 = 22. 210 = 17. 14 = 23. .03( )

2

= 18. !1( )

3

= 24. 32 • 23 =

example:

!

8

24

= 8

16= 1

2

example:

!

63

62

= 216

36= 6

Problems 25-30: Simplify:

25.

!

6

32

= 28.

!

10

42•5

=

26.

!

25

8= 29.

!

23•2

4

25•2

=

27.

!

4•5

10= 30.

!

5•12

62•10

=

Problems 31-38: Find the value:

31. 32 + 42 = 35. 3.1( )2

! .03( )2

= 32. 52 = 36. 3.1( )

2

+ .03( )2

= 33. 32 + 42 +122 = 37. 33 + 43 + 53 = 34. 132 = 38. 63 =

B. Integer exponent laws:

Problems 39-40: Write the meaning (not the value):

39. 32 = 40. 34 = 41. Write as a power of 3: 3 • 3• 3 • 3 • 3• 3 = 42. Write the meaning: 32 • 34 = 43. Write your answer to 42 as a power of 3,

then find the value. 44. Now find each value and solve:32 • 34 = 45. So 32 • 34 = 36 . Circle each of the powers.

Note how the circled numbers are related. 46. How are they related?

Problems 47-52: Write each expression as a power of the same factor:

example: 32 • 34 = 36

47. 41 • 42 = 50. !1( )5

• !1( )4

= 48. 53 • 53 = 51. 10 • 104 = 49. 33 • 3 = 52. 10 • 10 =

53. Make a formula by filling in the brackets: ab

•ac= a

[ ] . This is an exponent rule.

Problems 54-56: Find the value:

54. 36 = 55. 34 = 56. 729 ÷81 =

note:

!

36

÷ 34

= 36

34

!

= 3•3•3•3•3•3

3•3•3•3

!

= 3

3•3

3•3

3•3

3• 3 • 3 =1• 1•1•1 • 3• 3 = 3

2

57. Circle the exponents:

!

36

34

= 32

58. How are the circled numbers related?

Problems 59-63: Write each expression as a power:

example:

!

36

34

= 32

59. 24 ÷ 24 = 60.

!

25

2=

11 61.

!

52

5= 63.

!

15

13

=

62.

!

"4( )7

"4( )2

=

64. Make a formula by filling in the brackets:

!

ab

ac

= a [ ] . This is another exponent rule.

Problems 65-67: Find each value:

65. 43 = 67. 43( )2

= 64( )2

= 66. 46 =

Problems 68-69: Write the meaning of each expression:

example: 43( )2

= 43

• 43 = 4 • 4 • 4 • 4 • 4 • 4 = 4 • 4 • 4 • 4 • 4 • 4 = 4

6

68. 32( )4

= 69. 51( )3

=

70. Circle the three exponents: 43( )2

= 46

71. What is the relation of the circled numbers?

72. Make a rule:

!

ab( )

c

= a [ ]

73. Write your three exponent rules below: I. ab •ac = II.

!

ab

ac

=

III. ab( )c

=

Problems 74-80: Use the rules to write each expression as a power of the factor, and tell which rule you’re using:

74. 34 • 36 = 78.

!

34

3=

75.

!

210

25

= 79. 51( )2

=

76. 25( )2

= 80. 104 •103 =

77. 34( )4

=

C. Scientific notation: Note that scientific form always looks like a !10

n , where 1 ! a <10 , and n is an integer power of 10.

example: 32800 = 3.2800 !104 if the zeros in the ten’s and one’s places are significant. If the one’s zero is not significant, write: 3.280 !10

4 ; if neither is significant: 3.28 !10

4 . example: .0040301 = 4.031!10"3 example: 2 !102 = 200 example: 9.9 ! 10"1 = .99

Problems 81-84: Write in scientific notation:

81. 93,000,000 = 83. 5.07 = 82. .000042 = 84. !32 = Problems 85-87: Write in standard notation:

85. 1.4030 !103 = 87. 4 !10"6 = 86. 9.11 !10"2 = To compute with numbers written in scientific form, separate the parts, compute, then recombine:

example: 3.14 !105( ) 2( ) = 3.14( ) 2( ) !105 = 6.28 !105

example:

!

4.28"106

2.14"102

!

= 4.28

2.14•10

6

102= 2.00 ! 10

4

Problems 88-95: Write answer in scientific notation:

88. 1040 ! 102 = 91.

!

3.6"105

1.8"103=

89.

!

1040

1010

= 92.

!

1.8"108

3.6"105=

90.

!

1.86"104

3"10= 93. 4 !103( )

2

=

94.

!

1.5 "102( ) " 5 "103( ) =

95.

!

1.25 "102( ) 4 "10#2( ) =

D. Square roots or perfect squares:

!

a = b means b2 = a , where b ! 0 . Thus

!

49 = 7 , because 72 = 49 . Also,

!

" 49 = "7 . Note:

!

49 does not equal –7, (even though !7( )

2 does = 49 ) because –7 is not ! 0 .

example: If

!

a =10 , then a = 100 , because 10

2

= a = 100

Problems 96-99: Find the value and tell why:

96. If

!

a = 5 then a = 97. If

!

x = 4 , then x = 98. If

!

36 = b , then b = 99. If

!

169 = y , then y = Problems 100-110: Find the value:

100.

!

81 = 106.

!

32

+ 42

+122

= 101. 82 = 107.

!

172"15

2=

102.

!

82

= 108.

!

132"12

2=

103.

!

"7( )2

= 109.

!

43

=

104.

!

62

+ 82

= 110.

!

34

= 105.

!

32

+ 42

=

12 Answers:

1. 9 2. 8 3. 9 4. –9 5. –9 6. –8 7. –8 8. 10,000 9. 4.41 10. –.001 11.

!

8

27

12.

!

" 8

27

13. 6 • 6 • 6 = 216 14. !4( ) !4( ) =16 15. 0 • 0 • 0 • 0 = 0 16. 7 = 7 17. 1• 1•1•1 =1 18. !1( ) !1( ) !1( ) = !1 19. .1( ) .1( ) .1( ) .1( )

= .0001 20.

!

2

3• 23• 23• 23

= 16

81

21.

!

3

2• 32

= 9

4= 2 1

4

22. 2 • 2 • 2 • 2 • 2•2 • 2 • 2 • 2 • 2

= 1024 23. .03( ) .03( ) = .0009 24. 3 • 3• 2 • 2 • 2 = 72 25.

!

2

3

26. 4 27. 2 28.

!

1

8

29. 2 30.

!

1

6

31. 25 32. 25 33. 169 34. 169 35. 9.6091 36. 9.6109 37. 216 38. 216

39. 3 • 3 40. 3 • 3• 3 • 3 41. 36 42. 3 • 3• 3 • 3 • 3• 3 43. 36 = 729 44. 9 • 81 = 729 45. 3 2( )

• 34( )= 3

6( ) 46. 2 + 4 = 6 47. 43 48. 56 49. 34 50. !1( )

9 51. 105 52. 102 53. ab •ac = a b+c[ ] 54. 729 55. 81 56. 9 57.

!

36( )

34( )

= 32( )

58. 6 ! 4 = 2 59. 20 60. 24 61. 51 62. !4( )

5 63. 12 (or any power of

1) 64.

!

ab

ac

= ab"c[ ]

65. 64 66. 4096 67. 4096 68. 32 • 32 • 32 • 32 69. 5 • 5 • 5 70. 4 3( )( )

2( )= 4

6( ) 71. 3 ! 2 = 6 72. ab( )

c

= abc[ ]

73. I.

!

ab

• ac

= ab+c

II.

!

ab

ac

= ab"c

III.

!

ab( )

c

= abc

74. 310 , rule I 75. 25 , rule II 76. 210 , rule III 77. 316 , rule III 78. 33 , rule II 79. 52 , rule III 80. 107 , rule I 81. 9.3 !107 82. 4.2 !10"5 83. 5.07 84. !3.2 " 10 85. 1403.0 86. 0.0911 87. .000004 88. 1042 89. 1030 90. 6.2 !102 91. 2.0 !102 92. 5.0 !102 93. 1.6 !107 94. 7.5 !105 95. 5 96. 25; 52 = 25 97. 16; 42 =16 98. 6; 62 = 36 99. 13; 132 =169 100. 9 101. 64 102. 8 103. 7 104. 10 105. 5 106. 13 107. 8 108. 5 109. 8 110.9

TOPIC 5: EQUATIONS and EXPRESSIONS A. Operations with literal symbols (letters): When letters are used to represent numbers, addition is shown with a “plus sign” (+), and subtraction with a “minus sign” (–).

Multiplication is often show by writing letters together: example: ab means a times b So do a • b, a ! b, and a( ) b( )

example: 7 • 8 = 7 !8 = 7( ) 8( )

13 example: a b + c( ) means a times b + c( )

Problems 1-4: What is the meaning of:

1. abc 3. c2 2. 2a 4. 3 a ! 4( )

To show division, fractions are often used: example: 3 divided by 6 may be shown: 3 ÷ 6

or

!

3

6, or

!

6 3) , and all have value

!

1

2, or .5.

5. What does

!

4a

3b mean?

Problems 6-15: Write a fraction form and reduce to find the value (if possible):

6. 36 ÷ 9 = 11. 12 ÷ 5y = 7. 4 ÷ 36 = 12. 6b ÷ 2a = 8.

!

10 36) = 13. 8r ÷ 10s =

9.

!

1.2 .06) = 14. a ÷ a = 10. 2x ÷ a = 15. 2x ÷ x =

In the above exercises, notice that the fraction forms can be reduced if there is a common (shared) factor in the top and bottom:

example:

!

36

9= 9•4

9•1= 9

9•4

1=1• 4 = 4

example:

!

4

36= 4•1

4 •9= 4

4•1

9=1• 1

9= 1

9

example:

!

36

10= 2•18

2•5= 18

5 (or

!

33

5)

example:

!

6b

2a= 2•3•b

2•a= 3b

a

example:

!

a

a=1

example:

!

2x

x= 2•x

1•x= 2

1•x

x= 2

1•1 = 2 •1 = 2

Problems 16-24: Reduce and simplify:

16.

!

3x

3= 21.

!

abc

3ac=

17.

!

3x

4x= 22.

!

6x8xy

=

18.

!

x

2x= 23.

!

15x2

10x=

19.

!

12x

3x= 24.

!

6x

5• 5 =

20.

!

12x

3= (Hint:

!

6x

5• 5 = 6x

5•5

1= 6x •5

5•1...)

The distributive property says a b + c( ) = ab + ac . Since equality =( ) goes both ways, the distributive property can also be written ab + ac = a b + c( ) . Another form it often takes is a + b( )c = ac + bc , or ac + bc = a + b( )c .

example: 3 x ! y( ) = 3x ! 3y . Comparing this with a b + c( ) = ab + ac , we see a = 3 , b = x , and c = !y

example: Compare 4x + 7x = 4 + 7( )x = 11x with ac + bc = a + b( )c ; a = 4, c = x, b = 7

example: 4 2 + 3( ) = 4 • 2 + 4 • 3 (The distributive property says this has value 20, whether you do 4 • 5 or 8 +12 .)

example: 4a + 6x ! 2 = 2 2a +3x !1( )

Problems 25-35: Rewrite, using the distributive property:

25. 6 x ! 3( ) = 27. 3 ! x( )2 = 26. 4 b + 2( ) = 28. 4b ! 8c =

29. 4x ! x = (Hint: think 4x !1x , or 4 cookies minus 1 cookie)

30. !5 a ! 1( ) = 33. 3a ! a = 31. 5a + 7a = 34. 5x ! x + 3x = 32. 3a ! a = 35. x x + 2( ) =

B. Evaluation of an expression by substitution:

example: Find the value of 7 ! 4x , if x = 3 : 7 ! 4x = 7 ! 4 • 3 = 7 !12 = !5

example: If a = !7 and b = !1, then a2

b = !7( )2

!1( ) = 49 !1( ) = !49 example: If x = !2 , then 3x 2 + x ! 5 = 3 !2( )

2

+ !2( ) ! 5 = 3 • 4 ! 2 ! 5= 12 ! 2 ! 5 = 5

example: 8 ! c = 12 Add c to each, giving 8 ! c + c = 12 + c , or 8 =12 + c . Then subtract 12, to get !4 = c, or c = !4 .

example:

!

a

5= 4

3 Multiply each by the lowest common

denominator 15:

!

a

5•15 = 4

3•15 , or a • 3 = 4 • 5

Then divide by 3:

!

a •3

3= 20

3, so

!

a = 20

3

Problems 36-46: Given x = !1, y = 3 and z = !3 , find the value:

36. 2x = 42. 2x + 4y = 37. ! z = 43. 2x2 ! x !1 = 38. xz = 44. x + z( )

2

= 39. y + z = 45. x2 + z2 = 40. z + 3 = 46. !x2 z = 41. y2 + z 2 =

Problems 47-54: Find the value, given a = !1, b = 2, c = 0, x = !3, y =1 , and z = 2 :

47.

!

6

b= 51.

!

4x"3y

3y"2x=

48.

!

x

a= 52.

!

b

c=

49.

!

x

3= 53.

!

" b

z=

50.

!

a"y

b= 54.

!

c

z=

14 C. Solving a linear equation in one variable: Add or subtract the same thing on each side of the equation and/or multiply or divide each side by the same thing, with the goal of getting the variable alone on one side. If there are fractions, you can eliminate them by multiplying both sides of the equation by a common denominator. If the equation is a proportion, you may wish to cross-multiply.

example: 3x = 10 Divide both sides by 3, to get

!

1x : 3x3

= 10

3, or

!

x = 10

3

example: 5 + a = 3 Subtract 5 from each side, to get 1a (which is a): 5 + a ! 5 = 3 ! 5 or a = !2

example:

!

y

3=12 Multiplying both sides by 3, to

get

!

y :y

3• 3 =12 • 3, which gives y = 36 .

example: b ! 4 = 7 Add 4, to get 1b : b ! 4 + 4 = 7 + 4 , or b = 11

Problems 55-65: Solve:

55. 2x = 94 61. 4x ! 6 = x 56.

!

3 = 6x

5 62.

!

x " 4 = x

2+1

57. 3x + 7 = 6 63. 6 ! 4x = x 58.

!

x

3= 5

4 64. 7x ! 5 = 2x +10

59. 5 ! x = 9 65. 4x + 5 = 3 ! 2x 60.

!

x = 2x

5+1

Problems 66-70: Substitute the given value, then solve for the other variable:

example: If n = r +3 and r = 5 find the value of n: Replacing r with 5 gives n = 5 + 3 = 8 .

66. n = r +3, n = 5 69. 5x = y !3, x = 4 67. n = r +3, n = 1 70. 5x = y !3, y = 3 68.

!

a

2= b, b = 6

D. Word Problems: If an object moves at a constant rate of speed r, the distance d it travels in time t is given by the formula d = rt .

example: If t = 5 and d = 50 , find r: Substitute the given values in d = rt and solve: 50 = r • 5 , giving r = 10 .

Problems 71-72: In d = rt , substitute, then solve for the variable:

71. t = 5, r = 50; d = 72. d = 50, r = 4; t = 73. On a 40 mile hike, a strong walker goes 3

miles per hour. How much time will the person hike? Write an equation, then solve it.

74. “Product” 75. “Quotient” 76. “Difference” 77. “Sum”

78. The sum of two numbers is 43. One of the two numbers is 17. What is the other?

79. Write an equation which says that the sum of a number n and 17 is 43.

80. Write an equation which says the amount of simple interest A equals the product of the invested principle P, the rate of interest r, and the time t.

81. Use the equation of problem 80: P = $200, r = 7% and t = 5 years. Find the amount of interest A.

Problems 82-83: In a rectangle which has two sides of length a and two sides of length b, the perimeter P is found by adding all the side lengths, or P = 2a + 2b .

82. If a = 5 and b = 8 , find P. 83. If a = 7 and P = 40 , find b. Problems 84-85: The difference of two numbers x and 12 is 5.

84. If x is the larger, an equation, which says this same thing could be x !12 = 5 . Write an equation if x is the smaller of the two numbers x and 12.

85. Find the two possible values of x by solving each equation in problem 84.

Problems 86-87: Write an equation, which says:

86. n is 4 more than 3. 87. 4 less than x is 3. 88. Solve the two equations you wrote for

problems 86 and 87.

Answers: 1. a times b times c 2. 2 times a 3. c times c 4. 3 times a ! 4( )

5. 4a divided by 3b 6.

!

36

9= 4

7.

!

4

36= 1

9

8.

!

36

10= 18

5

9.

!

.06

1.2= 6

120= 1

20

10.

!

2x

a

11.

!

12

5y

15 12.

!

6b

2a= 3b

a

13.

!

8r

10s= 4 r

5s

14.

!

a

a=1

15.

!

2x

x= 2

16. x 17.

!

3

4

18.

!

1

2

19. 4 20. 4x 21.

!

b

3

22.

!

3

4y

23.

!

3x

2

24. 6x 25. 6x !18 26. 4b + 8 27. 6 ! 2x 28. 4 b ! 2c( ) 29. 4 !1( )x = 3x 30. !5a + 5 31. 5 + 7( )a =12a 32. 3 ! 2( )a = 1a = a 33. 2a 34. 7x 35. x2 + 2x 36. –2

37. 3 38. 3 39. 0 40. 0 41. 18 42. 10 43. 2 44. 16 45. 10 46. 3 47. 3 48. 3 49. –1 50. –1 51.

!

" 5

3

52. no value (undefined) 53. –1 54. 0 55. x = 47 56.

!

x = 5

2

57.

!

x = " 1

3

58.

!

x = 15

4

59. x = !4 60.

!

x = 5

3

61. x = 2 62. x =10 63.

!

x = 6

5

64. x = 3 65.

!

x = " 1

3

66. 5 = r + 3; r = 2 67. 1 = r +3; r = !2 68.

!

a

2= 6; a =12

69. 5 • 4 = y! 3; y = 23 70. 5x = 3 ! 3; x = 0 71. d = 50 • 5 ; d = 250 72.

!

50 = 4t;

!

t = 25

2

73.

!

40 = 3t; t = 40

3 hours

74. Multiply 75. Divide 76. Subtract 77. Add 78. 26 79. n +17 = 43 80. A = Prt 81. A = $70 82. P = 26 83. b = 13 84. 12 ! x = 5 85. x =17 or 7 86. n = 4 + 3 87. x ! 4 = 3 88. n = 7; x = 7

TOPIC 6: GEOMETRY A. Formulas for perimeter P and area A of rectangles, squares, parallelograms, and triangles: Rectangle with base b and altitude (height) h: P = 2b + 2h A = bh If a wire is bent in this shape, the perimeter P is the length of the wire, and the area A is the number of square units enclosed by the wire.

example: A rectangle with b = 7 and h = 8 : P = 2b + 2h = 2 •7 + 2 • 8 = 14 +16 = 30 units A = bh = 7• 8 = 56 square units

A square is a rectangle with all sides equal, so the rectangle formulas apply (and simplify). If the side length is s: P = 4s s A = s

2

example: A square with side s = 11cm has P = 4s = 4 !11 = 44 cm

A = s2

=112

=121cm 2 (sq. cm)

A parallelogram with base b and height h and other side a: A = bh h a P = 2a + 2b b example: A parallelogram has sides 4 and 6; 5 is

the length of the altitude perpendicular to the side 4. P = 2a + 2b = 2 • 6 + 2 • 4 5 6 = 12 +8 = 20units A = bh = 4 • 5 = 20 square units 4

In a triangle with side lengths a, b, and c, and altitude height h to side b: P = a + b + c a c

!

A = 1

2bh = bh

2 h

b s

b

h

16

example: P = a + b + c

= 6 + 8 + 10

= 24 units

!

A = 1

2bh = 1

210( ) 4.8( ) = 24 square units

Problems 1-8: Find P and A:

1. Rectangle with sides 5 and 10. 2. Rectangle with sides 1.5 and 4. 3. Square with sides 3 miles. 4. Square with sides

!

3

4yards.

5. Parallelogram with sides 36 and 24, and height 10 (on side 36).

6. Parallelogram, all sides 12, altitude 6. 7. Triangle with sides 5, 12, and 13.

Side 5 is the altitude 13 5 on side 12.

12 8. Triangle 5 3

shown: 4

B. Formulas for circumference C and area A of a circle: A circle with radius r (and diameter d = 2r ) has a distance around (circumference) C = !d = 2!r (If a piece of wire is bent into a circular shape, the circumference is the length of the wire.) example: A circle with radius r = 70 has d = 2r = 140 and exact circumference C = 2!r = 2 •! • 70 = 140! units

If ! is approximated by

!

22

7,

!

C =140" #140 22

7( ) # 440units (approx.) If ! is approximated by 3.1, C ! 140 3.1( ) = 434 units The area of a circle is A = !r2 example: If r = 8 , exact area is

A = !r2= ! • 8

2= 64! square units

Problems 9-11: Find the exact C and A for a circle with:

9. radius r = 5 units 10. r = 10 feet 11. diameter d = 4km

Problems 12-14: A circle has area 49! :

12. What is its radius length? 13. What is the diameter? 14. Find its circumference.

Problems 15-16: A parallelogram has area 48 and two sides each of length 12:

15. How long is the altitude to those sides? 16. How long are each of the other two sides? 17. How many times the P and A of a 3cm

square are the P and A of a square with sides all 6 cm?

18. A rectangle has area 24 and one side 6. Find the perimeter.

Problems 19-20: A square has perimeter 30:

19. How long is each side? 20. What is its area? 21. A triangle has base and height each 7. What

is its area? C. Pythagorean theorem:

In any triangle with a 90° (right) angle, the sum of the squares of the legs equals the square of hypotenuse.

(The legs are the two shorter sides; the hypotenuse is the longest side.)

If the legs have lengths a and b, and c is the hypotenuse length, then a2 + b2 = c2 . In words: “ In a right triangle, leg squared plus leg squared equals hypotenuse squared.”

example: A right triangle has hypotenuse 5 and one leg 3. Find the other leg. Since leg2 + leg2 = hyp2 ,

32 + x2 = 52 9 + x 2 = 25 x2 = 25 ! 9 = 16

!

x = 16 = 4

Problems 22-24: Find the length of the third side of the right triangle:

22. one leg: 15, hypotenuse: 17 23. hypotenuse: 10, one leg: 8 24. legs: 5 and 12

Problems 25-26: Find x:

25. 26. 27.

r

17

28. In right !RST with right angle R, SR =11 and TS = 61 . Find RT. (Draw and label a triangle to solve.)

29. Would a triangle with sides 7, 11, and 13 be a right triangle? Why or why not?

Similar triangles are triangles which are the same shape. If two angles of one triangle are equal respectively to two angles of another triangle, then the triangles are similar.

example: !ABC and !FED are similar: The pairs of sides which correspond are AB and FE ,BC and ED , AC and FD .

A B

C

36

53

Problems 30-32: Use this figure:

30. Find and name two similar triangles. 31. Draw the triangles separately and label them. 32. List the three pairs of corresponding sides.

If two triangles are similar, any two corresponding sides have the same ratio (fraction value): example: the ratio a to x, or

!

a

x, is the same as

!

by

and

!

c

z. Thus

!

ax

= by

,

!

a

x= c

z, and

!

by

= cz. Each of

these equations is called a proportion.

33. Draw the similar triangles separately, label them, and write proportions for the corresponding sides.

Problems 34-37: Solve for x:

example: Find x by writing and solving a proportion:

!

2

5= 3

x, so cross multiply and get 2x = 15 or

!

x = 7 12

34.

!

AC =10; EC = 7

!

BC = 4; DC = x

35. 37. 36. D. Graphing on the number line:

Problems 38-45: Name the point with given coordinate:

38. 0 42.

!

"1.5 39.

!

1

2 43. 2.75

40.

!

" 12

44.

!

" 3

2

41.

!

4

3 45.

!

1.3 Problems 46-51: On the number line above, what is the distance between the listed points? (Remember that distance is always positive.)

46. D and G 49. B and C 47. A and D 50. B and E 48. A and F 51. F and G

Problems 52-55: On the number line, find the distance from:

52. –7 to –4 54. –4 to 7 53. –7 to 4 55. 4 to 7

Problems 56-59: Draw a sketch to help find the coordinate of the point…:

56. Halfway between points with coordinates 4 and 14.

57. Midway between –5 and –1. 58. Which is the midpoint of the segment

joining –8 and 4. 59. On the number line the same distance from

–6 as it is from 10.

E. Coordinate plane graphing:

To locate a point on the plane, an ordered pair of numbers is used, written in the form

!

x, y( ) .

Problems 60-63: Identify coordinates x and y in each ordered pair:

60.

!

3,0( ) 62.

!

5,"2( ) 61.

!

"2,5( ) 63.

!

0,3( )

To plot a point, start at the origin and make the moves, first in the x-direction (horizontal) and

10 9 3

8 4

5

E D

A B C

55 55

x 30

20 15

x 5 3

4 2

10 4

7 x 3

10 4 x

4

-2 -1 0 1 2 3

A B C D E F G

A

E

D

A B C

E

D

F36

53

18

then the y-direction (vertical) indicated by the ordered pair.

example: (-3, 4) Start at the origin, move left 3 (since x = !3),

then (from there), up 4 (since y = 4 ),

put a dot there to indicate the point (-3, 4)

64. On graph paper, join these points in order: (–3,–2), (1,–4), (3, 0), (2, 3), (–1, 2), (3, 0), (–3, –2), (–1, 2), (1,–4).

65. Two of the lines drawn in problem 64 cross each other. What are the coordinates of the crossing point?

66. In what quadrant is the point

!

a,b( ) if

!

a > 0 and

!

b < 0?

Problems 67-69:

!

ABCD is a square, with C

!

5,"2( ) and D

!

"1,"2( ) . Find:

67. the length A B of each side.

68. the coordinates of A. x 69. the coordinates of

the midpoint of

!

DC . D C

Problems 70-72: Given

!

A 0,5( ) ,

!

B 12,0( ) :

70. Sketch a graph. Draw

!

AB . Find its length. 71. Find the midpoint of

!

AB and label it C. Find the coordinates of C.

72. What is the area of the triangle formed by A, B, and the origin?

Answers:1. 30 units, 50

!

units2

!

(units2means square units)

2. 11units, 6

!

units2

3. 12 miles, 9

!

miles2

4. 3 yards,

!

9

16yards

2 5. 120 un.,

!

360un.2

6. 48 un.,

!

72un.2

7. 30 un.,

!

30un.2

8. 12 un.,

!

6un.2

9.

!

10"un.,

!

25"un.2

10.

!

20" ft.,

!

100" ft.2

11.

!

4"km,

!

4"km2

12. 7 13. 14 14.

!

14" 15. 4 16. Cannot tell 17. P is 2 times, A is 4 times 18. 20 19.

!

7 12

20.

!

225

4

21.

!

241

2

22. 8 23. 6 24. 13

25. 9 26. 41 27. 10 28. 60 29. No, because

!

72

+112"13

2 30.

!

"ABE ~ "ACD 31.

32.

!

AB ,

!

AC ;

!

AE ,

!

AD ;

!

BE ,

!

CD 33.

!

3

9= 5

15= 4

12

34.

!

14

5 or

!

2 45

35.

!

2 45 or

!

14

5

36.

!

45

2

37.

!

40

7

38. D 39. E 40. C 41. F 42. B 43. G 44. B 45. F 46. 2.75 47. 2 48.

!

3 13

49. 1 50. 2 51.

!

17

12

52. 3 53. 11 54. 11 55. 3 56. 9 57. –3 58. –2 59. 2 60.

!

x = 3,y = 0 61.

!

x = "2,y = 5 62.

!

x = 5,y = "2 63.

!

x = 0,y = 3 64. 65. (0,–1) 66. IV 67. 6 68.

!

"1,4( ) 69.

!

2,"2( ) 70. 13 71.

!

6, 52( )

72. 30

y

E

A B

D

A C

1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE

Directions: Study the examples, work the problems, then check your answers at the end of each topic. If you don’t get the answer given, check your work and look for mistakes. If you have trouble, ask a math teacher or someone else who understands this topic.

TOPIC 1: ARITHMETIC OPERATIONS

A. Fractions:

Simplifying fractions:

example: Reduce 2736: 27

36= 9•39•4 = 9

9• 34

=1• 34

= 34

(Note that you must be able to find a common factor; in this case 9; in both the top and the bottom in order to reduce.)

Problems 1-3: Reduce:

1. 1352

= 2. 2665

= 3. 3+63+9 =

Equivalent fractions:

example: 34 is equivalent to how many eighths?

34

=8

34

=1• 34

= 22

• 34

= 2•32•4 = 6

8

Problems 4-5: Complete:

4. 49

=72 5. 3

5=20

How to get the lowest common denominator (LCD) by finding the least common multiple (LCM) of all denominators:

example: 56 and 8

15. First find LCM of 6 and 15:

6 = 2• 3 15 = 3•5 LCM = 2• 3•5 = 30 , so 5

6= 2530, and 8

15= 1630

Problems 6-7: Find equivalent fractions with the LCD:

6. 23 and 2

9 7. 3

8 and 7

12

8. Which is larger, 57 or 3

4?

(Hint: find the LCD fractions)

Adding, subtracting fractions: if the denominators are the same combine the numerators:

example: 710

− 110

= 7−110

= 610

= 35

Problems 9-11: Find the sum or difference (reduce if possible):

9. 47

+ 27

= 10. 56

+ 16

= 11. 78

− 58

=

If the denominators are different, find equivalent fractions with common denominators, then proceed as before:

example: 45

+ 23

= 1215

+ 1015

= 2215

=1 715

example: 12

− 23

= 36

− 46

= 3−46

= −16

12. 35

− 23

= 13. 58

+ 14

=

Multiplying fractions: multiply the top numbers, multiply the bottom numbers, reduce if possible.

example: 34

• 25

= 3•24•5 = 6

20= 310

14. 23

• 38

= 16. 34( )2 =

15. 12

• 13

= 17. 2 12( )2 =

Dividing fractions: make a compound fraction, then multiply the top and bottom (of the big fraction) by the LCD of both:

example: 3

4÷ 23

=34

23

=34 •1223 •12

= 98

example: 7

23

− 12

= 7•623

− 12( )•6

= 42

4 − 3= 421

= 42

18. 32

÷ 14

= 21. 234

=

19. 11 38

÷ 34

= 22. 23

4

20. 34

÷ 2 =

B. Decimals:

Meaning of places: in 324.519, each digit position has a value ten times the place to its right. The part to the left of the point is the whole number part. Right of the point, the places have values: tenths, hundredths, etc.,

So, 324.519 = 3×100( )+ 2×10( )+ 4 ×1( )+ 5× 1

10( )+ 1× 1100( )+ 9× 1

1000( ).

23.Which is larger: .59 or .7?

To add or subtract decimals, like places must be combined (line up the points).

example: 1.23 − .1 =1.13 example: 4 + .3 = 4.3 example: 6.04 − 2−1.4( )= 6.04 − .6 = 5.44

24. 5.4 + .78 = 25. .36 − .63 =

2 26. 4 − .3+ .001− .01+ .1= 27. $3.54 −$1.68 =

Multiplying decimals:

example: .3× .5 = .15 example: .3× .2 = .06 example: .03( )2 = .0009

28. 3.24 ×10 = 30. .51( )2 = 29. .01× .2 = 31. 5 × .4 =

Dividing decimals: change the problem to an equivalent whole number problem by multiplying both by the same power of ten.

example: .3 ÷ .03 Multiply both by 100, to get 30 ÷ 3 =10

example: .014.07 Multiply both by 1000, get

1470

=14 ÷ 70 = .2

32. .013 ÷100 = 34. 3403.4

= 33. .053 ÷ .2 =

C. Positive integer exponents and square

roots of perfect squares:

Meaning of exponents (powers):

example: 34 = 3• 3• 3• 3 = 81 example: 43 = 4 • 4 • 4 = 64

Problems 35-44: Find the value:

35. 32 = 40. 100

2 = 36. −3( )2=

41. 2.1( )2 = 37. − 3( )2 = 42. −.1( )3 =

38. −32 = 43. 23( )3 =

39. −2( )3 = 44. − 23( )3 =

a is a non-negative real number if a ≥ 0

a = b means b2 = a , where b ≥ 0 . Thus 49 = 7 , because 72 = 49 . Also, − 49 = −7 .

45. 144 = 49. 1.44 = 46. − 144 = 50. .09 = 47. −144 = 51. 4

9=

48. 8100 =

D. Fraction-decimal conversion:

Fraction to decimal: divide the top by the bottom.

example: 34

= 3÷ 4 = .75 example: 20

3= 20 ÷ 3= 6.66666...= 6.6

example: 3 25

= 3+ 25

= 3+ 2÷ 5( )= 3+ .4 = 3.4

Problems 52-55: Write each as a decimal. If the decimal repeats, show the repeating block of digits:

52. 58

= 54. 4 13

= 53. 3

7= 55. 3

100=

Non-repeating decimals to fractions: read the number as a fraction, write it as a fraction, reduce if possible:

example: .4 = four tenths= 410

= 25

example: 3.76 = three and seventy six hundredths = 3 76

100= 3 19

25

Problems 56-58: Write as a fraction:

56. .01= 57. 4.9 = 58. 1.25 =

E. Percents:

Meaning of percent: translate ‘percent’ as ‘hundredths’:

example: 8% means 8 hundredths or .08 or 8100

= 225

To change a decimal to percent form, multiply by 100: move the point 2 places right and write the % symbol.

example: .075= 7.5% example: 1 1

4=1.25 =125%

Problems 59-60: Write as a percent:

59. .3= 60. 4 =

To change a percent to decimal form, move the point 2 places left and drop the % symbol.

example: 8.76% = .0876 example: 67% = .67

Problems 61-62: Write as a decimal:

61. 10% = 62. .03% =

To solve a percent problem which can be written in this form: a % of b is c

First identify a,b,c :

Problems 63-65: If each statement were written (with the same meaning) in the form of a % of b is c , identify a, b, and c :

63. 3% of 40 is 1.2 64. 600 is 150% of 400 65. 3 out of 12 is 25%

Given a and b, change a% to decimal form and multiply (since ‘of’ can be translated ‘multiply’).

Given c and one of the others, divide c by the other (first change percent to decimal, or if the answer is a , write it as a percent).

3

example: What is 9.4% of $5000? (a% of b is c : 9.4% of $5000 is ? )

9.4% = .094 .094× $5000 = $470 (answer) example: 56 problems correct out of 80 is what percent? (a % of b is c : ? % of 80 is 56) 56 ÷80 = .7= 70% (answer) example: 5610 people vote in an election, which is 60% of the registered voters. How many are registered?

(a % of b is c : 60 % of ? is 5610); 60% = .6 ; 5610 ÷ .6 = 9350 (answer)

66. 4% of 9 is what? 67. What percent of 70 is 56? 68. 15% of what is 60?

F. Estimation and approximation:

Rounding to one significant digit:

example: 3.67 rounds to 4 example: .0449 rounds to .04 example: 850 rounds to either 800 or 900

Problems 69-71: Round to one significant digit.

69. 45.01 70. 1.09 71. 00083

To estimate an answer, it is often sufficient to round each given number to one significant digit, then compute.

example: .0298× .000513 Round and compute: .03× .0005 = .000015 .000015 is the estimate

Problems 72-75: Select the best approximation of the answer:

72. 1.2346825× 367.0003246 = (4, 40, 400, 4000, 40000) 73. .0042210398÷ .0190498238 = (.02, .2, .5, 5, 20, 50) 74. 101.7283507+ 3.141592653 = (2, 4, 98, 105, 400)

75. 4.36285903( )3 = (12, 64, 640, 5000, 12000)

Answers: 1. 1

4

2. 25

3. 34

4. 32 5. 12 6. 6

9, 29

7. 924 ,

1424

8. 34 (because

2028 < 21

28 )

9. 67

10. 1 11. 14

12. − 115

13. 7 8

14. 14

15. 16

16. 916

17. 254

18. 6 19. 15 16

20. 3 8

21. 8 3

22. 16

23. .7 24. 6.18 25. -.27

26. 3.791 27. $1.86 28. 32.4 29. .002 30. .2601 31. 2 32. .00013 33. .265 34. 100 35. 9 36. 9 37. –9 38. –9 39. –8 40. 10000 41. 4.41 42. -.001 43. 8 27

44. − 827

45. 12 46. –12 47. not a real number 48. 90 49. 1.2 50. .3 51. 2 3

52. .625

53. .428571 54. 4.3 55. .03 56. 1100

57. 4 910

= 4910

58. 1 14

= 54

59. 30% 60. 400% 61. .1 62. .0003 a b c 63. 3 40 1.2 64. 150 400 600 65. 25 12 3 66. .36 67. 80% 68. 400 69. 50 70. 1 71. .0008 72. 400 73. .2 74. 105 75. 64

TOPIC 2: POLYNOMIALS

A. Grouping to simplify polynomials:

The distributive property says: a b + c( ) = ab + ac

example: 3 x − y( )= 3x − 3y(a = 3, b = x, c = −y)

4

example: 4x + 7x = 4 + 7( )x =11xa = x, b = 4, c = 7( )

example: 4a + 6x − 2 = 2 2a +3x −1( )

Problems 1-3: Rewrite, using the distributive property:

1. 6 x − 3( )= 3. −5 a − 1( ) = 2. 4x − x =

Commutative and associative properties are also used in regrouping:

example: 3x + 7− x = 3x − x + 7 = 2x + 7 example: 5− x + 5 = 5+ 5 − x = 10 − x example: 3x + 2y − 2x + 3y

= 3x − 2x + 2y + 3y = x + 5y

Problems 4-9: Simplify:

4. x + x = 7. 4x +1+ x − 2 = 5. a + b − a + b = 8. 180 − x − 90 = 6. 9x − y + 3y − 8x = 9. x − 2y + y − 2x =

B. Evaluation by substitution:

example: If x = 3 , then 7 − 4x = 7 − 4(3) = 7 −12 = −5

example: If a = −7 and b = −1, then a2b = −7( )2 −1( ) = 49 −1( ) = −49

example: If x = −2 , then 3x 2 − x − 5 = 3 −2( )2 − −2( )− 5 = 3• 4 + 2 − 5 = 12 + 2 − 5 = 9

Problems 10-19: Given x = −1, y = 3, and z = −3 . Find the value:

10. 2x = 15. 2x + 4y = 11. − z = 16. 2x

2 − x −1 = 12. xz = 17. x + z( )2 = 13. y + z = 18. x

2 + z2 =

14. y2 + z 2 = 19. −x2 z =

C. Adding, subtracting polynomials:

Combine like terms:

example: 3x2 + x +1( )− x −1( )=

3x2 + x +1 − x +1 = 3x2 + 2

example: x −1( ) + x2 + 2x − 3( )=

x −1 + x2 + 2x − 3 = x

2 + 3x − 4 example: x

2 + x −1( )− 6x2 − 2x +1( )=

x2 + x −1 − 6x2 + 2x −1 = −5x2 + 3x − 2

Problems 20-25: Simplify:

20. x 2 + x( ) − x +1( )= 21. x − 3( )+ 5− 2x( ) =

22. 2a2 − a( )+ a

2 + a −1( )= 23. y

2 − 3y − 5( )− 2y2 − y + 5( )=

24. 7 − x( )− x − 7( ) = 25. x

2 − x2 + x −1( )=

D. Monomial times polynomial:

Use the distributive property:

example: 3 x − 4( )= 3• x + 3 −4( )= 3x + −12( )= 3x −12

example: 2x + 3( )a = 2ax +3a example: −4x x 2 −1( )= −4x3 + 4x

26. − x − 7( ) = 30. a 2x − 3( ) = 27. −2 3− a( )= 31. x2 −1( ) −1( )=

28. x x + 5( )= 32. 8 3a2 + 2a − 7( )= 29. 3x −1( )7 =

E. Multiplying polynomials:

Use the distributive property: a b+ c( ) = ab + ac

example: 2x +1( ) x − 4( ) is a b + c( ) if: a = 2x +1( ), b = x , and c = −4

So a b+ c( ) = ab + ac = 2x +1( )x + 2x +1( ) −4( ) = 2x2 + x − 8x − 4 = 2x2 − 7x − 4

Short cut to multiply above two binomials (see example above): FOIL (do mentally and write the answer)

F: First times First: 2x( ) x( ) = 2x2

O: multiply ‘Outers’: 2x( ) −4( ) = −8x I: multiply ‘Inners’: 1( ) x( ) = x

L: Last times Last: 1( ) −4( ) = −4 Add, get 2x

2 − 7x − 4

example: x + 2( ) x + 3( ) = x2 + 5x + 6

example: 2x −1( ) x + 2( )= 2x2 + 3x − 2 example: x − 5( ) x + 5( )= x

2 − 25 example: −4 x − 3( )= −4x +12 example: 3x − 4( )2 = 9x 2 − 24x +16 example: x + 3( ) a− 5( )= ax − 5x + 3a−15

Problems 33-41: Multiply:

33. x + 3( )2 = 38. −6x 3− x( ) =

34. x − 3( )2 = 39. x − 12( )2 =

35. x + 3( ) x − 3( )= 40. x −1( ) x + 3( )= 36. 2x + 3( ) 2x − 3( )= 41. x2 −1( ) x2 + 3( )= 37. x − 4( ) x − 2( )=

5 F. Special products:

These product patterns (examples of FOIL) should be remembered and recognized:

I. a + b( ) a − b( ) = a2 − b2

II. a + b( )2 = a2 + 2ab + b2

III. a − b( )2 = a2 − 2ab + b2

Problems 42-44: Match each pattern with its example:

a. 3x −1( )2 = 9x2 − 6x +1 b. x + 5( )2 = x

2 + 10x + 25 c. x + 8( ) x − 8( )= x2 − 64

42. I: 43. II: 44. III:

Problems 45-52: Write the answer using the appropriate product pattern:

45. 3a+1( ) 3a−1( )= 49. 3a− 2( ) 3a− 2( )= 46. y −1( )2 = 50. x − y( )2 = 47. 3a + 2( )2 = 51. 4x +3y( )2 = 48. 3a +2( ) 3a −2( )= 52. 3x + y( ) 3x − y( )=

G. Factoring:

Monomial factors: ab + ac = a b+ c( )

example: x2 − x = x x −1( )

example: 4x2y + 6xy = 2xy 2x + 3( )

Difference of two squares:

a2 − b2 = a + b( ) a − b( )

example: 9x 2 − 4 = 3x + 2( ) 3x − 2( )

Trinomial square:

a2 + 2ab+ b2 = a + b( )2 a2 − 2ab+ b2 = a − b( )2

example: x2 − 6x + 9 = x − 3( )2

Trinomial:

example: x2 − x − 2 = x − 2( ) x +1( )

example: 6x2 − 7x − 3 = 3x +1( ) 2x − 3( )

Problems 53-67: Factor:

53. a2 + ab = 61. x

2 − 3x −10 = 54. a

3 − a2b + ab2 = 62. 2x2 − x = 55. 8x

2 − 2 = 63. 8x3 + 8x2 + 2x =

56. x2 −10x + 25 = 64. 9x

2 +12x + 4 = 57. −4xy +10x2 = 65. 6x

3y2 − 9x 4y =

58. 2x2 −3x − 5 = 66. 1 − x − 2x2 =

59. x2 − x − 6 = 67. 3x

2 −10x + 3 = 60. x

2y − y2x =

Answers:

1. 6x −18 2. 3x 3. −5a + 5 4. 2x 5. 2b 6. x + 2y 7. 5x −1 8. 90 − x 9. −x − y

10. –2 11. 3 12. 3 13. 0 14. 18 15. 10 16. 2 17. 16 18. 10 19. 3

20. x2 −1

21. 2 − x 22. 3a

2 −1 23. −y 2 − 2y −10 24. 14 − 2x

25. −x +1 26. −x + 7 27. −6 + 2a 28. x

2 + 5x 29. 21x − 7 30. 2ax − 3a 31. −x 2 +1 32. 24a

2 +16a− 56 33. x

2 + 6x + 9 34. x

2 − 6x + 9 35. x

2 − 9 36. 4x

2 −9 37. x

2 − 6x + 8 38. −18x + 6x 2 39. x 2 − x + 1

4

40. x2 + 2x − 3

41. x4 + 2x 2 − 3

42. c 43. b 44. a

45. 9a2 −1

46. y2 − 2y +1

47. 9a2 +12a + 4

48. 9a2 − 4

49. 9a2 −12a+ 4

50. x2 − 2xy + y

2

51. 16x2 + 24xy + 9y2

52. 9x2 − y 2

53. a a + b( ) 54. a a2 − ab+ b2( ) 55. 2 2x +1( ) 2x −1( ) 56. x −5( )2 57. −2x 2y − 5x( ) 58. 2x −5( ) x +1( ) 59. x − 3( ) x + 2( ) 60. xy x − y( ) 61. x −5( ) x + 2( ) 62. x 2x −1( ) 63. 2x 2x +1( )2 64. 3x + 2( )2 65. 3x

3y 2y − 3x( )

66. 1−2x( ) 1+ x( ) 67. 3x −1( ) x − 3( )

6 TOPIC 3: LINEAR EQUATIONS and INEQUALITIES

A. Solving one linear equation in one variable:

Add or subtract the same value on each side of the equation, or multiply or divide each side by the same value, with the goal of placing the variable alone on one side. If there are one or more fractions, it may be desirable to eliminate them by multiplying both sides by the common denominator. If the equation is a proportion, you may wish to cross-multiply.

Problems 1-11: Solve:

1. 2x = 9 7. 4x − 6 = x

2. 3 = 6x5 8. x − 4 = x

2+1

3. 3x + 7 = 6 9. 6 − 4x = x

4. x3

= 54 10. 7x − 5 = 2x +10

5. 5− x = 9 11. 4x + 5 = 3 − 2x 6. x = 2x

5+1

To solve a linear equation for one variable in terms of the other(s), do the same as above:

example: Solve for F : C = 59F − 32( )

Multiply by 95: 9

5C = F − 32

Add 32: 95C + 32 = F

Thus, F = 95C + 32

example: Solve for b : a + b = 90 Subtract a : b = 90 − a

example: Solve for x : ax + b = c Subtract b : ax = c − b Divide by a : x = c−b

a

Problems 12-19: Solve for the indicated variable in terms of the other(s):

12. a + b = 180 16. y = 4 − x b = x = 13. 2a + 2b =180 17. y = 2

3x +1

b = x = 14. P = 2b + 2h 18. ax + by = 0 b = x = 15. y = 3x − 2 19. by − x = 0 x = y =

B. Solution of a one-variable equation

reducible to a linear equation:

Some equations which do not appear to be linear can be solved by using a related linear equation:

example: x+13x

= −1 Multiply by 3x : x +1 = −3x Solve: 4x = −1

x = − 14

(Be sure to check answer in the original equation.)

example: 3x+3x+1 = 5

Think of 5 as 51 and cross-multiply:

5x + 5 = 3x + 3 2x = −2 x = −1 But x = −1 does not make the original equation true (thus it does not check), so there is no solution.

Problems 20-25: Solve and check:

20. x−1x+1 = 6

7 23. x+3

2x= 2

21. 3x2x+1 = 5

2 24. 1

3= x

x+8

22. 3x−22x+1 = 4 25. x−2

4−2x = 3

example: 3− x = 2 Since the absolute value of both 2 and −2 is 2, 3− x can be either 2 or −2 . Write these two equations and solve each: 3− x = 2 or 3− x = −2 −x = −1 −x = −5 x =1 or x = 5

Problems 26-30: Solve:

26. x = 3 29. 2 − 3x = 0 27. x = −1 30. x + 2 =1 28. x −1 = 3

C. Solution of linear inequalities:

Rules for inequalities:

If a > b, then: a + c > b+ c a − c > b− c ac > bc (if c > 0 ) ac < bc (if c < 0 ) ac

> bc (if c > 0 )

ac

< bc (if c < 0 )

If a < b , then: a + c < b+ c a − c < b− c ac < bc (if c > 0 ) ac > bc (if c < 0 ) ac

< bc (if c > 0 )

ac

> bc (if c < 0 )

example: One variable graph: solve and graph on a number line: 1− 2x ≤ 7 (This is an abbreviation for x :1 − 2x ≤ 7{ })

Subtract 1, get −2x ≤ 6 Divide by −2 , x ≥ −3 Graph:

0 1 2 3-1-2-3-4

Problems 31-38: Solve and graph on a number line:

31. x − 3> 4 32. 4x < 2

7 33. 2x + 1≤ 6 36. 5− x > x − 3 34. 3< x − 3 37. x > 1 + 4 35. 4 − 2x < 6 38. 6x + 5 ≥ 4x −3

D. Solving a pair of linear equations in two

variables:

The solution consists of an ordered pair, an infinite number of ordered pairs, or no solution.

Problems 39-46: Solve for the common solution(s) by substitution or linear combinations:

39. x + 2y = 7 43. 2x − 3y = 5 3x − y = 28 3x + 5y =1 40. x + y = 5 44. 4x −1 = y

x − y = −3 4x + y =1 41. 2x − y = −9 45. x + y = 3 x = 8 x + y =1 42. 2x − y =1 46. 2x − y = 3 y = x −5 6x −9 = 3y

Answers:1. 9 2

2. 5 2

3. − 13

4. 154

5. −4 6. 5 3

7. 2 8. 10 9. 6 5

10. 3 11. − 1

3

12. 180 − a 13. 90 − a 14.

P−2h( )2

15. y+2( )3

16. 4 − y

17. 3y−3( )2

= 3 y−1( )2

18. − bya

19. x b

20. 13 21. − 5

4

22. − 65

23. 1 24. 4 25. no solution 26. –3, 3 27. no solution 28. –2, 4 29. 2 3

30. –3, −1 31. x > 7

0 7 32. x < 1

2

10 12

33. x ≤ 52

0 2 3 34. x > 6

0 6

35. x > −1

0-1 36. x < 4

0 4 37. x > 5

0 5 38. x ≥ −4

0-4 39. 9,−1( ) 40. (1, 4) 41. (8, 25)

42. −4,−9( ) 43. 28

19,−13

19( ) 44. 1

4 ,0( ) 45. no solution 46. any ordered pair of the form

a, 2a− 3( ) where a is any number. One example: 4, 5( ) . Infinitely many solutions.

TOPIC 4: QUADRATIC EQUATIONS

A. ax2 ++++ bx ++++ c ==== 0:

A quadratic equation can always be written so it

looks like ax2 + bx + c = 0 where a, b , and c

are real numbers and a is not zero.

example: 5 − x = 3x 2 Add x: 5 = 3x 2 + x

Subtract 5: 0 = 3x 2 + x − 5 or 3x

2 + x −5 = 0 So a = 3, b = 1, c = −5

example: x2 = 3

Rewrite: x2 − 3= 0

(Think of x2 + 0x − 3= 0)

So a = 1, b = 0, c = −3

Problems 1-4: Write each of the following in the

form ax2 + bx + c = 0 and identify a, b, c :

1. 3x + x2 − 4 = 0 3. x2 = 3x −1

2. 5− x2 = 0 4. x = 3x 2

5. 81x2 =1

B. Factoring:

Monomial factors:

ab+ ac = a b + c( )

example: x2 − x = x x −1( )

example: 4x2y + 6xy = 2xy 2x + 3( )

Difference of two squares:

a2 −b2 = a + b( ) a − b( )

example: 9x2 − 4 = 3x + 2( ) 3x −2( )

8 Trinomial square:

a2 + 2ab+ b2 = a+ b( )2 a2 −2ab + b2 = a− b( )2

example: x2 − 6x + 9 = x − 3( )2

Trinomial:

example: x2 − x −2 = x − 2( ) x +1( )

example: 6x2 − 7x − 3 = 3x +1( ) 2x − 3( )

Problems 6-20: Factor:

6. a2 + ab = 14. x

2 − 3x −10 = 7. a

3 − a2b+ ab2 = 15. 2x2 − x

8. 8x2 −2 = 16. 2x

3 + 8x2 + 8x = 9. x

2 −10x + 25 = 17. 9x2 +12x + 4 =

10. −4xy +10x 2 = 18. 6x3y2 −9x 4y =

11. 2x2 − 3x − 5 = 19. 1− x −2x2 =

12. x2 − x −6 = 20. 3x

2 −10x + 3 = 13. x

2y − y

2x =

C. Solving factored quadratic equations:

The following statement is the central principle: If ab = 0 , then a = 0 or b = 0 .

First, identify a and b in ab = 0 : example: 3− x( ) x + 2( ) = 0 Compare this with ab = 0 a = 3− x( ) ; b = x + 2( )

Problems 21-24: Identify a and b in each of the following:

21. 3x 2x −5( ) = 0 23. 2x −1( ) x − 5( )= 0 22. x − 3( )x = 0 24. 0 = x −1( ) x +1( )

Then, because ab = 0 means a = 0 or b = 0 , we can use the factors to make two linear equations to solve:

example: If 2x 3x − 4( ) = 0 then 2x( )= 0 or 3x − 4( ) = 0 and so x = 0 or 3x = 4 ; x = 4

3.

Thus, there are two solutions: 0 and 43

example: If 3 − x( ) x + 2( ) = 0 then 3 − x( ) = 0 or x + 2( )= 0 and thus x = 3 or x = −2 .

example: If 2x + 7( )2 = 0 then 2x + 7 = 0

2x = −7 x = − 7

2 (one solution)

Note: there must be a zero on one side of the equation to solve by the factoring method.

Problems 25-31: Solve:

25. x +1( ) x −1( ) = 0 29. x −6( ) x − 6( )= 0 26. 4x x + 4( )= 0 30. 2x − 3( )2 = 0 27. 0 = 2x − 5( )x 31. x x + 2( ) x − 3( ) = 0 28.0 = 2x + 3( ) x −1( )

D. Solving quadratic equations by factoring: Arrange the equation so zero is on one side (in

the form ax2 + bx + c = 0), factor, set each

factor equal to zero, and solve the resulting linear equations.

example: Solve: 6x2 = 3x

Rewrite: 6x2 − 3x = 0

Factor: 3x 2x −1( ) = 0 So 3x = 0 or 2x −1( ) = 0 Thus, x = 0 or x = 1

2

example: 0 = x2 − x −12

0 = x − 4( ) x + 3( ) x − 4 = 0 or x + 3 = 0 x = 4 or x = −3

Problems 32-43: Solve by factoring:

32. x x − 3( ) = 0 38. 0 = x + 2( ) x − 3( ) 33. x

2 − 2x = 0 39. 2x +1( ) 3x −2( ) = 0 34. 2x

2 = x 40. 6x2 = x + 2

35. 3x x + 4( )= 0 41. 9+ x2 = 6x

36. x2 = 2 − x 42. 1− x = 2x2

37. x2 + x = 6 43. x

2 − x −6 = 0

Another problem form: if a problem is stated in this form: ‘One of the solutions of

ax2 + bx + c = 0 is d’, solve the equation as

above, then verify the statement.

example: Problem: One of the solutions of

10x2 −5x = 0 is A. –2 B. −12

C. 12

D. 2 E. 5

Solve 10x2 −5x = 0 by factoring:

5x 2x −1( ) = 0 so 5x = 0 or (2x −1) = 0 thus x = 0 or x = 1

2.

Since x = 12 is one solution, answer C is correct.

9

44. One of the solutions of x −1( ) 3x + 2( ) = 0 is A. −32

B. −2 3

C. 0 D. 2 3

E. 32

45. One solution of x2 − x −2 = 0 is

A. –2 B. –1 C. −12

D. 12

E. 1

Answers: a b c

1. x 2 + 3x − 4 = 0 1 3 -4

2. −x 2 +5 = 0 -1 0 5

3. x 2 − 3x +1= 0 1 -3 1

4. 3x 2 − x = 0 3 -1 0

5. 81x 2 −1= 0 81 0 -1 Note: all signs could be the opposite.

6. a a + b( ) 7. a a2 − ab+ b2( ) 8. 2 2x+1( ) 2x -1( ) 9. x −5( )2 10. −2x 2y −5x( ) 11. 2x −5( ) x +1( ) 12. x − 3( ) x +2( ) 13. xy x − y( )

14. x −5( ) x +2( ) 15. x 2x −1( ) 16. 2x x +2( )2 17. 3x +2( )2 18. 3x 3y 2y − 3x( ) 19. 1−2x( ) 1+ x( ) 20. 3x −1( ) x − 3( )

a b

3x 2x −5 x − 3 x

2x −1 x − 5

21.

22.

23.

24. x −1 x +1 25. –1, 1 26. – 4, 0 27. 0, 5 2

28. −32 , 1

29. 6 30. 3 2

31. –2, 0, 3 32. 0, 3 33. 0, 2 34. 0, 12

35. –4, 0 36. –2, 1 37. –3, 2 38. –2, 3 39. −1

2 , 23

40. −12 ,

23

41. 3 42. –1, 12

43. –2, 3 44. B 45. B

TOPIC 5: GRAPHING

A. Graphing a point on the number line:

Problems 1-7: Select the letter of the point on the number line with coordinate:

-2 -1 0 1 2 3

A B C D E F G

1. 0 5. -1.5

2. 12 6. 2.75

3. − 12 7. − 3

2

4. 43

Problems 8-10: Which letter best locates the given number:

8. 59 9. 3

4 10. 2

3

Problems 11-13: Solve each equation and graph the solution on the number line:

example: x + 3=1 x = −2

11. 2x − 6 = 0 13. 4 − x = 3+ x 12. x = 3x + 5

B. Graphing a linear inequality (in one

variable) on the number line:

Rules for inequalities: If a > b, then: a + c > b + c a − c > b − c ac > bc (if c > 0) ac <<<< bc (if c < 0) ac

> bc (if c > 0)

ac

< bc (if c < 0)

If a < b, then: a + c < b + c a − c < b − c ac < bc (if c > 0) ac > bc (if c < 0) ac

< bc (if c > 0)

ac

> bc (if c < 0)

example: One variable graph: solve and graph on a number line: 1−−−− 2x ≤≤≤≤ 7

(This is an abbreviation for x :1−−−− 2x ≤≤≤≤ 7{{{{ }}}}) Subtract 1, get −−−−2x ≤≤≤≤ 6 Divide by -2, x ≥≥≥≥ −−−−3 Graph:

-3 -2 -1 0 1 -4 2 3

10 Problems 14-20: Solve and graph on a number line:

14. x −−−− 3 >>>> 4 18. 4 −−−− 2x <<<< 6 15. 4x <<<< 2 19. 5 −−−− x >>>> x −−−− 3 16. 2x +1≤ 6 20. x >1+ 4 17. 3 < x − 3

example: x > −3 and x <1 The two numbers -3 and 1 splits the number line into three parts: x > −3, −3< x <1, and x <1. Check each part to see if both x > −3 and x <1 are true:

part x values x > −3? x <1? both true?

1 2 3

x < −3 −3< x <1 x >1

no yes yes

yes yes no

no yes (solution) no

Thus the solution is −3< x <1 and the graph is:

example: x ≤ −3 or x <1 (‘or’ means ‘and/or’)

part x values x ≤ −3? x <1? at least one true?

1 2 3

x ≤ −3 −3≤ x <1 x >1

yes no no

yes yes no

yes (solution) yes (solution) no

So x ≤ −3 or −3 ≤ x <1; these cases are both covered if x <1. Thus the solution is x <1 and the graph is:

Problems 21-23: Solve and graph:

21. x <1 or x > 3 22. x ≥ 0 and x > 2 23. x >1 and x ≤ 4

C. Graphing a point in the coordinate plane:

If two number lines intersect at right angles so that: 1) one is horizontal with positive to the

right and negative to the left, 2) the other is vertical with positive up and

negative down, and 3) the zero points coincide

Then they form a coordinate plane, and 1) the horizontal number line is called the

x-axis, 2) the vertical line is the y-axis, 3) the common zero point is the origin, 4) there are four

quadrants, numbered as shown:

To locate a point on the plane, an ordered pair of numbers is used, written in the form (x, y). The x-coordinate is always given first.

Problems 24-27: Identify x and y in each ordered pair:

24. (3, 0) 26. (5, -2) 25. (-2, 5) 27. (0, 3)

To plot a point, start at the origin and make the moves, first in the x-direction (horizontal) and the in the y-direction (vertical) indicated by the ordered pair.

example: (-3, 4) Start at the origin, move left 3 (since x = −3), then (from there), up 4 (since y = 4 ) Put a dot there to indicate the point (-3, 4)

28. Join the following points in the given order: (-3, -2), (1, -4), (-3, 0), (2, 3), (-1, 2), (3, 0), (-3, -2), (-1, 2), (1, -4)

29. Two of the lines you drew cross each other. What are the coordinates of this crossing point?

30. In what quadrant does the point (a, b) lie, if a > 0 and b < 0?

Problems 31-34: For each given point, which of its coordinates, x or y, is larger?

D. Graphing linear equations on the

coordinate plane: The graph of a linear equation is a line, and one way to find the line is to join points of the line. Two points determine a line, but three are often plotted on a graph to be sure they are collinear (all in a line).

Case I: If the equation looks like x = a, then there is no restriction on y, so y can be any number. Pick 3 numbers for values of y, and make 3 ordered pairs so each has x = a. Plot and join.

II I

III IV

y

x

33

32

34

31

11

-1

1

example: x = −2 Select three y’s, say -3, 0, and 1 Ordered pairs: (-2, -3), (-2, 0), (-2, 1)

Plot and join:

Note the slope formula

gives −3−0−2−(−2) = −3

0,

which is not defined: a vertical line has no slope.

Case II: If the equation looks like y = mx + b , where either m or b (or both) can be zero, select any three numbers for values of x, and find the corresponding y values. Graph (plot) these ordered pairs and join.

example: y = −2 Select three x’s, say -1, 0, and 2 Since y must be -2, the pairs are: (-1, -2), (0, -2), (2, -2)

The slope is −2−(−2)

−1−0 = 0−1 = 0

And the line is horizontal.

example: y = 3x −1 Select 3 x’s, say 0, 1, 2: If x = 0 , y = 3•0 −1= −1 If x =1, y = 3•1−1= 2 If x = 2 , y = 3•2 −1= 5

Ordered pairs: (0, -1), (1, 2), (2, 5)

Note the slope is 2− −1( )1−0 = 3

1= 3,

And the line is neither horizontal nor vertical.

Problems 35-41: Graph each line on the number plane and find its slope (refer to section E below if necessary):

35. y = 3x 39. x = −2 36. x − y = 3 40. y = −2x 37. x =1− y 41. y = 1

2x +1

38. y =1

E. Slope of a line through two points:

Problems 42-47: Find the value of each of the following:

42. 36

= 45. 0−1−1−4 =

43. 5−21−(−1) = 46. 0

3=

44. −6−(−1)5−10 = 47. −2

0=

The line joining the points P1 x1, y1( ) and P2 x2, y2( ) has slope y2−y1

x2−x1.

example: A (3, -1), B (-2, 4)

slope of AB = 4−(−1)−2−3 = 5

−5 = −1

Problems 48-52: Find the slope of the line joining the given points:

48. (-3, 1) and (-1, -4) 49. (0, 2) and (-3, -5) 50. (3, -1) and (5, -1)

51. 52.

Answers:1. D 2. E 3. C 4. F 5. B 6. G 7. B 8. Q 9. T 10. S 11. 3

12. −52

13. ½

14. x > 7 15. x < 1

2

16. x ≤ 52

17. x > 6 18. x > −1 19. x < 4 20. x > 5 21. x <1 or x > 3

22. x > 2 23. 1< x ≤ 4 x y 24. 3 0 25. -2 5 26. 5 -2 27. 0 3 28.

29. (0,-1) 30. IV

31. x 32. y

33. y

34. x 35. 3

36. 1

37. -1

38. 0

39. none

0 3

-3 -2 0

0 1

0 7

0 ½ 1

0 2 3

0 6

-1 0

0 4

0 5

0 1 3

0 2

0 1 4 -3

3

1

-2

12 40. -2 41. ½

42. ½ 43. 3 2

44. 1 45. 15

46. 0 47. none (undefined)

48. −52

49. 7 3

50. 0 51. −3

5

52. 3 4

TOPIC 6: RATIONAL EXPRESSIONS

A. Simplifying fractional expressions:

example: 2736

= 9•39•4 = 9

9• 34

=1• 34

= 34

(note that you must be able to find a common factor—in this case 9 –in both the top and bottom in order to reduce a fraction.)

example: 3a12ab

= 3a•13a•4b = 3a

3a• 14b

=1• 14b

= 14b

(common factor: 3a)

Problems 1-12: Reduce:

1. 1352

= 7. 5a+b5a+c =

2. 2665

= 8. x−44−x =

3. 3+63+9 = 9.

2 x+4( ) x−5( )x−5( ) x−4( ) =

4. 6axy

15by= 10. x

2−9xx−9 =

5. 19a2

95a= 11.

8 x−1( )2

6 x 2−1( ) =

6. 14x−7y7y

= 12. 2x2−x−1

x 2−2x+1=

example: 3x

• y

15• 10x

y 2= 3•10•x•y

15•x•y 2=

33

• 55

• 21

• xx

• y

y• 1y

=1•1•2•1•1• 1y

= 2y

Problems 13-14: Simplify:

13. 4x6

• xy

y 2• 3y

2= 14. x

2−3xx−4 • x x−4( )

2x−6 =

B. Evaluation of fractions:

example: If a = −1 and b = 2 , find the value of a+3

2b−1

Substitute: −1+32(2)−1 = 2

3

Problems 15-22: Find the value, given a = −1, b = 2 , c = 0 , x = −3, y =1, z = 2:

15. 6b

= 19. 4x−5y3y−2x =

16. xa

= 20. bc

= 17. x

3= 21. − b

z=

18. a−yb

= 22. cz

=

C. Equivalent fractions:

example: 34 is equivalent to how many eighths?

34

=8, 3

4=1• 3

4= 22

• 34

= 2•32•4 = 6

8

example: 65a

=5ab

65a

= bb

• 65a

= 6b5ab

example: 3x+2x+1 =

4 x+1( )

3x+2x+1 = 4

4• 3x+2

x+1 = 12x+84x+4

example: x−1x+1 =

x+1( ) x−2( )

x−1x+1 = x−2( ) x−1( )

x−2( ) x+1( ) = x 2−3x+2x+1( ) x−2( )

Problems 23-27: Complete:

23. 49

=72 26. 30−15a

15−15b =1+b( ) 1−b( )

24. 3x7

=7y 27. x−6

6−x = −2

25. x+3x+2 =

x−1( ) x+2( )

How to get the lowest common denominator (LCD) by finding the least common multiple (LCM) of all denominators:

example: 5 6 and 815 .

First find LCM of 6 and 15:

6 = 2• 3 15 = 3•5 LCM= 2• 3•5 = 30, so 6

5= 2530, and 8

15= 1630

example: 34 and 1

6a:

4 = 2•2 6a = 2• 3• a LCM= 2•2• 3• a =12a, so 3

4= 9a12a

, and 16a

= 212a

example: 3x+2 and

−1x−2

LCM= x + 2( ) x − 2( ), so 3

x+2 = 3• x−2( )x+2( ) x−2( ) , and

−1x−2 = −1• x+2( )

x+2( ) x−2( )

Problems 28-33: Find equivalent fractions with the lowest common denominator:

28. 23 and 2

9 31. 3

x−2 and 42−x

29. 3x and 5 32. −4

x−3 and 5x+3

30. x3 and −4

x+1 33. 1x and 3x

x+1

1 -2

13 D. Adding and subtracting fractions: If denominators are the same, combine the numbers:

example: 3xy

− xy

= 3x−xy

= 2xy

Problems 34-38: Find the sum or difference as indicated (reduce if possible):

34. 47

+ 27

= 37. x+2x 2+2x

− 3y 2

xy 2=

35. 3x−3 − x

x−3 = 38. 3ab

+ 2b

− ab

= 36. b−a

b+a − a−bb+a =

If denominators are different, find equivalent fractions with common denominators, then proceed as before (combine numerators):

example: a2

− a4

= 2a4

− a4

= 2a−a4

= a4

example: 3x−1 + 1

x+2

= 3 x+2( )x−1( ) x+2( ) + x−1( )

x−1( ) x+2( )

= 3x+6+x−1x−1( ) x+2( ) = 4x+5

x−1( ) x+2( )

Problems 39-51: Find the sum or difference:

39. 3a

− 12a

= 46. a − 1a

= 40. 3

x− 2

a= 47. x

x−1 + x1−x =

41. 45

− 2x

= 48. 3x−2x−2 − 2

x+2 = 42. 2

5+ 2 = 49. 2x−1

x+1 − 2x−1x−2 =

43. ab

− 2 = 50. xx−2 − 4

x 2−2x=

44. a − cb

= 51. xx−2 − 4

x 2−4=

45. 1a

+ 1b

=

E. Multiplying fractions: Multiply the tops, multiply the bottoms, reduce if possible:

example: 34

• 25

= 620

= 310

example: 3 x+1( )x−2 • x

2−4x 2−1

= 3 x+1( ) x+2( ) x−2( )x−2( ) x+1( ) x−1( ) = 3x+6

x−1

52. 23

• 38

= 56. 2 12( )2 =

53. ab

• cd

= 57. 2a 3

5b( )3 =

54. 27a

• ab12

= 58. 3 x+4( )5y

• 5y 3

x 2−16=

55. 34( )2 = 59. x+3

3x• x 2

2x+6 =

F. Dividing fractions: A nice way to do this is to make a compound fraction and then multiply the top and bottom (of the big fraction) by the LCD of both:

example: ab

÷ cd

=ab

cd

=ab

•bdcd

•bd= ad

bc

example: 7

23

− 12

= 7•623

− 12( )•6

= 424−3 = 42

1 = 42

example: 5x2y

÷ 2x =5x2y

2x=

5x2y

•2y2x •2y

= 5x4xy

= 54y

60. 3423

= 66. a − 43a

− 2=

61. 11 38

÷ 34

= 67.

x+7x 2−91x−3

=

62. 34

÷ 2 = 68. 234

=

63. ab

÷ 3= 69.

23

4=

64. 3a

÷ b3

= 70. ab

c=

65. 2a− b

12

= 71. abc

=

Answers:1. 1

4

2. 25

3. 34

4. 2ax5b

5. a5

6. 2x−yy

7. 5a+b5a+c

8. -1

9. 2 x+4( )x−4

10. x

11. 4 x−1( )3 x+1( )

12. 2x+1x−1

13. x 2 14. x

2

2

15. 3 16. 3 17. -1 18. -1 19. −17

9

20. undefined 20

21. -1

22. 0 23. 32 24. 3xy

25. x 2 +2x − 3 or x −1( ) x + 3( ) 26. 2+2b− a − ab or 1+ b( )2− a( ) 27. 2

28. 69, 29

29. 3x, 5x

x

30. x x+1( )3 x+1( ) ,

−123 x+1( )

31. 3x−2 ,

−4x−2

14

32. −4 x+3( )x−3( ) x+3( ) ,

−5 x−3( )x−3( ) x+3( )

33. x+1x x+1( ) ,

3x 2

x x+1( )

34. 6 7

35. -1

36. 2b−2ab+a

37. −2x

38. 2a+2b

39. 52a

40. 3a−2xax

41. 4x−105x

42. 125

43. a−2bb

44. ab−cb

45. a+bab

46. a2−1a

47. 0

48. 3x2 +2xx 2−4

49. −3 2x−1( )x+1( ) x−2( )

50. x+2x

51. x2 +2x−4x 2−4

52. 14

53. acbd

54. b42

55. 916

56. 254

57. 8a 9

125b 3

58. 3y 2

x−4

59. x 6

60. 9 8

61. 916

62. 3 8

63. a3b

64. 9ab

65. 4a − 2b 66. a

2−4a3−2a

67. x+7x+3

68. 8 3

69. 16

70. abc

71. acb

TOPIC 7: EXPONENTS and SQUARE ROOT

A. Positive integer exponents:

ab means use a as a factor b times. (b is the exponent or power of a.)

example: 25 means 2•2•2•2•2, and has value 32. example: c •c •c = c 3

Problems 1-14: Find the value:

1. 23 = 8. .2( )2 =

2. 32 = 9. 1 12( )2 =

3. −42 = 10. 210 = 4. −4( )2 = 11. −2( )9 =

5. 04 = 12. 2 23( )2 =

6. 14 = 13. −1.1( )3 = 7. 2

3( )4 = 14. 32 •23 =

example: Simplify:

a•a•a•a•a = a5

Problems 15-18: Simplify:

15. 32 • x4 = 17. 42 −x( ) −x( ) −x( )= 16. 24 •b•b•b = 18. −y( )4 =

B. Integer exponents:

I. ab • ac = ab+c

II. ab

a c = ab−c

III. a b( )c = abc

IV. ab( )c = a c •bc

V. ab( )c = a

c

b c

VI. a0 =1 (if a ≠ 0) VII. a−b = 1

a b

Problems 19-28: Find x:

19. 23 •24 = 2x 24. 8 = 2x 20. 2

3

24= 2x 25. a3 • a = ax

21. 3−4 = 1

3x 26. b

10

b 5= bx

22. 52

52= 5x 27. 1

c −4 = c x

23. 23( )4 = 2x 28. a3y−2

a 2y−3 = ax

Problems 29-41: Find the value:

29. 7x0 = 36. xc+3

x c−3 =

30. 3−4 = 37. 2x−3

6x −4 =

31. 23 •24 = 38. ax+3( )x−3=

32. 05 = 39. x3( )2 =

33. 50 = 40. 3x3( )2 =

34. −3( )3 − 33 = 41. −2xy 2( )3 = 35. x c+3 • x c−3 =

C. Scientific notation:

example: 32800 = 3.2800 ×104 if the zeros in the ten’s and one’s places are significant. If

the one’s zero is not, write 3.280 ×104 ; if neither is significant: 3.28 ×104 .

example: .004031= 4.031×10−3

15

example: 2 ×102 = 200 example: 9.9 ×10−1 = .99

Note that scientific form always looks like a ×10n where 1≤ a <10 , and n is an integer power of 10.

Problems 42-45: Write in scientific notation:

42. 93,000,000 = 44. 5.07 = 43. .000042 = 45. −32 =

Problems 46-48: Write in standard notation:

46. 1.4030 ×103 = 48. 4 ×10−6 = 47. −9.11×10−2

To compute with numbers written in scientific form, separate the parts, compute, then recombine.

example: 3.14 ×105( ) 2( )=

3.14( ) 2( )×105 = 6.28×105 example: 4.28×106

2.14×10−2 = 4.282.14

× 106

10−2 = 2.00 ×108

example: 2.01×10−3

8.04×10−6 =

.250 ×103 = 2.50 ×102

Problems 49-56: Write answer in scientific notation:

49. 1040 ×10−2 = 53. 1.8×10−8

3.6×10−5 =

50. 10−40

10−10 = 54. 4 ×10−3( )2 =

51. 1.86×104

3×10−1 = 55. 2.5×102( )−1=

52. 3.6×10−5

1.8×10−8 = 56. −2.92×103( ) 4.1×107( )

−8.2×10−3 =

D. Simplification of square roots:

ab = a • b if a and b are both non-negative ( a ≥ 0 and b ≥ 0).

example: 32 = 16 • 2 = 4 2

example: 75 = 3 • 25 = 3 •5 = 5 3

example: If x ≥ 0 , x6 = x3

If x < 0 , x6 = x 3

Note: a = b means (by definition) that 1) b

2 = a , and 2) b ≥ 0

Problems 57-69: Simplify (assume all square roots are real numbers):

57. 81 = 59. 2 9 = 58. − 81 = 60. 4 9 =

61. 40 = 66. x5 = 62. 3 12 = 67. 4x 6 = 63. 52 = 68. a2 = 64. 9

16= 69. a3 =

65. .09 =

E. Adding and subtracting square roots:

example: 5 + 2 5 = 3 5

example: 32 − 2 = 4 2 − 2 = 3 2

Problems 70-73: Simplify:

70. 5 + 5 = 72. 3 2 + 2 = 71. 2 3 + 27 − 75 = 73. 5 3 − 3 =

F. Multiplying square roots:

a • b = ab if a ≥ 0 and b ≥ 0 .

example: 6 • 24 = 6•24 = 144 =12 example: 2 • 6 = 12 = 4 • 3 = 2 3

example: 5 2( ) 3 2( )=15 4 =15•2 = 30

Problems 74-79: Simplify:

74. 3 • 3 = 77. 9( )2 =

75. 3 • 4 = 78. 5( )2 =

76. 2 3( ) 3 2( )= 79. 3( )4 =

Problems 80-81: Find the value of x:

80. 4 • 9 = x 81. 3 2 • 5 = 3 x

G. Dividing square roots:

a ÷ b = a

b= a

b, if a ≥ 0 and b > 0 .

example: 2 ÷ 64 = 2

64= 2

8 (or 1

82 )

Problems 82-86: Simplify:

82. 3 ÷ 4 = 85. 36 ÷ 4 = 83. 9

25= 86. −8

16=

84. 492

=

If a fraction has a square root on the bottom, it is sometimes desirable to find an equivalent fraction with no root on the bottom. This is called rationalizing the denominator.

example: 58

= 5

8= 5

8• 2

2= 10

16= 10

4

example: 1

3= 1

3• 3

3= 3

9= 3

3

16 Problems 87-94: Simplify:

87. 94

= 89. 49

=

88. 18

9= 90. 3

2=

91. 1

5= 93. a

b=

92. 3

3= 94. 2 + 1

2=

Answers:1. 8 2. 9 3. -16 4. 16 5. 0 6. 1 7. 16

81

8. .04 9. 9

4

10. 1024 11. -512 12. 64 9

13. -1.331 14. 72

15. 9x 4 16. 16b3

17. −16x 3 18. y 4

19. 7 20. -1 21. 4 22. 0 23. 12 24. 3 25. 4 26. 5 27. 4 28. y +1 29. 7 30. 181

31. 128 32. 0 33. 1

34. -54

35. x 2c 36. x 6 37. x 3

38. a x2−9

39. x 6

40. 9x 6

41. −8x 3y 6 42. 9.3×107 43. 4.2×10−5 44. 5.07

45. −3.2×10 46. 1403.0 47. -.0911 48. .000004

49. 1×1038 50. 1×10−30

51. 6.2×104 52. 2.0×103 53. 5.0×10−4

54. 1.6×10−5

55. 4.0×10−3

56. 1.46×1013 57. 9 58. -9 59. 6 60. 12

61. 2 10

62. 6 3

63. 2 13 64. 3 4

65. .3

66. x 2 x

67. 2 x 3

68. a

69. a a

70. 2 5 71. 0

72. 4 2

73. 4 3 74. 3

75. 2 3

76. 6 6 77. 9 78. 5 79. 9 80. 36 81. 10

82. 32

83. 3 5

84. 7 2

85. 3 2

86. -2 87. 3 2

88. 2 89. 2 9

90. 62

91. 55

92. 3

93. abb

94. 3 2

2

TOPIC 8: GEOMETRIC MEASUREMENT

A. Intersecting lines and parallels: If two lines intersect as shown, adjacent angles

add to 180°. For example, a + d =180°. Non-adjacent angles are equal: for example, a = c .

If two lines, a and b, are parallel and are cut by a third line c, forming angles w, x, y, z as shown, then x = z , w + y =180°, so z + y =180° .

example: If a = 3x and c = x , find the measure of c.

b = c , so b = x . a + b =180, so 3x + x =180 , giving 4x =180 , or x = 45

Thus c = x = 45°

Problems 1-4: Given x =127°, find the measures of the other angles:

1. t 3. z 2. y 4. w

a

b

c

x y

w

z

a b c

d a b c

t y w z

x

17 5. Find x:

B. Formulas for perimeter P and area A of

triangles, squares, rectangles, and

parallelograms:

Rectangle, base b, altitude (height) h:

P = 2b + 2h A = bh

If a wire is bent in the shape, the perimeter is the length of the wire, and the area is the number of square units enclosed by the wire.

example: Rectangle with b = 7 and h = 8 : P = 2b + 2h = 2• 7 + 2•8 =

14 +16 = 30 units A = bh = 7•8 = 56 sq. units

A square is a rectangle with all sides equal, so the formulas are the same (and simpler if the side length is s):

P = 4s A = s2

example: Square with side 11cm has

P = 4s = 4 •11= 44cm A = s2 =112 =121cm2

(sq. cm)

A parallelogram with base b and height h has A = bh . If the other side length is a, then P = 2a + 2b .

example: Parallelogram has sides 4 and 6, and 5 is the length of the altitude perpendicular to the side 4. P = 2a + 2b = 2•6 + 2• 4 =12 + 8 = 20 units A = bh = 4 •5 = 20 sq. units

In a triangle with side lengths a, b, c and h is the altitude to side b, P = a+ b+ cA = 1

2bh = bh

2

example:

P = a + b + c = 6+ 8+10 = 24 units A = 1

2bh = 1

210( ) 4.8( )= 24 sq. units

Problems 6-13: Find P and A for each of the following figures:

6. Rectangle with sides 5 and 10. 7. Rectangle, sides 1.5 and 4. 8. Square with side 3 mi.

9. Square, side 34 yd.

10. Parallelogram with sides 36 and 24, and height 10 (on side 36).

11. Parallelogram, all sides 12, altitude 6. 12. Triangle with sides 5, 12, 13, and 5 is the height on side 12.

13. The triangle shown:

C. Formulas for circle area A and

circumference C:

A circle with radius r (and diameter d = 2r) has distance around (circumference)

C = πd or C = 2πr

(If a piece of wire is bent into a circular shape, the circumference is the length of wire.)

example: A circle with radius r = 70 has d = 2r =170 and exact circumference C = 2πr = 2•π • 70 =140π units.

If π is approximated by 227,

C =140π =140 227( )= 440 units approximately.

If π is approximated by 3.1, the approximate C =140(3.1) = 434 units.

The area of a circle is A = πr2 :

example: If r = 8 A = πr2 = π •82 = 64π sq. units

Problems 14-16: Find C and A for each circle:

14. r = 5 units 16. d = 4 km 15. r =10 feet

D. Formulas for volume V:

A rectangular solid (box) with length l, width w, and height h, has volume V = lwh .

example: A box with dimensions 3, 7, and 11 has what volume?

V = lwh = 3• 7•11= 231 cu. units

A cube is a box with all edges equal. If the edge is e the

volume V = e3

example: A cube has edge 4 cm.

V = e3 = 43 = 64cm3 (cu. cm)

A (right circular) cylinder with radius r and

altitude h has V = πr2h

example: A cylinder has r =10 and h =14 .

2x 3x

s

s

5 6

4

13 5

12

5 3

4

r

18 The exact volume is

V = πr2h = π •102 •14 =1400π cu. units If π is approximated by 22

7,

V =1400• 227

= 4400 cu. units If π is approximated by 3.14, V =1400 3.14( )= 4396 cu. units

A sphere (ball) with radius r has

volume V = 43

πr3

example: The exact volume of a sphere with

radius 6 in. is V = 43πr3 = 4

3•π •63

= 43π 216( )= 288π in3

Problems 17-24: Find the exact volume of each of the following solids:

17. Box, 6 by 8 by 9.

18. Box, 1 23 by 5

6 by 2 2

5.

19. Cube with edge 10. 20. Cube, edge .5 . 21. Cylinder with r = 5, h =10 22. Cylinder, r = 3 , h = 2 23. Sphere with radius r = 2. 24. Sphere with radius r = 3

4.

E. Sum of the interior angles of a triangle: The three angles of any triangle add to 180°.

example: Find the measures of angles C and A: ∠C (angle C) is marked to show its measure is 90°.

∠B + ∠C = 36 + 90 =126, so ∠A =180 −126 = 54°

Problems 25-29: Given two angles of a triangle, find the measure of the third angle:

25. 30°, 60° 28. 82°, 82° 26. 115°, 36° 29. 68°, 44° 27. 90°, 17°

F. Isosceles triangles:

An isosceles triangle is defined to have at least two sides with equal measure. The equal sides may be marked:

Or the measures may be given:

Problems 30-35: Is the triangle isosceles?

30. Sides 3, 4, 5 31. Sides 7, 4, 7

32. Sides 8, 8, 8 34.

33. 35.

The angles which are opposite the equal sides also have equal measures (and all three angles add to 180°).

example: Find the measures of ∠A and ∠C , given ∠B = 65° :

∠A + ∠B + ∠C =180 , and ∠A = ∠B = 65 , so ∠C = 50°

36. Find measures of

∠A and ∠B , if ∠C = 30°.

37. Find measures of ∠B and ∠C , if ∠A = 30°.

38. Find measure of ∠A .

39. If the angles of a triangle are 30°, 60°, and 90°, can it be isosceles?

40. If two angles of a triangle are 45° and 60°, can it be isosceles?

If a triangle has equal angles, the sides opposite these angles also have equal measures.

example: Find the measures of ∠B , AB and AC , given this figure, and ∠C = 40°:

∠B = 70°(because all angles add to 180°)

Since ∠A = ∠B, AC = BC =16 . AB can be found with trig -- later.

41. Can a triangle be isosceles and have a 90°? 42. Given ∠D = ∠E = 68°

and DF = 6 . Find the measure of ∠F and length of FE :

G. Similar triangles: If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.

example: ∆ABC and ∆FED are similar:

The pairs of corresponding

sides are AB and FE

BC and ED, and AC and FD .

r

36°

A

B C

12

7 12

C B

A

7

7

A

C

B B

6 6

A C

B

8 8

A C

8

A B

16

C

70°

D

F

E

C D

A B F E

53

36 53

36

19 43. Name two similar triangles and list the pairs of corresponding sides.

If two triangles are similar, any two corresponding sides have the same ratio (fraction value):

example: the ratio a to x, or ax,

is the same as by and c

z. Thus,

ax

= by, ax

= cz,

and by

= cz.

Each of these equations is called a proportion

Problems 44-45: Write proportions for the two similar triangles:

44. 45.

example: Find x:

Write and solve a proportion:

25

= 3x, so 2x =15 , x = 7 1

2

Problems 46-49: Find x:

46. 48. 47. 49.

50. Find x and y:

H. Pythagorean theorem:

In any triangle with a 90° (right) angle,

the sum of the squares of the legs equals the square of the hypotenuse. (The legs are the two shorter sides; the hypotenuse is the longest side.) If the legs have lengths a and b,

and the hypotenuse length is c, then a2 + b2 = c2 (In words, ‘In a right triangle, leg squared plus leg squared equals hypotenuse squared.’)

example: A right triangle has hypotenuse 5 and one leg 3. Find the other leg.

Since leg2 + leg2 = hyp2 , 32 + x2 = 52

9 + x 2 = 25 x2 = 25 − 9 =16 x = 16 = 4

Problems 51-54: Each line of the chart lists two sides of a right triangle. Find the length of the third side:

leg leg hypotenuse

51. 15 17 52. 8 10 53. 5 12 54. 2 3

Problems 55-56: Find x:

55. 56. If the sum of the squares of two sides of a triangle is the same as the square of the third side, the triangle is a right triangle.

example: Is a triangle with sides 20, 29, 21 a right triangle?

202 + 212 = 292, so it is a right triangle.

Problems 57-59: Is a triangle right, if it has sides:

57. 17, 8, 15 59. 60, 61, 11 58. 4, 5, 6

Answers:

1. 127° 2. 53° 3. 53° 4. 127° 5. 36°

6. 30 un 50 un2 7. 11 un 6 un 2 8. 12 mi 9 mi2

9. 3 yd 916 yd2

10. 120 un 360 un2 11. 48 un 72 un2 12. 30 un 30 un2 13. 12 un 6 un2

C A 14. 10π un 25π un2 15. 20π ft 100π ft2 16. 4πkm 4πkm2

17. 432 18. 103

19. 1000 20. .125

P A

20

21. 250π 22. 6π 23. 32π

3

24. 9π16

25. 90° 26. 29° 27. 73° 28. 16° 29. 68° 30. no 31. yes 32. yes 33. yes 34. yes 35. can’t tell

36. 75° each 37. 120°, 30° 38. 60° 39. no 40. no

41. yes:

42. 44°, 6 43. ∆ABE , ∆ACD

AB, AC AE, AD BE, CD

44. 39

= 515

= 412

45. dc

= aa+b = f

f +e

46. 14 5

47. 9 4

48. 14 5

49. 452

50. 407 , 163

51. 8 52. 6 53. 13

54. 5 55. 9

56. 41 57. yes 58. no 59. yes

TOPIC 9: WORD PROBLEMS

A. Arithmetic, percent, and average:

1. What is the number, which when multiplied by 32, gives 32• 46?

2. If you square a certain number, you get 92 . What is the number?

3. What is the power or 36 that gives 362? 4. Find 3% of 36. 5. 55 is what percent of 88? 6. What percent of 55 is 88? 7. 45 is 80% of what number? 8. What is 8.3% of $7000? 9. If you get 36 on a 40-question test, what percent is this?

10. The 3200 people who vote in an election are 40% of the people registered to vote. How many are registered?

Problems 11-13: Your wage is increased by 20%, then the new amount is cut by 20% (of the new amount):

11. Will this result in a wage which is higher than, lower than, or the same as the original wage?

12. What percent of the original wage is this final wage?

13. If the above steps were reversed (20% cut followed by 20% increase), the final wage would be what percent of the original wage?

Problems 14-16: If A is increased by 25%, it equals B:

14. Which is larger, B or the original A? 15. B is what percent of A? 16. A is what percent of B?

17. What is the average of 87, 36, 48, 59, and 95? 18. If two test scores are 85 and 60, what minimum score on the next test would be needed for an overall average of 80?

19. The average height of 49 people is 68 inches. What is the new average height if a 78-inch person joins the group?

B. Algebraic substitution and evaluation:

Problems 20-24: A certain TV uses 75 watts of power, and operates on 120 volts:

20. Find how many amps of current it uses, from the relationship: volts times amps equals watts.

21. 1000 watts = 1 kilowatt (kw). How many kilowatts does the TV use?

22. Kw times hours = kilowatt-hours (kwh). If the TV is on for six hours a day, how many kwh of electricity are used?

23. If the set is on for six hours every day of a 30-day month, how many kwh are used for the month?

24. If the electric company charges 8¢ per kwh, what amount of the month’s bill is for TV power?

Problems 25-33: A plane has a certain speed in still air, where it goes 1350 miles in three hours:

25. What is its (still air) speed? 26. How far does the plane go in 5 hours? 27. How far does it go in x hours? 28. How long does it take to fly 2000 miles? 29. How long does it take to fly y miles? 30. If the plane flies against a 50 mph headwind, what is its ground speed?

21 31. If the plane flies against a headwind of z

mph, what is its ground speed? 32. If it has fuel for 7.5 hours of flying time, how far can it go against the headwind of 50 mph?

33. If the plane has fuel for t hours of flying time, how far can it go against the headwind of z mph?

C. Ratio and proportion:

Problems 34-35: x is to y as 3 is to 5:

34. Find y when x is 7. 35. Find x when y is 7.

Problems 36-37: s is proportional to P, and P = 56 when s =14 :

36. Find s when P =144 . 37. Find P when s =144 .

Problems 38-39: Given 3x = 4y :

38. Write the ratio x : y as the ratio of two

integers 39. If x = 3, find y.

Problems 40-41: x and y are numbers, and two x’s equal three y’s:

40. Which of x or y must be larger? 41. What is the ratio of x to y?

Problems 42-44: Half of x is the same as one-third of y:

42. Which of x and y is the larger? 43. Write the ratio x : y as the ratio of two

integers. 44. How many x’s equal 30 y’s?

D. Problems leading to one linear equation:

45. 36 is three-fourths of what number? 46. What number is 34 of 36?

47. What fraction of 36 is 15? 48. 2 3 of

16 of

34 of a number is 12. What is

the number? 49. Half the square of a number is 18. What is the number?

50. 81 is the square of twice what number? 51. Given a positive number x. Two times a positive number y is at least four times x. How small can y be?

52. Twice the square root of half a number is 2x. What is the number?

Problems 53-55: A gathering has twice as many women as men. W is the number of women and M is the number of men:

53. Which is correct: 2M =W or M = 2W ? 54. If there are 12 women, how many men are there?

55. If the total number of men and women present is 54, how many of each are there?

56. $12,000 is divided into equal shares. Babs gets four shares, and Ben gets the one remaining share. What is the value of one share?

E. Problems leading to two linear equations:

57. Two science fiction coins have values x and y. Three x’s and five y’s have of 75¢, and one x and two y’s have a value of 27¢. What is the value of each?

58. In mixing x gm of 3% and y gm of 8% solutions to get 10 gm of 5% solution, these equations are used:

.03x + .08y = .05 10( ), and x + y =10 How many gm of 3% solution are needed?

F. Geometry:

59. Point x is on each of two given intersecting lines. How many such points x are there?

60. On the number line, points P and Q are two units apart. Q has coordinate x. What are the possible coordinates of P?

Problems 61-62:

61. If the length of chord AB is x and the length of CB is 16, what is AC?

62. If AC = y and CB = z , how long is AB (in terms of y and z)?

Problems 63-64: The base of a rectangle is three times the height:

63. Find the height if the base is 20. 64. Find the perimeter and area.

65. In order to construct a square with an area which is 100 times the area of a given square, how long a side should be used?

B

C A

O

22 Problems 66-67: The length of a rectangle is increased by 25% and its width is decreased by 40%.

66. Its new area is what percent of its old area? 67. By what percent has the old area increased or decreased?

68. The length of a rectangle is twice the width. If both dimensions are increased by 2 cm, the

resulting rectangle has 84cm2 more area.

What was the original width? 69. After a rectangular piece of knitted fabric shrinks in length 1 cm and stretches in width

2 cm, it is a square. If the original area was

40cm2, what is the square area?

70. This square is cut into two smaller squares and two non-square rectangles as shown. Before being cut, the large square had area

a+ b( )2. The two smaller squares have areas a2 and b2 . Find the total area of the two non-square rectangles. Show that the areas of the 4 parts add up to the area of the original square.

Answers:1. 46 2. 9 3. 2 4. 1.08 5. 62.5% 6. 160% 7. 56.25 8. $581 9. 90% 10. 8000 11. lower 12. 96% 13. same (96%)

14. B 15. 125% 16. 80% 17. 65 18. 95 19. 68.2 20. .625 amps 21. .075 kw 22. .45 kwh 23. 13.5 kwh 24. $1.08 25. 450 mph

26. 2250 mi 27. 450x mi 28. 409 hr

29. y 450hr

30. 400 mph 31. 450− z mph 32. 3000 mi

33. 450− z( )t mi 34. 353

35. 215

36. 36 37. 576

38. 4 : 3 39. 9 4

40. x 41. 3 : 2 42. y

43. 3 : 2 44. 45 45. 48 46. 27 47. 512

48. 144 49. 6

50. 9 2

51. 2x 52. 2x 2 53. 2M =W 54. 6 55. 18 men, 36 women 56. $1500 57. x :15¢, y : 6¢

58. 6 gm 59. 1 60. x − 2 , x + 2 61. x −16 62. y + z 63. 203

64. P = 1603 , A = 400

3

65. 10 times the original side

66. 75% 67. 25% decrease

68. 5cm 69. 49

70. 2ab

a2 +2ab+ b2 = a + b( )2

a b

a

b

1 INTERMEDIATE ALGEBRA READINESS DIAGNOSTIC TEST PRACTICE

Directions: Study the examples, work the problems, then check your answers at the end of each topic. If

you don’t get the answer given, check your work and look for mistakes. If you have trouble, ask a math

teacher or someone else who understands this topic.

TOPIC 1: ELEMENTARY OPERATIONS

A. Algebraic operations, grouping, evaluation:

To evaluate an expression, first calculate the

powers, then multiply and divide in order from

left to right, and finally add and subtract in order

from left to right. Parentheses have preference.

example: 14 − 32 = 14 − 9 = 5 example: 2• 4 + 3•5 = 8+15 = 23 example: 10 − 2• 32 =10 − 2•9 =10 −18 = −8 example: (10− 2)• 32 = 8•9 = 72

Problems 1-7: Find the value:

1. 23 = 5. 0

4 = 2. −24 = 6. (−2)4 = 3. 4 + 2•5 = 7. 1

5 = 4. 32 − 2• 3+1=

Problems 8-13: Find the value if a = −3 , b = 2 , c = 0, d = 1, and e = −3 :

8. a −e = 11. ed

+ ba

− 2de

=

9. e2 + (d − ab)c = 12. b

e=

10. a − (bc − d) + e = 13. dc

=

Combine like terms when possible:

example: 3x + y2 − (x + 2y2) = 3x − x + y 2 − 2y2 = 2x − y2

example: a − a2 + a = 2a − a

2

Problems 14-20: Simplify:

14. 6x + 3 − x − 7 = 18. 3a −2 4(a −2b) − 3a( )= 15. 2(3− t) = 19. 3(a + b) −2(a − b) = 16. 10r −5(2r − 3y) = 20. 1+ x −2x + 3x − 4x = 17. x2 − (x − x2) =

B. Simplifying fractional expressions:

example: 2736

= 9•39•4 = 9

9• 34

=1• 34

= 34

(note that you must be able to find a common factor - in this case 9 - in both the top and bottom in order to reduce a fraction.)

example: 3a12ab

= 3a•13a•4b = 3a

3a• 14b

=1• 14b

= 14b

(common factor: 3a)

Problems 21-32: Reduce:

21. 1352

= 22. 2665

=

23. 3+63+9 = 28. x−4

4−x =

24. 6axy

15by= 29.

2(x+4)(x−5)(x−5)(x−4 ) =

25. 19a2

95a= 30. x

2−9xx−9 =

26. 14x−7y7y

= 31. 8(x−1)2

6(x 2−1)=

27. 5a+b5a+c = 32. 2x

2−x−1x 2−2x+1

=

example: 3x

• y

15• 10x

y 2= 3•10•x•y

15•x•y 2=

33

• 55

• 21

• xx

• y

y• 1

y=

1•1•2•1•1• 1y

= 2y

Problems 33-34: Simplify:

33. 4x6

• xy

y 2• 3y

2= 34. x

2−3xx−4 • x x−4( )

2x−6 =

C. Laws of integer exponents:

I. ab • ac = ab+c

II. ab

a c = ab−c

III. (ab)c = a

bc

IV. ab( )c = ac •bc

V. ab( )c = a c

b c

VI. a0 =1 (if a ≠ 0 )

VII. a−b = 1

a b

Problems 35-44: Find x:

35. 23 •24 = 2x 40. 8 = 2x

36. 23

24

= 2x 41. ax = a3 • a

37. 3−4 = 1

3x 42. b

10

b5

= bx

38. 52

52

= 5x 43. 1

c−4 = c

x

39. 24( )3 = 2

x 44. a

3y−2

a2y−3 = a

x

Problems 45-59: Simplify:

45. 8x0 = 50. −3( )3 − 33 =

46. 3−4 = 51. 2x • 4x−1 =

47. 23 •24 = 52. 2c+3

2c−3 =

48. 05 = 53. 2c+3 •2c−3 =

49. 50 = 54. 8x

2x−1 =

2

55. 2x−3

6x−4 = 58. −2a2( )4 ab

2( )=

56. ax+3( )x = 59. 2 4 xy

2( )−1−2x−1

y( )2 =

57. a3x−2

a2x−3 =

D. Scientific notation:

example: 32800= 3.2800×104 if the zeros in the ten’s and one’s places are significant. If

the one’s zero is not, write 3.280×104 ; if neither is significant: 3.28×104

example: .004031= 4.031×10−3

example: 2×102 = 200 example: 9.9×10−1 = .99

Note that scientific form always looks like a ×10n

where 1 ≤ a <10 , and n is an integer power of 10.

Problems 60-63: Write in scientific notation:

60. 93,000,000 = 62. 5.07 =

61. .000042 = 63. −32 =

Problems 64-66: Write in standard notation:

64. 1.4030×103 = 66. 4 ×10−6 = 65. −9.11×10−2 =

To compute with numbers written in scientific form,

separate the parts, compute, and then recombine.

example: 3.14 ×105( ) 2( )= 3.14( ) 2( )×105

= 6.28×105 example: 4.28×106

2.14×10−2 = 4.282.14

× 106

10−2 = 2.00×108

example: 2.01×10−3

8.04×10−6 = .250×103 = 2.50×102

Problems 67-74: Write answer in scientific notation:

67. 1040 ×10−2 = 71. 1.8×10−8

3.6×10−5 =

68. 10−40

10−10 = 72. 4 ×10−3( )2 =

69. 1.86×104

3×10−1 = 73. 2.5×102( )−1 =

70. 3.6×10−5

1.8×10−8 = 74. −2.92×103( ) 4.1×107( )

−8.2×10−3 =

E. Absolute value:

example: 3 = 3 example: −3 = 3 example: a depends on a

if a ≥ 0, a = a

if a < 0, a = −a example: −−3 = −3

Problems 75-78: Find the value:

75. 0 = 77. 3 + −3 =

76. a

a= 78. 3 − −3 =

Problems 79-84: If x = −4 , find:

79. x +1 = 82. x + x = 80. 1− x = 83. −3x = 81. − x = 84. (x − (x − x ) ) =

Answers:

1. 8

2. –16

3. 14

4. 4

5. 0

6. 16

7. 1

8. 0

9. 9

10. –5

11. –3

12. − 23

13. no value (undefined)

14. 5x − 4 15. 6 − 2t 16. 15y

17. 2x2 − x

18. a +16b 19. a + 5b 20. 1− 2x

21. 14

22. 2 5

23. 3 4

24. 2ax5b

25. a 5

26. 2x−yy

27. 5a+b5a+c

28. -1

29. 2 x+4( )x−4

30. x

31. 4 x−1( )3 x+1( )

32. 2x+1x−1

33. x2

34. x2

2

35. 7

36. –1

37. 4

38. 0

39. 12

40. 3

41. 4

42. 5

43. 4

44. y +1 45. 8

46. 181

47. 128

48. 0

49. 1

50. –54

51. 23x−2

52. 64

53. 4c

54. 22x+1

55. x 3

3

56. ax 2 + 3x

57. ax+1

58. 16a9b2

59. 2x 3

60. 9.3×107 61. 4.2 ×10−5

62. 5.07

63. –3.2×10 64. 1403.0

65. -.0911

66. .000004

67. 1×1038 68. 1×10−30

69. 6.2×104 70. 2.0×103 71. 5.0×10−4

72. 1.6×10−5

73. 4.0 ×10−3

74. 1.46×1013 75. 0

76. 1 if a > 0; -1 if a < 0; (no value if a = 0 )

77. 6

78. 0

79. 3

80. 5

81. –4

82. 0

83. 12

84. 12

TOPIC 2: RATIONAL EXPRESSIONS

A. Adding and subtracting fractions:

If denominators are the same, combine the

numerators:

example: 3xy

− xy

= 3x−xy

= 2xy

Problems 1-5: Find the sum or difference as

indicated (reduce if possible):

1. 47

+ 27

= 4. x+2x 2+2x

− 3y 2

xy 2=

2. 3x−3 − x

x−3 = 5. 3ab

+ 2b

− ab

= 3. b−a

b+a − a−bb+a =

If denominators are different, find equivalent

fractions with common denominators:

example: 34 is equivalent to how many eighths? 34

=8; 3

4=1• 3

4= 2

2• 34

= 2•32•4 = 6

8

example: 65a

=5ab

; 65a

= bb

• 65a

= 6b5ab

example: 3x+2x+1 =

4 x+1( ) ; 3x+2x+1 = 4

4• 3x+2

x+1 = 12x+84x+4

example: x−1x+1 =

x+1( ) x−2( ) ;

x−1x+1 = x−2( ) x−1( )

x−2( ) x+1( ) = x 2−3x+2x+1( ) x−2( )

Problems 6-10: Complete:

6. 49

=72 9. 30−15a

15−15b =1+b( ) 1−b( )

7. 3x7

=7y 10. x−6

6−x = −2

8. x+3x+2 =

x−1( ) x+2( )

How to get the lowest common denominator

(LCD) by finding the least common multiple

(LCM) of all denominators:

example: 5 6 and 815 .

First find LCM of 6 and 15:

6 = 2• 3 15 = 3•5 LCM = 2• 3•5 = 30

so, 56

= 2530, and 8

15= 16

30

example: 34 and 16a :

4 = 2•2 6a = 2• 3• a LCM = 2•2• 3• a =12a

so, 34

= 9a12a, and 1

6a= 2

12a

example: 23 x+2( ) and

ax6(x+1)

3 x + 2( )= 3• x + 2( ) 6 x +1( )= 2• 3• x +1( ) LCM= 2• 3• x +1( )• x + 2( )

so, 23(x+2) = 2•2 x+1( )

2•3 x+1( ) x+2( ) = 4 x+1( )6 x+1( ) x+2( ) and

ax6 x+1( ) = ax x+2( )

6 x+1( ) x+2( )

Problems 11-16: Find equivalent fractions with

the lowest common denominator:

11. 23 and 2

9 14. 3

x−2 and 42−x

12. 3xand 5 15. x

15 x 2−2( ) and7x y−1( )10 x−1( )

13. x3and −4

x+1 16. 1x, 3xx+1 , and

x2

x 2+x

After finding equivalent fractions with common

denominators, proceed as before (combine numerators):

example: a2

− a4

= 2a4

− a4

= 2a−a4

= a4

example: 3x−1 + 1

x+2

= 3 x+2( )x−1( ) x+2( ) + x−1( )

x−1( ) x+2( )

= 3x+6+x−1x−1( ) x+2( ) = 4x+5

x−1( ) x+2( )

Problems 17-30: Find the sum or difference:

17. 3a

− 12a

= 23. 1a

+ 1b

= 18. 3

x− 2

a= 24. a − 1

a=

19. 45

− 2x

= 25. xx−1 + x

1−x = 20. 2

5+ 2 = 26. 3x−2

x−2 − 2x+2 =

21. ab

− 2 = 27. 2x−1x+1 − 2x−1

x−2 = 22. a − c

b=

4

28. 1x−1( ) x−2( ) + 1

x−2( ) x−3( ) − 2x−3( ) x−1( ) =

29. xx−2 − 4

x 2−2x= 30. x

x−2 − 4

x 2−4=

B. Multiplying fractions:

Multiply the top numbers, multiply the bottom

numbers, and reduce if possible.

example: 34

• 25

= 620

= 310

example: 3 x+1( )x−2 • x

2−4x 2−1

= 3 x+1( ) x+2( ) x−2( )x−2( ) x+1( ) x−1( ) = 3x+6

x−1

31. 23

• 38

= 33. 27a

• ab12

=

32. ab

• cd

= 34.3 x+4( )5y

• 5y3

x 2−16=

35. a+b( )3

(x−y )2• x−y( )

5− p( ) • p−5( )2

a+b( )2=

36. 34( )2 = 38. 2 1

2( )2 =

37. 2a3

5b( )3 =

C. Dividing fractions:

Make a compound fraction and then multiply the top

and bottom (of the big fraction) by the LCD of both:

example: ab

÷ cd

=ab

cd

=ab

•bdcd

•bd= ad

bc

example: 7

23

− 12

= 7•623

− 12( )•6

= 424−3 = 42

1 = 42

example: 5x2y

÷ 2x =5x2y

2x=

5x2y

•2y2x •2y

= 5x4xy

= 54y

39. 34

÷ 23

= 47. 234

=

40. 11 38

÷ 34

= 48.

23

4=

41. 34

÷ 2 = 49.

ab

c=

42. ab

÷ 3= 50. abc

=

43. 3a

÷ b3

= 51.

1a

− 1b

1a

+ 1b

=

44.

x+7x 2−91

x−3

= 52.

12a

− 1b

1a

− 12b

=

45. a− 43a

− 2= 53.

1a

− 1b

1ab

=

46. 2a− b

12

=

Answers:

1. 67

2. –1

3. 2b−2ab+a

4. − 2x

5. 2a+2b

6. 32

7. 3xy

8. x2 + 2x − 3

9. 2 + 2b − a − ab 10. 2

11. 69, 2

9

12. 3x, 5x

x

13. x x+1( )3 x+1( ) ,

−123 x+1( )

14. 3x−2 ,

−4x−2

15. 2x x−1( )

30 x 2−2( ) x−1( ), 21x y−1( ) x 2−2( )30 x 2−2( ) x−1( )

16. x+1x x+1( ) ,

3x2

x x+1( ) , x2

x x+1( )

17. 52a

18. 3a−2xax

19. 4x−105x

20. 125

21. a−2bb

22. ab−cb

23. a+bab

24. a2−1a

25. 0

26. 3x2 +2x

x 2−4

27. −3 2x−1( )x+1( ) x−2( )

28. 0

29. x+2x

30. x 2 +2x−4x 2−4

31. 14

32. acbd

33. b42

34. 3y 2

x−4

35. a+b( ) 5− p( )

x−y

36. 916

37. 8a 9

125b 3

38. 254

39. 9 8

40. 916

41. 3 8

42. a3b

43. 9ab

44. x+7x+3

45. a2−4a3−2a

46. 4a − 2b 47. 8 3

48. 16

49. abc

50. acb

51. b−ab+a

52. b−2a2b−a

53. b − a

5 TOPIC 3: EXPONENTS and RADICALS

A. Definitions of powers and roots:

Problems 1-20: Find the value:

1. 23 = 11. −1253 =

2. 32 = 12. 52 =

3. −42 = 13. −5( )2 =

4. −4( )2 = 14. x2 =

5. 04 = 15. a3

3 = 6. 1

4 = 16. 14

=

7. 64 = 17. .04 = 8. 643 = 18. 2

3( )4 =

9. 646 = 19. 81a84 =

10. − 49 = 20. 7 • 7 =

B. Laws of integer exponents:

I. ab •a c = ab+c

II. ab

a c = ab−c

III. ab( )c = a

bc

IV. ab( )c = ac •bc

V. ab( )c = a

c

b c

VI. a0 =1 (if a ≠ 0 )

VII. a−b = 1

a b

Problems 21-30: Find x:

21. 23 •24 = 2x 26. 8 = 2x 22. 2

3

24= 2x 27. a3 • a = ax

23. 3−4 = 1

3x 28. b

10

b 5 = bx

24. 52

52= 5x 29. 1

c −4 = c x

25. 23( )4 = 2x 30. a

3y−2

a 2y−3 = ax

Problems 31-43: Find the value:

31. 7x0 = 38. 2x • 4x−1 =

32. 3−4 = 39. x

c+3

x c−3 =

33. 23 •24 = 40. 8x

2x−1 =

34. 05 = 41. 2x

−3

6x −4 =

35. 50 = 42. a

x+3( )x−3=

36. −3( )3 − 33 = 43. a3x−2

a 2x−3 = 37. x c+3 • x c−3 =

Problems 44-47: Write given two ways:

Given No negative

powers

No

fraction

44. d−4

d 4

45. 3x3

y( )−2

46. a2bc

2ab 2c( )3

47. x 2y 3z −1

x 5y −6z −3

C. Laws of rational exponents, and radicals:

Assume all radicals are real numbers:

I. If r is a positive integer, p is an integer, and

a ≥ 0, then apr = a pr = ar( )p which is a

real number. (Also true if r is a positive odd integer and a < 0)

Think of p

r as

power

root

II. abr = ar • br , or (ab)1r = a

1r •b

1r

III. ab

r = ar

br, or a

b( )1r = a1r

b1r

IV. ars = asr = ars

or a1rs = a

1s( )

1r = a

1r( )

1s

Problems 48-53: Write as a radical:

48. 312 = 51. x

32 =

49. 423 = 52. 2x

12 =

50. 12( )13 = 53. 2x( )1

2 =

Problems 54-57: Write as a fractional power:

54. 5 = 56. a3 = 55. 23 = 57. 1

a=

Problems 58-62: Find x:

58. 4 • 9 = x 61. 643 = x

59. x = 4

9 62. x = 8

23

432

60. 643 = x

Problems 63-64: Write with positive exponents:

63. 9x 6y−2( )12 = 64. −8a6b−12( )− 2

3 =

D. Simplification of radicals:

example: 32 = 16 • 2 = 4 2

example: 723 = 83 • 93 = 2 93

6

example: 543 + 4 163 = 273 • 23 + 4 83 • 23 = 3 23 + 4 •2 23 = 3 23 + 8 23 =11 23

example: 8 − 2 = 2 2 − 2 = 2

Problems 65-82: Simplify (assume all radicals

are real numbers):

65. − 81 = 73. x 2x + 2 2x3 + 2x 2

2x=

66. 50 = 74. a2 = 67. 3 12 = 75. a3 = 68. 543 = 76. a5

3 = 69. 52 = 77. 3 2 + 2 = 70. 2 3 + 27 − 75 = 78. 5 3 − 3 = 71. x5 = 79. 9x2 − 9y2 =

72. 4x 6 = 80. 9x2 + 9y 2 =

81. 9 x + y( )2 = 82. 64 x + y( )33 =

E. Rationalization of denominators:

example: 58

= 5

8= 5

8• 2

2= 10

16= 10

4

example: 1

23= 1

23• 43

43= 43

83= 43

2

example: 3

3−1= 3

3−1• 3+1

3+1= 9+ 3

3−1 = 3+ 32

Problems 83-91: Simplify:

83. 23

= 88. 2 + 1

2=

84. 1

5= 89. 3

2+1=

85. 3

3= 90. 3

1− 3=

86. a

b= 91.

3+23−2

=

87. 23

3 =

Answers:

1. 8

2. 9

3. –16

4. 16

5. 0

6. 1

7. 8

8. 4

9. 2

10. –7

11. –5

12. 5

13. 5

14. x if x ≥ 0 –x if x < 0

15. a

16. 12

17. 0.2

18. 1681

19. 3a2

20. 7

21. 7

22. –1

23. 4

24. 0

25. 12

26. 3

27. 4

28. 5

29. 4

30. y +1 31. 7

32. 181

33. 128

34. 0

35. 1

36. –54

37. x2c

38. 23x−2

39. x6

40. 22x+1

41. x

3

42. ax 2 −9

43. ax+1

44. 1

d 8= d−8

45. y 2

9x 6= 9−1x−6y 2

46. a 3

8b 3= 8−1

a3b

−3

47. y 9z 2

x 3= x−3y 9z 2

48. 3

49. 163

50. 12

3 = 42

3

51. x3 = x x

52. 2 x

53. 2x

54. 512

55. 232

56. a13

57. a− 1

2 58. 36

59. 4 9

60. 4

61. 2

62. 12

63. 3 x

y

3

64. b 8

4a 4

65. –9

66. 5 2

67. 6 3

68. 3 23

69. 2 13 70. 0

71. x 2 x

72. 2 x3

73. 4x 2x 74. a if a ≥ 0,

-a if a < 0

75. a a

76. a a23

77. 4 2

78. 4 3

79. 3 x 2 − y2

80. 3 x 2 + y 2

81. 3 x + y

82. 4 x + y( ) 83. 6

3

84. 55

85. 3

7

86. abb

87. 183

3

88. 3 22

89. 3 2 − 3 90. 3+3

−2

91. −7− 4 3

TOPIC 4: LINEAR EQUATIONS and INEQUALITIES

A. Solving one linear equation in one variable:

Add or subtract the same value on each side of

the equation, or multiply or divide each side by

the same value, with the goal of placing the

variable alone on one side. If there are one or

more fractions, it may be desirable to eliminate

them by multiplying both sides by the common

denominator. If the equation is a proportion,

you may wish to cross-multiply.

Problems 1-15: Solve:

1. 2x = 9 9. x − 4 = x2

+1 2. 3 = 6x

5 10. 3x

2x+1 = 52

3. 3x + 7 = 6 11. 6 − 4x = x

4. x3

= 54 12. 3x−2

2x+1 = 4 5. 5 − x = 9 13. x+3

2x−1 = 2 6. x = 2x

5+1 14. 7x − 5 = 2x +10

7. 4x −6 = x 15. 13

= xx+8

8. x−1x+1 = 6

7

To solve a linear equation for one variable in

terms of the other, do the same as above:

example: Solve for F: C = 59F − 32( )

Multiply by 95: 9

5C = F − 32

Add 32: 95C + 32 = F

Thus, F = 95C + 32

example: Solve for b: a + b = 90 Subtract a: b = 90 − a

Problems 16-21: Solve for the indicated variable

in terms of the other(s):

16. a + b =180; b = 19. y = 3x − 2; x = 17. 2a +2b =180; b = 20. y = 4 − x; x = 18. P = 2b + 2h; b = 21. y = 2

3x +1; x =

B. Solving a pair of linear equations in two

variables:

The solution consists of an ordered pair, an

infinite number of ordered pairs, or no solution.

Problems 22-28: Solve for the common solution(s)

by substitution or linear combinations:

22. x + 2y = 73x − y = 28 23.

x + y = 5x − y = −3

24. 2x − y = −9x = 8 27.

4x −1 = y4x + y =1

25. 2x − y =1y = x −5 28.

x + y = 3x + y =1

26. 2x − 3y = 53x + 5y =1 29.

2x − y = 36x −9 = 3y

C. Analytic geometry of one linear equation

in two variables:

The graph of y = mx + b is a line with slope m and

y-intercept b. To draw the graph, find one point on

it (such as (0, b)) and then use the slope to find

another point. Draw the line joining the two points.

example: y = −32x +5 has

slope −32 and y-intercept 5.

To graph the line,

locate (0, 5). From that point,

go down 3 (top of slope fraction),

and over (right) 2

(bottom of fraction)

to find a second point.

Draw the line joining the points.

Problems 30-34: Find slope and y-intercept, and

sketch the graph:

30. y = x + 4 33. x − y = −1 31. y = − 1

2x − 3 34. x = −3y + 2

32. 2y = 4x − 8

A vertical line has no slope, and its equation can

be written so it looks like x = k (where k is a number). A horizontal line has zero slope, and

its equation looks like y = k .

example: Graph on the same graph:

x + 3 = −1 and 1+ y = −3 . The first equation is x = −4

The second is y = −4

Problems 35-36: Graph and write an equation for…

35. The line through (-1, 4) and (-1, 2)

36. The horizontal line through (4, -1)

8 D. Analytic geometry of two linear equations

in two variables:

Two distinct lines in a plane are either parallel or

intersecting. They are parallel if and only if they

have the same slope, and hence the equations of the

lines have no common solutions. If the lines have

unequal slopes, they intersect in one point and their

equations have exactly one common solution.

(They are perpendicular if and only if their slopes

are negative reciprocals, or one is horizontal and

the other is vertical.) If one equation is a multiple of

the other, each equation has the same graph, and

every solution of one equation is a solution of the

other.

Problems 37-44: For each pair of equations in

problems 22 to 29, tell whether the lines are

parallel, perpendicular, intersecting but not

perpendicular, or the same line:

37. Problem 22 41. Problem 26

38. Problem 23 42. Problem 27

39. Problem 24 43. Problem 28

40. Problem 25 44. Problem 29

E. Solution of a one-variable equation

reducible to a linear equation:

Some equations which do not appear to be linear

can be solved by using a related linear equation:

example: 3− x = 2 Since the absolute value of both 2 and –2 is

2, 3-x can be either 2 or –2. Write these two

equations and solve each:

3− x = 2 3− x = −2 −x = −1 or −x = −5 x =1 x = 5

Problems 45-49: Solve:

45. x = 3 48. 2− 3x = 0 46. x = −1 49. x +2 =1 47. x −1 = 3

example: 2x −1 = 5

Square both sides: 2x −1 = 25 Solve:

2x = 26x =13

Be sure to check answer(s):

2x −1 = 2•13−1 = 25 = 5 (Check)

example: x = −3 Square: x = 9 Check: x = 9 = 3≠ −3

There is no solution, since 9 doesn’t satisfy the

original equation (it is false that 9 = −3).

Problems 50-52: Solve and check:

50. 3− x = 4 52. 3 = 3x − 2 51. 2x +1 = x − 3

F. Linear inequalities:

Rules for inequalities:

if a > b, then :

a + c > b + c

a − c > b − c

ac > bc (if c > 0)ac < bc (if c < 0)ac

> bc(if c > 0)

ac

< bc(if c < 0)

if a < b, then :

a + c < b + c

a − c < b − c

ac < bc (if c > 0)ac > bc (if c < 0)ac

< bc(if c > 0)

ac

> bc(if c < 0)

example: One variable graph: solve and graph

on a number line: 1− 2x ≤ 7 (This is an abbreviation for {x: 1− 2x ≤ 7 })

Subtract 1, get −2x ≤ 6 Divide by –2, x ≥ −3

Graph:

Problems 53-59: Solve and graph on a number line:

53. x − 3 > 4 57. 4 − 2x < 6 54. 4x < 2 58. 5 − x > x − 3 55. 2x +1≤ 6 59. x >1+ 4 56. 3 < x − 3

Answers:

1. 92

2. 52

3. − 13

4. 154

5. – 4

6. 53

7. 2

8. 13

9. 10

10. − 54

11. 6 5

12. − 65

13. 5 3

14. 3

15. 4

16. 180− a

17. 90− a

18. P 2 − h = p−2h2

19. y+2( )

3

20. 4 − y

21. 3 y−1( )

2

22. (9, -1)

23. (1, 4)

0 1 2 3-1-2-3-4

9 24. (8, 25)

25. (-4, -9)

26. 2819 , −13

19( ) 27. 1

4 , 0( ) 28. no solution

29. a, 2a − 3( ), where a is any number; infinite

number of solutions

30. m =1, b = 4

31. m = −12 , b = −3

32. m = 2, b = −4

33. m =1, b =1

34. m = −13 , b = 2

3

35. x = −1

36. y = −1

37. intersecting, not⊥ 38. ⊥ 39. intersecting, not⊥ 40. intersecting, not⊥ 41. intersecting, not⊥ 42. intersecting, not⊥ 43. parallel

44. same line

45. –3, 3

46. no solution

47. –2, 4

48. 2 3

49. –3, -1

50. –13

51. no solution in real numbers

52. 113

53. x > 7 0 7

54. x < 12 10

55. x ≤ 52

2 30

56. x > 6 0 6

57. x > −1 -1 0

58. x < 4 40

59. x > 5 0 5

TOPIC 5: QUADRATIC POLYNOMIALS, EQUATIONS, and INEQUALITIES

A. Multiplying polynomials:

example: x + 2( ) x + 3( )= x2 + 5x + 6 example: 2x −1( ) x + 2( )= 2x2 + 3x − 2 example: x − 5( ) x + 5( )= x 2 − 25 example: −4 x − 3( )= −4x +12

example: x + 2( ) x 2 − 2x + 4( )= x3 + 8

example: 3x − 4( )2 = 9x 2 −24x +16 example: x + 3( ) a− 5( )= ax − 5x + 3a−15

Problems 1-10: Multiply:

1. x + 3( )2 = 6. −6x 3− x( )= 2. x − 3( )2 = 7. 2x −1( ) 4x 2 +2x +1( )=

3. x + 3( ) x − 3( )= 8. x − 12( )2 =

4. 2x + 3( ) 2x − 3( )= 9. x −1( ) x + 3( ) = 5. x − 4( ) x − 2( )= 10. x 2 −1( ) x2 + 3( )=

B. Factoring:

Monomial factors: ab+ ac = a b + c( ) example: x 2 − x = x x −1( ) example: 4x 2y +6xy = 2xy 2x + 3( )

Difference of two squares: a2 −b2 = a + b( ) a − b( )

example: 9x2 − 4 = 3x + 2( ) 3x −2( )

Trinomial square: a2 +2ab+ b2 = a + b( )2 a2 −2ab+ b

2 = a − b( )2

example: x2 − 6x + 9 = x − 3( )2

Trinomial:

example: x2 − x − 2 = x − 2( ) x +1( ) example: 6x2 − 7x − 3 = 3x +1( ) 2x − 3( )

Sum and difference of two cubes:

a3 + b

3 = a + b( ) a2 − ab+ b

2( ) a3 − b

3 = a − b( ) a2 + ab+ b

2( ) example: x3 − 64 = x3 − 43 = x − 4( ) x2 + 4x +16( )

Problems 11-27: Factor:

11. a2 + ab = 20. x2 − 3x −10 = 12. a3 − a2b + ab2 = 21. 2x 2 − x = 13. 8x2 − 2 = 22. 8x3 +8x2 +2x = 14. x2 −10x + 25 = 23. 9x 2 +12x + 4 = 15. −4xy +10x2 = 24. 6x3y2 − 9x 4y = 16. 2x2 − 3x − 5 = 25. 1− x − 2x2 = 17. x2 − x − 6 = 26. 3x2 −10x + 3 = 18. x2y − y 2x = 27. x4 + 3x2 − 4 = 19. 8x3 +1=

C. Solving quadratic equations by factoring:

If ab = 0 , then a = 0 or b = 0 .

example: if 3− x( ) x + 2( ) = 0 then 3 − x( ) = 0 or x + 2( )= 0 and thus, x = 3 or x = −2

Note: there must be a zero on one side of the

equation to solve by the factoring method.

4

-3

-4

1

2

3

-1

-1

10

example: 6x2 = 3x

Rewrite: 6x2 − 3x = 0

Factor: 3x 2x −1( ) = 0 So 3x = 0 or 2x −1( ) = 0 Thus x = 0 or x = 1

2

Problems 28-39: Solve by factoring:

28. x x − 3( ) = 0 34. x + 2( ) x − 3( ) = 0 29. x

2 − 2x = 0 35. 2x +1( ) 3x −2( ) = 0 30. 2x

2 = x 36. 6x2 = x + 2

31. 3x x + 4( )= 0 37. 9 + x2 = 6x

32. x2 = 2 − x 38. 1− x = 2x2

33. x2 + x = 6 39. x

2 − x −6 = 0

D. Completing the square:

x2 + bx will be the square of a binomial when c

is added, if c is found as follows: find half the x

coefficient and square it–this is c.

Thus, c = b2( )2 = b 2

4, and

x2 + bx + c = x2 + bx + b 2

4= x + b

2( )2

example: x2 + 5x

Half of 5 is 5 2 , and 52( )2 = 25

4 , which must

be added to complete the square:

x2 + 5x + 254

= x + 52( )2

If the coefficient of x2 is not 1, factor it so that it is.

example: 3x2 − x = 3(x2 − 13x)

Half of − 13 is− 1

6 , and − 16( )2 = 1

36 , so

x 2 − 13x + 1

36( )= x − 16( )2 , and

3 x 2 − 13x + 1

36( )= 3x2 − x + 336

Thus 336 or 1

12( ) must be added to 3x2 − x to

complete the square.

Problems 40-43: Complete the square, and tell

what must be added:

40. x2 −10x 42. x2 − 3

2x

41. x2 + x 43. 2x

2 + 8x

E. The quadratic formula:

If a quadratic equation looks like ax2 + bx + c = 0 ,

then the roots (solutions) can be found by using the

quadratic formula: x = −b± b 2−4ac2a

Problems 44-49: Solve:

44. x2 − x −6 = 0 47. x

2 − 3x − 4 = 0 45. x

2 + 2x = −1 48. x2 + x − 5 = 0

46. 2x2 − x − 2 = 0 49. x

2 + x =1

F. Quadratic inequalities:

example: Solve x2 − x < 6 . First make one side

zero: x2 − x −6 < 0

Factor: x − 3( ) x + 2( ) < 0 . If x − 3( )= 0 or x + 2( ) = 0 then x = 3 or x = −2 .

These two numbers (3 and –2) split the real numbers

into three sets (visualize the number line):

x x − 3( ) x + 2( ) x − 3( )× x + 2( ) Solution ?

x < −2 −2 < x < 3 x > 3

neg

neg

pos

neg

pos

pos

pos

neg

pos

no

yes

no

Therefore, if x − 3( ) x + 2( )< 0 , then −2 < x < 3 Note that this solution means that x > −2 and x < 3

Problems 50-54: Solve, and graph on a number line:

50. x2 − x −6 > 0 53. x > x

2

51. x2 + 2x < 0 54. 2x

2 + x −1> 0 52. x

2 − 2x < −1

G. Complex numbers:

−1 is defined to be i, so i2 = −1

example: i 3 = i 2 • i = −1• i = −i

55. Find the value of i4.

A complex number is of the form a + bi , where a and b are real numbers. a is called the real part and b

is the imaginary part. If b is zero, a + bi is a real number. Ifa = 0 , then a + bi is pure imaginary.

Complex number operations:

example: 3 + i( )+ 2 − 3i( )= 5 − 2i example: 3 + i( )− 2 − 3i( )= 1 + 4i example: 3 + i( ) 2 − 3i( ) = 6 − 7i− 3i2

= 6 − 7i + 3 = 9 − 7i example: 3+i

2−3i = 3+i2−3i • 2+3i

2+3i = 6+11i−34+9 = 3+11i

13= 313

+ 1113i

Problems 56-65: Write each of the following so

the answer is a+bi:

56. 3+ 2i( ) 3− 2i( )= 61. i6 = 57. 3+2i( )+ 3−2i( )= 62. i

7 = 58. 3+2i( )− 3−2i( )= 63. i

8 = 59. 3+2i( )÷ 3−2i( )= 64. i

1991 = 60. i

5 = 65. 1i

=

Problems 66-67: Solve and write the answer as a+bi:

66. x2 + 2x + 5 = 0 67. x

2 + x + 2 = 0

11 Answers:

1. x 2 +6x +9 2. x

2 −6x +9 3. x 2 −9 4. 4x 2 −9 5. x

2 −6x +8 6. −18x +6x 2 7. 8x 3 −1 8. x 2 − x + 1

4

9. x2 +2x − 3

10. x4 +2x 2 − 3

11. a a + b( ) 12. a a

2 − ab+ b2( )

13. 2 2x +1( ) 2x −1( ) 14. x −5( )2 15. 2x −2y +5x( ) 16. 2x −5( ) x +1( ) 17. x − 3( ) x +2( ) 18. xy(x − y)

19. 2x +1( ) 4x 2 −2x +1( ) 20. x −5( ) x +2( ) 21. x 2x −1( ) 22. 2x 2x +1( )2 23. 3x +2( )2

24. 3x 3y 2y − 3x( ) 25. 1−2x( ) 1+ x( ) 26. 3x −1( ) x − 3( ) 27. x

2 + 4( ) x +1( ) x −1( ) 28. 0, 3

29. 0, 2

30. 0, 12

31. –4, 0

32. –2, 1

33. –3, 2

34. –2, 3 35. − 1

2 , 23

36. − 12 ,

23

37. 3

38. –1, 12

39. –2, 3

40. x −5( )2, add 25 41. x + 1

2( )2 , add 14 42. x − 3

4( )2, add 916 43. 2 x +2( )2 , add 8 44. –2, 3

45. –1

46. 1± 17

4

47. –1, 4

48. −1± 21

2

49. −1± 5

2

50. x < −2 or x > 3 51. −2 < x < 0 52. no solution, no graph

53. 0 < x <1 54. x < −1 or x > 1

2

55. 1

56. 13

57. 6

58. 4i

59. 513

+ 1213i

60. i

61. –1

62. –i

63. 1

64. –i

65. –i

66. −1±2i 67. − 1

2± 7

2i

TOPIC 6: GRAPHING and the COORDINATE PLANE

A. Graphing points:

1. Join the following points in the given order:

(-3, -2), (1, -4), (3, 0), (2, 3), (-1, 2), (3, 0),

(-3, -2), (-1, 2), (1, -4)

2. In what quadrant does the point a, b( ) lie, if a > 0 and b < 0?

Problems 3-6: For each given point, which of its

coordinates, x or y, is larger?

B. Distance between points: The distance between the points P1 x1, y1( ) and P2 x2, y2( ) is found by using the Pythagorean Theorem, which gives

P1P2 = x2 − x1( )2 + y2 − y1( )2 .

example: A(3, -1), B(-2, 4)

AB = 4 − −1( )( )2 + −2 − 3( )2 =

52 + −5( )2 = 50 = 25 2 = 5 2

Problems 7-10: Find the length of the segment

joining the given points:

7. (4, 0), (0, -3) 9. (2, -4), (0, 1)

8. (-1, 2), (-1, 5) 10. − 3, − 5( ), 3 3, − 6( ) C. Linear equations in two variables, slope,

intercepts, and graphing: The line joining the points P1 x1, y1( ) and P2 x2, y2( ) has slope y2−y1

x2−x1.

example: A(3,-1), B(-2,4) slope of

AB____

= 4− −1( )−2−3 = 5

−5 = −1

Problems 11-15: Find the slope of the line

joining the given points:

11. (-3, 1), and (-1, -4) 13. (3, -1), and (5, -1)

12. (0, 2), and (-3, 5)

6

3

4

5

-2 30

-2 0

0 1

12 14. 15.

To find the x-intercept (x-axis crossing) of an

equation, let y be zero and solve for x. For the y-

intercept, let x be zero and solve for y.

example: 3y − 4x = 12 if x = 0 , y = 4 so

y-intercept is 4. If y = 0 , x = −3 so x-intercept is –3.

The graph of y = mx + b is a line with slope m and

y-intercept b. To draw the graph, find one point on

it (such as (0, b)) and then use the slope to find

another point. Draw the line joining the two.

example: y = −32x + 5 has slope − 3

2 and y-

intercept 5. To graph the line, locate (0, 5).

From that point, go down 3 (top of slope

fraction), and over (right) 2 (bottom of

fraction) to find a second point. Join.

Problems 16-20: Find the slope and y-intercept,

and sketch the graph:

16. y = x + 4 19. x − y = −1 17. y = − 1

2x − 3 20. x = −3y + 2

18. 2y = 4x − 8

To find an equation of a non-vertical line, it is

necessary to know its slope and one of its points.

Write the slope of the line through x, y( ) and the known point, then write an equation which

says that this slope equals the known slope.

example: Find an equation of the line through

(-4, 1) and (-2, 0).

Slope = 1−0−4+2 = 1−2

Using (-2, 0) and x, y( ), Slope =

y−0x+2 = 1

−2 ; cross multiply, get

−2y = x + 2, or y = − 12x −1

Problems 21-25: Find an equation of the line:

21. Through (-3, 1) and (-1, -4)

22. Through (0, -2) and (-3, -5)

23. Through (3, -1) and (5, -1)

24. Through (8, 0), with slope –1

25. Through (0, -5), with slope 2 3

A vertical line has no slope, and its equation can

be written so it looks like x = k (where k is a number). A horizontal line has zero slope, and

its equation looks like y = k .

example: Graph on the same graph:

x + 3 = −1 and 1+ y = −3. The first equation is x = −4 ; the second is y = −4 .

Problems 26-27: Graph and write equation for:

26. The line through (-1, 4) and (-1, 2)

27. Horizontal line through (4, -1)

D. Linear inequalities in two variables:

example: Two variable graph: graph solution on

a number plane: x − y > 3 (This is an abbreviation for { x, y( ): x − y > 3}. Subtract x, multiply by –1, get y < x − 3 . Graph y = x − 3 , but draw a dotted

line, and shade the

side where y < x − 3 :

28. y < 3 31. x < y +1 29. y > x 32. x + y < 3 30. y ≥ 2

3x + 2 33. 2x − y >1

E. Graphing quadratic equations:

The graph of y = ax2 + bx + c is a parabola, opening

upward (if a > 0) or downward (if a < 0), and with line of symmetry. x = −b

2a, also called axis of

symmetry. To find the vertex V(h, k) of the parabola,

h = −b2a (since V is on the axis of symmetry), and k is

the value of y when h is substituted for x.

example: y = x2 −6x

a =1, b = −6, c = 0

Axis:

x = −b2a

= 62

= 3h = 3, k = 32 −18 = −9

Thus, vertex is (3, − 9)

13

example: y = 3 − x2 V(0, 3),

Axis: x = 0 , y-intercept: if x = 0 , y = 3 −02 = 3 x-intercept: if y = 0 , 0 = 3− x

2,

so 3 = x2, and x = ± 3

Problems 34-40: Sketch the graph:

34. y = x2 38. y = x +1( )2

35. y = −x2 39. y = x − 2( )2 −1 36. y = x

2 +1 40. y = x + 2( ) x −1( ) 37. y = x

2 − 3

Answers:

1.

2. IV

3. x 4. y

5. y

6. x 7. 5

8. 3

9. 29 10. 7

11. − 52

12. -1

13. 0

14. − 35

15. 3 4

16. m =1, b = 4

17. m = − 12 , b = −3

18. m = 2, b = −4

19. m =1, b =1

20. m = − 13 , b = 2

3

21. y = − 52 x −13

2

22. y = x − 2 23. y = −1 24. y = −x + 8 25. y = 2

3 x −5 26. x = −1

27. y = −1

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

TOPIC 7: LOGARITHMS and FUNCTIONS

A. Functions:

The area A of a square depends on its side

length s, and we say A is a function of s, and

write ‘A= f s( )’; for short, we read this as ‘A= f of s.’ There are many functions of s.

The one here is s2. We write this f s( ) = s

2 and

can translate: ‘the function of s we’re talking

about is s2’. Sometimes we write A s( ) = s

2.

This says the area A is a function of s, and

specifically, it is s2.

B. Function values and substitution:

If A s( ) = s2, A(3), read ‘A of 3’, means replace

every s in A s( ) = s2with 3, and find A when s is 3.

When we do this, we find A 3( )= 32 = 9 .

example: g x( ) is given: y = g x( )= πx2

4

-4

1

-1

-1

1

-1

(2,-1)

-2 1

-3

2

3

14

example: g(3) = π • 32 = 9π example: g 7( )= π • 72 = 49π example: g a( )= πa2 example: g x + h( )= π x + h( )2 = πx 2 +2πxh + πh 2

1. Given y = f x( )= 3x − 2 . Complete these ordered pairs: (3,___), (0,___),

( 12 ,___), (___, 10), (___,-1) ( x −1,___)

Problems 2-10: Given f x( ) = x2 − 4x + 2 . Find:

2. f 0( )=3. f 1( )=4. f −1( )=5. f −x( )=6. − f x( )=

7. f x( )− 2 =8. f x − 2( )=9. 2 f x( )=10. f 2x( )=

Problems 11-15: Given f x( )= xx+1 . Find:

11. f 1( )=12. f −2( )=13. f 0( )=

14. f −1( )=15. f x −1( )=

example: If k x( ) = x2 − 4x , for what x is

k x( ) = 0? If k x( ) = 0 , then x 2 − 4x = 0 and since x2 − 4x = x x − 4( ) = 0 , x can be either 0 or 4.

(These values of x: 0 and 4, are called ‘zeros of the

function’, because each makes the function zero.)

Problems 16-19: Find all real zeros of:

16. x x +1( )17. 2x2 − x − 3

18. x 2 −16x + 6419. x 2 + x + 2

Problems 20-23: Given f x( ) = x2 − 4x + 2 ,

find real x so that:

20. f x( )= −221. f x( )= 2

22. f x( )= −323. x is a zero of f x( )

Since y = f x( ), the values of y are the values of the function which correspond to specific values of x. The heights of the graph above (or below) the x-axis are the values of y and so also of the function. Thus for this graph, f 3( ) is the height (value) of the function at x = 3 and the value is 2: At x = −3 , the value (height) of f x( ) is zero; in other words, f −3( ) = 0 . Note that f 3( ) > f −3( ) , since 2 > 0 , and that f 0( ) < f −1( ), since f −1( )=1 and f 0( ) <1.

Problems 24-28: For this

graph, tell whether the

statement is true or false:

24. g −1( )= g 0( )25. g 0( )= g 3( )26. g 1( )> g −1( )

27. g −2( )> g 1( )28. g 2( )< g 0( )< g 4( )

C. Logarithms and exponents:

Exponential form: 23 = 8

Logarithmic form: log28 = 3

Both of the equations above say the same thing. ‘ log28 = 3’ is read ‘log base two of eight equals three’ and translates ‘the power of 2 which gives 8 is 3’.

Problems 29-32: Write the following information

in both exponential and logarithmic forms:

29. The power of 3 which gives 9 is 2.

30. The power of x which gives x3 is 3

31. 10 to the power –2 is 1100

.

32. 12 is the power of 169 which gives 13.

Problems 33-38: Write in logarithmic form:

33. 43 = 64 36. 1

10=10−1

34. 30 =1 37. a

b = c

35. 25 = 52 38. y = 3x

Problems 39-44: Write in exponential form:

39. log39 = 240. log31= 041. 5 = log2 32

42. 1= log4 443. y = loga x44. logb a = 2

Problems 45-50: Find the value:

45. 210 = 48. 610

310=

46. log4 410 = 49. log49 7 =

47. log6 6 = 50. log7 49 =

D. Logarithm and exponent rules:

exponent rules: log rules:

all quantities real) (base any positive real

number except 1)

ab • ac = ab+c log ab = log a + log b

ab

a c = ab−c log ab

= log a− log b

ab( )c = abc log ab = b log a

ab( )c = acbc loga ab = b

ab( )c = a c

b c aloga b( ) = b

a0 =1 (if a ≠ 0) loga b = logc b

logc a

a−b = 1

a b

ap

r = a pr = ar( )p (think of

p

r as

power

root) (base change rule)

15 Problems 51-52: Given log21024 =10 , find:

51. log210245 = 52. log2 1024 =

Problems 53-63: Solve for x in terms of y and z:

53. 3x = 3y • 3z 55. x 3 = y

54. 9y = 3z

3x 56. 3x = y

57. log x 2 = 3log y 58. log x = 2log y − log z 59. 3log x = log y 60. log x = log y + log z 61. log x + log y3 = log z 2 62. log7 3 = y; log72 = z; x = log32 63. y = loga 9; x = loga 3

E. Logarithmic and exponential equations:

Problems 64-93: Use the exponent and log rules

to find the value of x:

64. 62x = 63 66. 4 x = 8 65. 22x = 23 67. 9x = 27x−1

68. log3 x = log36 73. 52( )3 = 5x 69. log3 4x = log36 74. 5x+1 =1 70. 43 • 45 = 4 x

75. log3 37 = x

71. 3−2 = x 76. 6(log6 x ) = 8 72. 3

x

3= 30

77. log10 x = log10 4 + log10 2 78. log32x = log38 + log3 4 − 4 log32 79. logx 25 = 2 80. 3loga 4 = loga x 81. log 2x − 6( )= log 6 − x( )

82. logx 3

logx 4= log4 3 88. log4 64 = x

83. log3 27•3−4( )= x 89. 54 = 5x

84. log3 81

log3 27− log3 8127 = x 90. 27x = 1

9( )3 85. log4 303 = x log4 30 91. 4

10 = 2x 86. log2

132

= x 92. 2x = 3

87. log16 x = 32 93. 3•2x = 4

Answers:

1. 7, -2, − 12 , 4,

13 , 3x − 5

2. 2

3. –1

4. 7

5. x2 + 4x + 2

6. −x 2 + 4x − 2 7. x

2 − 4x 8. x

2 − 8x +14 9. 2x

2 −8x + 4 10. 4x

2 −8x + 2 11. 12

12. 2

13. 0

14. no value

15. x−1x

16. –1, 0

17. –1, 3 2

18. 8

19. none

20. 2

21. 0, 4

22. none

23. 2± 2 24. F

25. T

26. T

27. T

28. T

29. 32 = 9, log3 9 = 2

30. x 3 = x 3 , logx x3 = 3

31. 10−2 = 1100 , log10

1100 = −2

32. 1691/2

=13, log16913= 12

33. log4 64 = 3 34. log31 = 0 35. log525 = 2 36. log10

110 = −1

37. loga c = b

38. log3 y = x

39. 32 = 9

40. 30 =1

41. 25 = 32

42. 41 = 4

43. ay = x

44. b2 = a

45. 1024

46. 10

47. 1

48. 210 =1024

49. 12

50. 2

51. 50

52. 5

53. y + z

54. z − 2y

55. y13 = y3

56. log3 y

57. y32 = y y

58. y2

z

59. y3

60. yz

61. z 4

y23

62. zy

63. y2

64. 3 2

65. 3 2

66. 3 2

67. 3

68. 6

69. 3 2

70. 8

71. 19

72. 1

73. 6

74. –1

75. 7

76. 8

77. 8

78. 1

79. 5

16 80. 64

81. 4

82. any real number > 0

and ≠1 83. –1

84. 13

85. 13

86. –5

87. 64

88. 3

89. 14

90. –2

91. 20

92. log3

log2

93. log4− log 3

log 2 (any base; if

base=2, x = 2− log2 3)

TOPIC 8: WORD PROBLEMS

1. 2 3 of 16 of

34 of a number is 12. What is

the number?

2. On the number line, points P and Q and 2

units apart. Q has coordinate x. What are the

possible coordinates of P?

3. What is the number, which when multiplied

by 32, gives 32•46? 4. If you square a certain number, you get 9

2.

What is the number?

5. What is the power of 36 that gives 3612 ?

6. Point X is on each of two given intersecting

lines. How many such points X are there?

7. Point Y is on each of two given circles. How

many such points Y?

8. Point Z is on each of a given circle and a

given ellipse. How many such Z?

9. Point R is on the coordinate plane so its

distance from a given point A is less than 4.

Show in a sketch where R could be.

Problems 10-11:

10. If the length of chord AB is x and length of

CB is 16, what is AC?

11. If AC= y and CB= z , how long is AB (in

terms of y and z)?

12. This square is cut into two smaller squares

and two non-square rectangles as shown.

Before being cut, the large square had area

a + b( )2 . The two smaller squares have areas a2 and b

2. Find the total area of the two non-

square rectangles. Do the areas of the 4 parts

add up to the area of the original square?

13. Find x and y:

14. When constructing an equilateral triangle

with an area that is 100 times the area of a

given equilateral triangle, what length should

be used for a side?

Problems 15-16: x and y are numbers, and two

x’s equal three y’s.

15. Which of x or y must be larger?

16. What is the ratio of x to y?

Problems 17-21: A plane has a certain speed in

still air. In still air, it goes 1350 miles in 3 hours:

17. What is its (still air) speed?

18. How long does it take to fly 2000 miles?

19. How far does the plane go in x hours?

20. If the plane flies against a 50 mph

headwind, what is its ground speed?

21. If it has fuel for 7.5 hours of flying time,

how far can it go against this headwind?

Problems 22-32: Georgie and Porgie bake pies.

Georgie can complete 30 pies an hour:

22. How many can he make in one minute?

23. How many can he make in 10 minutes?

24. How many can he make in x minutes?

25. How long does he take to make 200 pies?

Problems 26-28: Porgie can finish 45 pies an hour:

26. How many can she make in one minute?

27. How many can she make in 20 minutes?

28. How many can she make in x minutes?

Problems 29-32: If they work together, how

many pies can they produce in:

29. 1 minute 31. 80 minutes

30. x minutes 32. 3 hours

Problems 33-41: A nurse needs to mix some

alcohol solutions, given as a percent by weight

of alcohol in water. Thus in a 3% solution, 3%

of the weight would be alcohol. She mixes x

grams of 3% solution, y grams of 10% solution,

and 10 grams of pure water to get a total of 140

grams of a solution which is 8% alcohol:

A B

C O

a b

a

b

17 33. In terms of x, how many grams of alcohol

are in the 3% solution?

34. The y grams of 10% solution would include

how many grams of alcohol?

35. How many grams of solution are in the final

mix (the 8% solution)?

36. Write an expression in terms of x and y for

the total number of grams in the 8% solution

contributed by the three ingredients (the 3%,

10%, and water).

37. Use your last two answers to write a ‘total

grams equation’.

38. How many grams of alcohol are in the 8%?

39. Write an expression in terms of x and y for

the total number of grams of alcohol in the

final solution.

40. Use the last two answers to write a ‘total

grams of alcohol equation’.

41. How many grams of each solution are needed?

42. Half the square of a number is 18. What is

the number?

43. If the square of twice a number is 81, what

is the number?

44. Given a positive number x. The square of a

positive number y is at least 4 times x. How

small can y be?

45. Twice the square of half of a number is x.

What is the number?

Problems 46-48: Half of x is the same as one-

third of y:

46. Which of x and y is the larger?

47. Write the ratio x : y as the ratio of two

integers.

48. How many x’s equal 30 y’s?

Problems 49-50: A gathering has twice as many

women as men. If W is the number of women

and M is the number of men:

49. Which is correct: 2M=W or M=2W?

50. If there are 12 women, how many men are

there?

Problems 51-53: If A is increased by 25%, it

equals B:

51. Which is larger, B or the original A?

52. B is what percent of A?

53. A is what percent of B?

Problems 54-56: If C is decreased by 40%, it

equals D:

54. Which is larger, D or the original C?

55. C is what percent of D?

56. D is what percent of C?

Problems 57-58: The length of a rectangle is

increased by 25% and its width is decreased by 40%:

57. Its new area is what percent of its old area?

58. By what percent has the old area increased

or decreased?

Problems 59-61: Your wage is increased by

20%, then the new amount is cut by 20% (of the

new amount):

59. Will this result in a wage which is higher

than, lower than, or the same as the original

wage?

60. What percent of the original wage is this

final wage?

61. If the above steps were reversed (20% cut

followed by 20% increase), the final wage

would be what percent of the original wage?

62. Find 3% of 36.

63. 55 is what percent of 88?

64. What percent of 55 is 88?

65. 45 is 3% of what number?

66. The 3200 people who vote in an election are

40% of the people registered to vote. How

many are registered?

67. If you get 36 on a 40-question test, what

percent is this?

68. What is the average of 87, 36, 48, 59, and 95?

69. If two test scores are 85 and 60, what

minimum score on the next test would be

needed for an overall average of 80?

70. The average height of 49 people is 68

inches. What is the new average height if a

78-inch person joins the group?

Problems 71-72: s varies directly as P, and

P = 56 when s = 14 :

71. Find s when P = 144 . 72. Find P when s = 144 .

Problems 73-74: A is proportional to r2, and

when r = 10 , A = 400π .

73. Find A when r = 15 . 74. Find r when A= 36π

75. If b is inversely proportional to h, and

b = 36 when h = 12 , find h when b = 3 . 76. If 3x = 4y , write the ratio x : y as the ratio

of two integers.

18 77. The length of a rectangle is twice the width.

If both dimensions are increased by 2 cm, the

resulting rectangle has area 84 cm2. What

was the original width?

78. After a rectangular piece of knitted fabric

shrinks in length 1 cm and stretches in width 2

cm, it is a square. If the original area was

40 cm2, what is the square area?

Answers:

1. 144

2. x +2, x −2 3. 46

4. 9

5. 12

6. 1

7. 0,1, or 2

8. 0, 1, 2, 3, or 4

9. Inside circle of radius

4 centered on A

10. x −16 11. y + z

12. 2ab, yes:

a + b( )2 =a2 + 2ab + b

2

13. x = 407 ; y = 16

3

14. 10 times the original side

15. x 16. 3 2

17. 450 mph

18. 4 49 hours

19. 450x miles

20. 400 mph

21. 3000 miles

22. 12

23. 5

24. x2

25. 400 min.

26. 3 4

27. 15

28. 3x 4

29. 5 4

30. 5x 4

31. 100

32. 225

33. .03x

34. .1y

35. 140

36. x + y +10 37. x + y +10 =140 38. 11.2

39. .03x + .1y 40. .03x + .1y =11.2 41. x = 180

7 ; y = 7307

42. 6, -6

43. 4.5, -4.5

44. 2 x

45. 2x 46. y

47. 3:2 or 3 2

48. 45

49. 2M=W

50. 6

51. B 52. 125%

53. 80%

54. C 55. 166 2

3%

56. 60%

57. 75%

58. 25% decrease

59. lower

60. 96%

61. same (96%)

62. 1.08

63. 62.5%

64. 160%

65. 1500

66. 8000

67. 90%

68. 65

69. 95

70. 68.2 inches

71. 36

72. 576

73. 900π 74. 3

75. 144

76. 4 : 3

77. 5 cm

78. 49

1 PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE

Directions: Study the samples, work the problems, then check your answers at the end of each topic. If

you don’t get the answer given, check your work and look for mistakes. If you have trouble, ask a math

teach or someone else who understands the topic.

TOPIC 1: ELEMENTARY OPERATIONS with NUMERICAL and ALGEBRAIC FRACTIONS

A. Simplifying fractions (by reducing):

example: 2736

= 9•39•4 = 9

9• 3

4=1• 3

4= 3

4

(note that you must be able to find a common

factor–in this case 9–in both the top and

bottom in order to reduce a fraction).

example: 3a12ab

= 3a•13a•4b = 3a

3a• 1

4b=1• 1

4b= 1

4b

(common factor: 3a)

Problems 1-8: Reduce:

1. 1352

= 5. 14x+7y

7y=

2. 2665

= 6. 5a+b5a+c =

3. 3+63+9 = 7. x−3

3−x = 4.

6axy

15by= 8.

4(x+2)(x−3)(x−3)(x−2) =

B. Equivalent fractions (equivalent ratios):

example: 34 is equivalent to how many eighths?

34

=8, 3

4=1• 3

4= 2

2• 3

4= 2•3

2•4 = 68

example: 65a

=5ab

, 65a

= bb

• 65a

= 6b5ab

example: 3x+2x+1 =

4 x+1( ) , 3x+2x+1 = 4

4• 3x+2

x+1 = 12x+84x+4

example: x−1x+1 =

x+1( ) x−2( ) ,

x−1x+1 = x−2( ) x−1( )

x−2( ) x+1( ) = x 2−3x+2x+1( ) x−2( )

Problems 9-13: Complete:

9. 49

=72 12. 30−15a

15−15b =1+b( ) 1−b( )

10. 3x7

=7y 13. x−6

6−x = −2

11. x+3x+2 =

x−1( ) x+2( )

C. Finding the lowest common denominator:

(LCD) by finding the least common multiple

(LCM) of all denominators:

example: 56 and 8

15. First find LCM of 6 and 15:

6 = 2• 315 = 3•5

LCM= 2• 3•5 = 30,

so 56

= 2530, and 8

15= 16

30

example: 34 and 1

6a:

4 = 2•2 6a = 2• 3• a

LCM= 2•2• 3• a =12a, so 3

4= 9a

12a, and 1

6a= 2

12a

example: 23 x+2( ) and

ax6 x+1( ) :

3 x + 2( )= 3• x + 2( )6 x +1( )= 2• 3• x +1( )

LCM= 2• 3• x +1( )• x + 2( ), so 2

3 x+2( ) = 2•2 x+1( )2•3 x+1( ) x+2( ) = 4 x+1( )

6 x+1( ) x+2( ) ,

and ax6 x+1( ) = ax x+2( )

6 x+1( ) x+2( )

Problems 14-18: Find equivalent fractions with

the lowest common denominator:

14. 23 and 2

9 17. 3

x−2 and 4

2−x

15. 3x and 5 18. x

15 x2−2( ) and

7x y−1( )10 x−1( )

16. x3 and −4

x+1

D. Adding and subtracting fractions:

If denominators are the same, combine the

numerators:

example: 3xy

− xy

= 3x−xy

= 2xy

If denominators are different, find equivalent

fractions with common denominators:

example: a2

− a4

= 2a4

− a4

= 2a−a4

= a4

example: 3x−1 + 1

x+2 = 3 x+2( )x−1( ) x+2( ) + x−1( )

x−1( ) x+2( )

= 3x+6+x−1x−1( ) x+2( ) = 4x+5

x−1( ) x+2( )

Problems 19-26: Find the sum or difference as

indicated (reduce if possible):

19. 47

+ 27

= 23. 3a

− 12a

= 20. 3

x−3 − xx−3 = 24. 3x−2

x−2 − 2x+2 =

21. b−ab+a − a−b

b+a = 25. 2x−1x+1 − 2x−1

x−2 =

22. xx−1 + x

1−x =

26. 1x−1( ) x−2( ) + 1

x−2( ) x−3( ) − 2x−3( ) x−1( ) =

2 E. Multiplying fractions:

Multiply the top numbers, multiply the bottom

numbers, reduce if possible:

example: 34

• 25

= 620

= 310

example: 3 x+1( )x−2 • x 2−4

x 2−1= 3 x+1( ) x+2( ) x−2( )

x−2( ) x+1( ) x−1( ) = 3x+6x−1

27. 27a

• ab12

= 29. a+b( )3

x−y( )2• x−y( )

5− p( ) • 5− p( )2

a+b( )2=

28. 3 x+4( )5y

• 5y 3

x 2−16=

F. Dividing fractions:

Make a compound fraction and then multiply the

top and bottom (of the larger fraction) by the

LCD of both:

example: ab

÷ cd

=ab

cd

=ab

•bdcd

•bd= ad

bc

example: 7

23

− 12

= 7•623

− 12( )•6

= 424−3 = 42

1= 42

example: 5x2y

÷ 2x =5x2y

2x=

5x2y

•2y2x •2y

= 5x4xy

= 54y

30. 3a

÷ b3

= 32. a − 43a

− 2=

31.

x+7x 2−91x−3

= 33. 2a− b

12

=

Answers:

1. 14

2. 2 5

3. 3 4

4. 2ax5b

5. 2x+yy

6. 5a+b5a+c (can’t reduce)

7. −1 8.

4 x+2( )x−2

9. 32

10. 3xy

11. x −1( ) x + 3( ) 12. 1+ b( ) 2− a( )

13. 2

14. 69, 2

9

15. 3x, 5x

x

16. x x+1( )3 x+1( ) ,

−123 x+1( )

17. 3x−2 ,

−4x−2

18. 2x x−1( )

30 x 2−2( ) x−1( ),

21x x 2−2( ) y−1( )30 x 2−2( ) x−1( )

19. 6 7

20. −1 21. 2b−2a

b+a

22. 0

23. 5 2a

24. 3x2 +2xx 2−4

25. −3 2x−1( )x+1( ) x−2( )

26. 0

27. b 42

28. 3y 2

x−4

29. a+b( ) 5− p( )

x−y

30. 9ab

31. x+7x+3

32. a2−4a3−2a

33. 4a − 2b

TOPIC 2: OPERATIONS with EXPONENTS and RADICALS

A. Definitions of powers and roots:

Problems 1-20: Find the value:

1. 23 = 11. −1253 =

2. 32 = 12. 52 =

3. −42 = 13. −5( )2 =

4. −4( )2 = 14. x2 = 5. 0

4 = 15. a33 =

6. 14 = 16. 1

4=

7. 64 = 17. .04 = 8. 643 = 18. 2

3( )4 =

9. 646 = 19. 81a84 =

10. − 49 = 20. 7 • 7 =

B. Laws of integer exponents:

I. ab • ac = ab+c

II. ab

a c = ab−c

III. ab( )c = a

bc

IV. ab( )c = ac •bc

V. ab( )c = a

c

b c

VI. a0 =1 (if a≠ 0)

VII. a−b = 1

a b

Problems 21-30: Find x:

21. 23 •24 = 2x 24. 52

52 = 5x

22. 23

24 = 2x 25. 23( )4 = 2x

23. 3−4 = 1

3x 26. 8 = 2

x

3

27. a3 • a = ax 29. 1

c −4 = c x

28. b10

b 5 = bx 30. a3y−2

a 2y−3 = ax

Problems 31-43: Find the value:

31. 7x0 = 38. 2x • 4x−1 =

32. 3−4 = 39. x

c+3

x c−3 =

33. 23 •24 = 40. 8x

2x−1 =

34. 05 = 41. 2x

−3

6x −4 =

35. 50 = 42. a

x+3( )x−3=

36. −3( )3 −33 = 43. a3x−2

a 2x−3 =

37. x c+3 • x c−3 =

Problems 44-47: Write two given ways:

Given No negative powers No fraction

44. d−4

d 4

45. 3x3

y( )−2

46. a2bc

2ab 2c( )3

47. x 2y 3z −1

x 5y −6z −3

C. Laws of rational exponents, radicals:

Assume all radicals are real numbers:

I. If r is a positive integer, p is an integer, and

a ≥ 0, then apr = a

pr = ar( )p , which is a real number. (Also true if r is a positive odd integer

and a < 0).

II. abr = ar • br , or ab( )1r = a 1

r •b 1r

III. ab

r = ar

br, or a

b( )1r = a

1r

b1r

IV. ars = asr = ars , or a1rs = a

1s( )

1r = a

1r( )

1s

Problems 48-53: Write as a radical:

48. 312 = 51. x

32 =

49. 423 = 52. 2x

12 =

50. 12( )

13 = 53. 2x( )1

2 =

Problems 54-57: Write as a fractional power:

54. 5 = 56. a3 = 55. 23 = 57. 1

a=

Problems 58-62: Find x:

58. 4 • 9 = x 61. 643 = x

59. x = 4

9 62. x = 8

23

432

60. 643 = x

Problems 63-64: Write with positive exponents:

63. 9x 6y−2( )12 = 64. −8a 6b−12( )− 2

3 =

D. Simplification of radicals:

example: 32 = 16 • 2 = 4 2

example: 723 = 83 • 93 = 2 93

example: 543 + 4 163 = 273 • 23 + 4 83 • 23

= 3 23 + 4 •2 23

= 3 23 + 8 23 =11 23

example: 8 − 2 = 2 2 − 2 = 2

Problems 65-78: Simplify (assume all radicals

are real numbers):

65. − 81 = 72. 4x6 = 66. 50 = 73. x 2x +2 2x 3 + 2x 2

2x=

67. 3 12 = 74. a2 =

68. 543 = 75. a3 = 69. 52 = 76. a5

3 =

70. 2 3 + 27 − 75 = 77. 3 2 + 2 =

71. x5 = 78. 5 3 − 3 =

E. Rationalization of denominators:

example: 58

= 5

8= 5

8• 2

2= 10

16= 10

4

example: 1

23= 1

23• 43

43 = 43

83= 43

2

example: 3

3−1 = 3

3−1• 3+1

3+1= 9+ 3

3−1 = 3+ 3

2

Problems 79-87: Simplify:

79. 23

= 84. 2 + 1

2=

80. 1

5= 85. 3

2+1=

81. 3

3= 86.

3

1− 3=

82. a

b= 87.

3+23−2

=

83. 23

3 =

Answers:

1. 8

2. 9

3. −16

4. 16

5. 0

6. 1

7. 8

8. 4

9. 2

4 10. −7 11. –5

12. 5

13. 5

14. x if x ≥ 0; −x if x < 0

15. a

16. 12

17. .2

18. 1681

19. 3a2

20. 7

21. 7

22.−1 23. 4

24. 0

25. 12

26. 3

27. 4

28. 5

29. 4

30. y +1 31. 7

32. 181

33. 128

34. 0

35. 1

36. −54 37. x

2c

38. 23x−2

39. x6

40. 22x+1

41. x 3

42. ax 2 −9

43. ax+1

44. 1

d 8= d−8

45. y 2

9x 6= 9−1x−6y 2

46. a 3

8b 3= 8

−1a3b

−3

47. y 9z 2

x 3= x−3y 9z 2

48. 3

49. 163

50. 12

3 = 43

2

51. x3 = x x

52. 2 x

53. 2x

54. 512

55. 232

56. a13

57. a− 1

2

58. 36

59. 4 9

60. 4

61. 2

62. 12

63. 3 x

3

y

64. b 8

4a 4

65. –9

66. 5 2

67. 6 3

68. 3 23

69. 2 13

70. 0

71. x 2 x

72. 2 x3

73. 4x 2x

74. a if a ≥ 0; −a if a < 0

75. a a

76. a a23

77. 4 2

78. 4 3

79. 63

80. 55

81. 3

82. abb

83. 183

3

84. 3 22

85. 3 2 − 3

86. 3+3−2

87.−7− 4 3

TOPIC 3: LINEAR EQUATIONS and INEQUALITIES

A. Solving linear equations:

Add or subtract the same value on each side of the

equation, or multiply or divide each side by the same

value, with the goal of placing the variable alone on

one side. If there are one or more fractions, it may

be desirable to eliminate them by multiplying both

sides by the common denominator. If the equation

is a proportion, you may wish to cross-multiply.

Problems 1-15: Solve:

1. 2x = 9 9. x − 4 = x2

+1 2. 3 = 6x

5 10. 3x

2x+1 = 52

3. 3x + 7 = 6 11. 6 − 4x = x 4. x

3= 5

4 12. 3x−2

2x+1 = 4

5. 5 − x = 9 13. x+32x−1 = 2

6. x = 2x5

+1 14. 7x − 5 = 2x +10 7. 4x −6 = x 15. 1

3= x

x+8

8. x−1x+1 = 6

7

B. Solving a pair of linear equations:

The solution consists of an ordered pair, an

infinite number of ordered pairs, or no solution.

Problems 16-23: Solve for the common solution(s)

by substitution or linear combinations:

16. x + 2y = 7 20. 2x − 3y = 5

3x − y = 28 3x + 5y =1 17. x + y = 5 21. 4x −1= y

x − y = −3 4x + y =1 18. 2x − y = −9 22. x + y = 3

x = 8 x + y =1 19. 2x − y =1 23. 2x − y = 3

y = x − 5 6x − 9 = 3y

C. Analytic geometry of one linear equation:

The graph of y = mx + b is a line with slope m and y-intercept b. To draw the graph, find one point on

it (such as (0, b)) and then use the slope to find

another point. Draw the line joining the two points.

5

example: y = −32x + 5

has slope − 32 and y-intercept 5.

To graph the line, locate (0, 5).

From that point, go down 3

(top of slope fraction), and

over (right) 2 (bottom of fraction)

to find a second point.

Draw the line joining the points.

Problems 24-28: Find slope and y-intercept, and

sketch graph:

24. y = x + 4 27. x − y = −1 25. y = − 1

2x − 3 28. x = −3y + 2

26. 2y = 4x − 8

To find an equation of a non-vertical line, it is

necessary to know its slope and one of its points.

Write the slope of the line through (x, y) and the

known point, then write an equation which says

that this slope equals the known slope.

example: Find an equation of the line through

(-4, 1) and (-2, 0).

Slope= 1−0−4+2 = 1

−2

Using (-2, 0) and (x, y),

Slope= y−0x+2 = 1

−2 ; cross multiply,

get −2y = x + 2, or y = − 12x −1

Problems 29-33: Find an equation of line:

29. Through (-3, 1) and (-1, -4)

30. Through (0, -2) and (-3, -5)

31. Through (3, -1) and (5, -1)

32. Through (8, 0), with slope –1

33. Through (0, -5), with slope 2 3

A vertical line has no slope, and its equation can

be written so it looks like x = k (where k is a number). A horizontal line has zero slope, and

its equation looks like y = k .

example: Graph on the same graph:

x − 3 =1 and 1+ y = −3 . The first equation is x = −4 The second is y = −4

Problems 34-35: Graph and write equation for…

34. The line through (-1, 4) and (-1, 2)

35. The horizontal line through (4, -1)

D. Analytic geometry of two linear equations:

Two distinct lines in a plane are either parallel or

intersecting. They are parallel if and only if they

have the same slope, and hence the equations of the

lines have no common solutions. If the lines have

unequal slopes, they intersect in one point and their

equations have exactly one common solution.

(They are perpendicular if their slopes are negative

reciprocals, or one is horizontal and the other is

vertical.) If one equation is a multiple of the other,

each equation has the same graph, and every

solution of one equation is a solution of the other.

Problems 36-43: For each pair of equations in

problems 16 to 23, tell whether the lines are

parallel, perpendicular, intersecting but not

perpendicular, or the same line:

36. Problem 16 40. Problem 20

37. Problem 17 41. Problem 21

38. Problem 18 42. Problem 22

39. Problem 19 43. Problem 23

E. Linear inequalities:

example: One variable graph: solve and graph

on a number line: 1− 2x ≤ 7 (This is an abbreviation for {x: 1− 2x ≤ 7 })

Subtract 1, get −2x ≤ 6 Divide by –2, x ≥ −3 Graph:

Problems 44-50: Solve and graph on a number line:

44. x − 3 > 4 48. 4 − 2x < 6 45. 4x < 2 49. 5 − x > x − 3 46. 2x +1≤ 6 50. x >1+ 4 47. 3 < x − 3

example: Two variable graph: graph solution on

a number plane: x − y > 3

(This is an abbreviation for x, y( ): x − y > 3{ }) Subtract x, multiply by –1, get y < x − 3 .

Graph y = x − 3 ,

but draw a dotted line,

and shade the side

where y < x − 3 :

Problems 51-56: Graph on a number plane:

51. y < 3 54. x < y +1 52. y > x 55. x + y < 3

53. y ≥ 23x + 2 56. 2x − y >1

0 1 2 3-1-2-3-4

6 F. Absolute value equations and inequalities:

example: 3− x = 2

Since the absolute value of both 2 and –2 is

2, 3− x can be either 2 or –2. Write these two

equations and solve each:

3− x = 2 3− x = −2 −x = −1 or −x = −5 x =1 x = 5

Graph:

-1 0 1 2 3 4 5 6

Problems 57-61: Solve and graph on number line:

57. x = 3 60. 2 − 3x = 0

58. x = −1 61. x + 2 =1 59. x −1 = 3

example: 3− x < 2

The absolute value of an number between –2

and 2 (exclusive) is less than 2. Write this

inequality and solve: −2 < 3− x < 2 .

Subtract 3

Multiply by –1, get 5 > x >1. (Note that this says x > 1 and x < 5).

Graph:

-1 0 1 2 3 4 5 6

example: 2x +1 ≥ 3. The absolute value is

greater than or equal to 3 for any number ≥ 3 or ≤ −3 . So,

2x +1≥ 3 2x +1≤ −3 2x ≥ 2 or 2x ≤ −4 x ≥1 x ≤ −2

Graph:

-4 -3 -2 -1 0 1 2 3

Problems 62-66: Solve and graph on number line:

62. x < 3 65. 1≤ x + 3

63. 3 < x 66. 5 − x < 5

64. x + 3 <1

Answers:

1. 92

2. 52

3. − 13

4. 154

5. –4

6. 53

7. 2

8. 13

9. 10

10. − 54

11. 65

12. − 65

13. 53

14. 3

15. 4

16. (9, -1)

17. (1, 4)

18. (8, 25)

19. (-4, -9)

20. ( 2819 , −1319)

21. ( 14 , 0)

22. no solution

23. a, 2a − 3( ), where a is any number;

infinite number of

solutions

24. m = 1, b = 4

25. m = − 12 , b = −3

26. m = 2, b = −4 27. m = 1, b =1

28. m = − 13 , b = 2

3

29. y = − 5

2 x −132

30. y = x − 2

31. y = −1 32. y = −x + 8

33. y = 23 x −5

34. x = −1 35. y = −1 36. intersecting, not⊥

37. ⊥ 38. intersecting, not⊥

39. intersecting, not⊥

40. intersecting, not⊥

41. intersecting, not⊥

42. parallel

43. same line

44. x > 7 0 7

45. x < 12 10

46. x ≤ 52

2 30

47. x > 6 0 6

48. x > −1 -1 0

49. x < 4 40

50. x > 5 0 5 51.

52.

53.

54.

55.

4

-3

-4

1

2

3

-1

-1

7 56.

57. x = ±3 (on number line)

58. no solution

59. x = −2, 4

60. x = 23

61. x = −1, − 3 62. −3 < x < 3

-3 30 63. x > 3 or x < −3

-3 30

64. −4 < x < −2

-4 0-2 65. x ≥ −2 or x ≤ −4

-4 0-2 66. 0 < x <10

0 10

TOPIC 4: POLYNOMIALS and POLYNOMIAL EQUATIONS

A. Solving quadratic equations by factoring:

If ab = 0 , then a = 0 or b = 0

example: If 3− x( ) x + 2( )= 0

then 3 − x( )= 0 or x + 2( ) = 0 and thus

x = 3 or x = −2

Note: there must be a zero on one side of the

equation to solve by the factoring method.

example: 6x2 = 3x

Rewrite: 6x2 − 3x = 0

Factor: 3x 2x −1( ) = 0

So 3x = 0 or 2x −1( ) = 0 .

Thus, x = 0 or x = 12 .

Problems 1-12: Solve by factoring:

1. x x − 3( ) = 0 7. x2 − x −6 = 0

2. x2 − 2x = 0 8. x

2 = 2− x 3. 2x

2 = x 9. 6x2 = x + 2

4. 3x(x + 4)= 0 10. x2 + x = 6

5. x + 2( ) x − 3( ) = 0 11. 9+ x2 = 6x

6. 2x +1( ) 3x −2( )= 0 12. 1− x = 2x2

B. Monomial factors:

The distributive property says ab + ac = a b + c( )

example: x2 − x = x(x −1)

example: 4x2y + 6xy = 2xy(2x + 3)

Problems 13-17: Factor:

13. a2 + ab 16. x

2y − y

2x =

14. a3 − a2b+ ab2 = 17. 6x

3y2 −9x 4

y =

15. −4xy +10x 2

C. Factoring: x − a( ) x + b( )2 + x − a( ) x + c( ): The distributive property says jm+ jn = j m + n( ) . Compare this equation with the following:

x +1( ) x + 3( )2 + x +1( ) x − 4( ) = x +1( ) x + 3( )2 + x − 4( )( )

Note that j = x +1, m = (x + 3)2, and n = (x − 4) ,

and we get x +1( ) x2 + 6x + 9+ x − 4( ) = x +1( ) x2 + 7x + 5( )

Problems 18-20: Find P which completes the

equation:

18. (x − 2)(x −1)2 − (x − 2)(x + 3) = (x − 2)( P )

19. x + 4( ) x − 3( )2 + x − 4( ) x − 3( ) = x − 3( ) P( )

20. 2x −1( ) x +1( )− 2x −1( )2 x + 3( ) = 2x −1( ) P( )

D. The quadratic formula:

If a quadratic equation looks like ax2 + bx + c = 0 ,

then the roots (solutions) can be found by using the

quadratic formula: x = −b± b 2−4ac2a

example: 3x2 + 2x −1= 0, a = 3 , b = 2 , and

c = −1

x = −2± 22−4 3( ) −1( )2•3 = −2± 4+12

6= −2± 16

6

= −2±46

= −66

= −1 or 26

= 13

example: x2 − x −1= 0 , a = 1, b = −1,

c = −1 x = 1± 1+4

2= 1± 5

2 So there are two roots:

1+ 5

2 and 1− 5

2

Problems 21-24: Solve:

21. x2 − x −6 = 0 23. 2x

2 − x − 2 = 0

22. x2 + 2x = −1 24. x

2 − 3x − 4 = 0

E. Quadratic inequalities:

example: Solve x2 − x < 6 . First make one side

zero: x2 − x −6 < 0 .

Factor: x − 3( ) x + 2( ) < 0 .

If x − 3( )= 0 or x + 2( ) = 0 , then x = 3 or

x = −2 .

8

These two numbers (3 and –2) split the real

numbers into three sets (visualize the number

line):

x x − 3( ) x +2( ) x − 3( ) x + 2( )

solution?

x < −2 −2< x < 3 x > 3

negative

negative

positive

negative

positive

positive

positive

negative

positive

no

yes

no

Therefore, if x − 3( ) x + 2( )< 0 , then −2 < x < 3

Note that this solution means that x > −2 and x < 3

Problems 25-29: Solve, and graph on a number line:

25. x2 − x −6 > 0 28. x > x

2

26. x2 + 2x < 0 29. 2x

2 + x −1> 0

27. x2 − 2x < −1

F. Completing the square:

x2 + bx will be the square of a binomial when c

is added, if c is found as follows: find half the

coefficient of x, and square it–this is c. Thus

c = b2( )2 = b

2

4, and x

2 + bx + c

= x2 + bx + b 2

4= x + b

2( )2

example: x2 + 5x

Half of 5 is 5 2 , and 52( )2 = 25

4 , which must

be added to complete the square:

x2 + 5x + 254

= x + 52( )2

If the coefficient of x2 is not 1, factor so it is.

example: 3x2 − x = 3 x2 − 13x( ) Half of − 1

3 is

− 16 , and − 1

6( )2 = 136 , so

x2 − 13x + 1

36( )= x − 16( )2 , and

3 x2 − 13x + 1

36( )= 3x2 − x + 336. Thus, 3

36 (or 1

12)

must be added to 3x2 − x to complete the square.

Problems 30-33: Complete the square, and tell

what must be added:

30. x2 −10x 32. x2 − 3

2x

31. x2 + x 33. 2x

2 + 8x

G. Graphing quadratic functions:

Problems 34-40: Sketch the graph:

34. y = x2 38. y = (x +1)2

35. y = −x2 39. y = (x − 2)2 −1

36. y = x2 +1 40. y = x + 2( )(x −1)

37. y = x2 − 3

Answers:

1. 0, 3

2. 0, 2

3. 0, 12

4. 0, –4

5. –2, 3

6. − 12 ,

23

7. 3, –2

8. –2, 1

9. 23 , − 1

2

10. –3, 2

11. 3

12. –1, 12

13. a a + b( ) 14. a a

2 − ab+ b2( ) 15. 2x −2y +5x( ) 16. xy x − y( ) 17. 3x 3y(2y − 3x)

18. x 2 − 3x −2

19. x 2 +2x −16 20. −2x 2 − 4x + 4

21. –2, 3

22. –1

23. 1± 17

4

24. –1, 4

25. x < −2 or x > 3

26. −2 < x < 0

27. no solution, no graph

28. 0 < x <1

29. x < −1 or x > 12

30. (x −5)2, add 25 31. x + 1

2( )2 , add 14 32. x − 3

4( )2 , add 916

33. 2(x +2)2 , add 8 34.

35.

36.

37.

38.

39.

40.

-2 30

-2 0

0 1

1-3

-1

-2 1

(2,-1)

9 TOPIC 5: FUNCTIONS

A. What functions are and how to write them:

The area of a square depends on the side length

s, and given s, we can find the area A for that

value of s. The side and area can be thought of

as an ordered pair: (s, A). For example, (5, 25)

is an ordered pair. Think of a function as a set of

ordered pairs with one restriction: no two

different ordered pairs may have the same first

element. Thus {(s,A) : A is the area of the

square with side length s} is a function

consisting of an infinite set of ordered pairs.

A function can also be thought of as a rule: for

example, A = s2 is the rule for finding the area of

a square, given a side. The area depends on the

given side and we say the area is a function of the

side. ' A = f (s)' is read ' A is a function of s' , or

'A = f of s'. There are many functions of s , the

one here is s2. We write this f (s) = s 2 and can

translate: ‘the function of s is s2’. Sometimes we

write A(s)= s2. This says the area is a function

of s, and specifically, it is s2.

In some relations, as x2 + y

2 = 25 , y is not a

function of x, since both (3, 4) and (3, -4) make

the relation true.

Problems 1-7: Tell whether or not each set of

ordered pairs is a function:

1. 1, 3( )−1, 3( ) 0,−1( ){ } 3. 0, 5( ){ } 2. 3,1( ) 3,−1( )−1,0( ){ }

4. {(x, y) : y = x 2 and x is any real number}

5. {(x, y) : x = y 2 and y is any real number}

6. {(x, y) : y = −3 and x is any real number}

7. {(x, y) : x = 4 and y is any real number}

Problems 8-11: Is y a function of x?

8. y = 4x + x2 10. x + y( )2 = 36

9. y2 = x

2 11. y = x

B. Function values and substitution:

If A(s) = s2, A(3), read ‘A of 3’, means replace

every s in A(s)= s2 with 3, and find the area when

s is 3. When we do this, we find A(3) = 32 = 9 .

example: g x( ) is given: y = g(x) = πx 2

example: g(3) = π • 32 = 9π

example: g(7) = π •72= 49π

example: g(a) = πa2 example: g(x + h) = π (x + h)2 = πx2 + 2πxh + πh2

12. Given y = f x( )= 3x − 2; complete these

ordered pairs:

(3, __), (0, __), ( 12 , __), (__, 10), (__, -1),

( x −1 , ___)

Problems 13-17: Given f (x) = xx+1 Find:

13. f 1( ) = 16. f −1( ) =

14. f −2( )= 17. f x −1( ) =

15. f 0( ) =

C. Composition of functions:

example: If f (x) = x2, and g x( ) = x − 3 ,

f g x( )( ) is read ‘f of g of x’, and means replace

every x in f (x) = x2with g x( ) giving

f g x( )( )= g x( )( )2, which equals (x − 3)

2 = x2 − 6x + 9 .

example: g f 12( )( )= g 1

2( )2( )

= g 14( )= 1

4− 3 = −2 3

4

Problems 18-26: Use f and g as above:

18. g f x( )( )= 23. f x( )− g x( )=

19. f g 1( )( )= 24. f x( )g x( ) =

20. g g x( )( )= 25. g x2( )=

21. f x( )+ g x( )= 26. g x( )( )2 = 22. f x( )•g x( )=

example: If k x( )= x2 − 4x , for what x is

k x( )= 0? If k x( )= 0, then x2 − 4x = 0 and

since x2−4x = x x − 4( )= 0 , x can be either 0

or 4. (These values of x: 0 and 4, are called

‘zeros of the function’, because each makes the

function zero.)

Problems 27-30: Find x so:

27. k x( )= −4 29. x is a zero of x x +1( ) 28. k x( )= 5 30. x is a zero of (2x 2 − x − 3)

D. Graphing functions:

An easy way to tell whether a relation between two

variables is a function or not is by graphing it: if a

vertical line can be drawn which has two or more

10 points in common with the graph, the relation is not

a function. If no vertical line touches the graph

more than once, then it is a function.

example: x = y2 has this graph:

Not a function (the vertical

line hits it more than once).

Problems 31-39: Tell whether or not each of the

following is a function:

31. 36.

32. 37.

33. 38.

34. 39.

35.

Since y = f x( ), the values of y are the values of the function, which correspond to specific values

of x. The heights of the graph above (or below)

the x-axis are the values of y and so also of the

function. Thus for this graph,

f 3( ) is the height (value) of the function at x = 3 and value is 2: At x = −3 , the value (height) of f x( ) is zero; in other words, f −3( ) = 0 . Note that

f 3( )> f −3( ), since 2 > 0 , and that

f 0( ) < f −1( ), since f −1( ) =1 and f 0( ) <1 .

Problems 40-44: For this graph,

tell whether the statement

is true or false:

40. g −1( )= g 0( ) 43. g −2( )> g 1( ) 41. g 0( ) = g 3( ) 44. g 2( )< g 0( ) < g 4( ) 42. g 1( )> g −1( )

To graph y = f x( ), determine the degree of f x( ) if it is a polynomial. If it is linear (first degree) the

graph is a line, and you merely plot two points

(select any x and find the corresponding y) and

draw their line. If f x( )is quadratic (second degree), its graph is a parabola, opening upward if the coefficient of x

2 is

positive, downward if negative.

To plot any graph, it can be helpful to find the following:

a) The y-intercept (find f 0( ) to locate y-axis crossing)

b) The x-intercept (find x for which

f x( ) = 0 – x-axis crossing)

c) What happens to y when x is very large

(positive) or very small negative?

d) What happens to y when x is very close to

a number which makes the bottom of a

fraction zero?

e) Find x in terms of y, and find what happens

to x as y approaches a number which makes

the bottom of a fraction zero.

(d, e, and sometimes c above will help find

vertical and horizontal asymptotes.)

example: y = 3− x 2 analysis: a quadratic

function (due to x2)

y-intercept: f 0( ) = 3

x-intercepts: y = 3 − x 2 = 0 so x = ± 3

example: y = −2x+1

a) y-intercepts: f 0( ) = −2 b) x-intercept: none, since there is no solution to

y = −2x+1 = 0

c) Large x: negative y approaches zero; very

negative x makes y positive and going to

zero. (So y = 0 , the x-axis, is an asymptote

line.)

d) The bottom of the fraction, x +1 , is zero if x = −1 . As x moves to –1 from the left, y

gets very large positive, and if x approaches –

1 from above, y becomes very negative. (The

line x = −1 is an asymptote.)

11

Graph:

e) To solve for x, multiply by x +1 and divide by y to get x +1= − 2

y

or x = −2

y−1

= −2−yy

Note that y close to zero results in a very

positive or negative x and means y = 0 is an

asymptote, which we already found above in

part c.

Problems 45-54: Sketch the graph:

45. f x( )= −x 50. y = 2x

46. y = 4 − x 51. f x( )= −2x

47. y = x 52. y = 1x−2

48. y = x − 2 53. y = 1

x 2

49. y = x2 + 3x − 4 54. y = 3x−1

Answers:

1. yes

2. no

3. yes

4. yes

5. no

6. yes

7. no

8. yes

9. no (2 intersecting lines)

10. no (2 parallel lines)

11. yes

12. 7, -2, − 12 , 4,

13 , 3x − 5

13. 12

14. 2

15. 0

16. none (undefined)

17. x−1x

18. x 2 − 3 19. 4

20. x − 6

21. x2 + x − 3

22. x3 − 3x

2

23. x2 − x + 3

24. x 2

x−3

25. x2 − 3

26. x2 −6x +9

27. 2

28. –1, 5

29. –1, 0

30. –1, 3 2

31. yes

32. no

33. yes

34. no

35. yes

36. yes

37. yes

38. no

39. yes

40. F

41. T

42. T

43. T

44. T

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

TOPIC 6: TRIGONOMETRY

A. Trig functions in right triangles:

The sine ratio for an acute angle of a right

triangle is defined to be the length of the

opposite leg to the length of the hypotenuse.

Thus the sine ratio

for angle B, abbreviated

sin B, is b c .

The reciprocal of the sine ratio is the cosecant

(csc), so csc B= cb .

The other four trig ratios (all functions) are

cosine= cos = adjacent leg

hypotenuse

secant= sec = 1cos

= hypotenuse

adjacent leg

tangent= tan = opposite leg

adjacent leg

cotangent= cot = ctn = adjacent leg

opposite leg

Problems 1-8: For this right triangle, give the

following ratios:

1. tan x =

2. sin x

cos x=

3. cos θ =

A

B

C

c

b

a

θ

10

8

6

x

4

2

12 4. sinθ •cosθ =

5. cos2x , which means (cos x)

2 =

6. 1 − sin2x = 8. sinθ

cosθ = 7. cosx

sinθ =

B. Circular trig definitions:

Given a circle with radius r, centered on (0, 0).

Draw the radius connecting a line from the

vertex to any point on the circle, making an

angle θ with the positive x-axis (θ may be any

real number, positive measure is counter-

clockwise). The coordinates (x, y) of point P

together with radius r are used to define the

functions:

sinθ = yr , cosθ = x

r ,

tanθ = yx ,

and the reciprocal

functions as before.

(Note that for 0<θ < π2 ,

these definitions agree

with the right triangle definitions. Also note that

−1 ≤ sinθ ≤1, −1 ≤ cosθ ≤ 1, and tanθ can

be any real number.)

Problems 9-12: For the point (-3, 4) on the

above circle, give:

9. x = y = r =

10. tanθ =

11. cosθ =

12. cotθ • sinθ =

Note that for any given value of a trig function, (in

its range), there are infinitely many values of θ .

Problems 13-14: Find two positive and two

negative values θ for which:

13. sinθ = −1 14. tanθ = tan45°

Problems 15-16: Given sin θ = 35 and

π2 < θ < π , then:

15. tanθ = 16. cosθ =

C. Pythagorean relations (identities):

a2 + b 2 = c

2 (or x

2 + y2 = r 2 )

above, can be divided by c2 (or r

2)

to give a2

c 2 + b 2

c 2 = c 2

c 2 , or

sin2A+ cos

2A =1, (or sin2θ + cos

2θ =1) , called an identity because it is true for all values

of A for which it is defined.

Problems 17-18: Get a similar identity by

dividing a2 + b2 = c 2

by:

17. b2 18. a

2

D. Similar triangles:

If ∆ABC ~ ∆DEF , and if tan A = 34 , then

tanD = 34 also, since EF :DF = BC : AC = 3

4 .

19. Find DC, given DB = 5 and sin E = .4

E. Radians and degrees:

For angle θ , there is a point P on the circle, and

an arc from A counter-clockwise to P. The length

of the arc is θ °360° •C = θ

360•2πr , and the ratio of

the length of arc to radius is π180

•θ , where θ is

the number of degrees (and the ratio has no

units). This is the radian measure associated with

point P. So P can be located two ways:

by giving the central angle θ

in degrees, or in a number of

radii to be wrapped around

the circle from point A

(the radian measure).

Converting: radians= π180

•degrees or degrees= 180

π •radians.

example: π3 (radians) = π

3• 180

π = 1803

= 60° example: 420° = 420• π

180= 7

3π (radians)

(which means that it would take a little over 7

radii to wrap around the circle from A to 420°.)

Problems 20-23: Find the radian measure for a

central angle of:

20. 36° = 22. 180° =

21. −45° = 23. 217° =

Problems 24-26: Find the degree measure, which

corresponds to radian measure of:

24. 3π2

= 25. −3 = 26. − 7π6

=

Problems 27-29: Find the following values by

sketching the circle, central angle, and a vertical

segment from point P to the x-axis. (Radian

r y

x

P x, y( ) θ

13 measure if no units are given.) Use no tables or

calculator.

27. cos 5π6

= 29. sin(−225°) =

28. tan(−315°)=

Problems 30-33: Sketch to evaluate without

table or calculator:

30. sec180° = 32. sinπ =

31. cot(− 3π2) = 33. cos 3π

2=

F. Trigonometric equations:

example: Solve, given that

0 ≤θ < 2π : tan2θ − tanθ = 0 . Factoring,

we get tanθ(tanθ −1) = 0 , which means that

tanθ = 0 or tanθ =1. Thus θ = 0 (degrees or radians) plus any multiple of 180° (or π ),

which is n •180° (or n •π ), or θ = 45° ( π4

radians), or 45 + n •180° (or π4 + n • π ).

Thus, θ can be 0, π4 , π , or 5π

4 , which all

check in the original equation.

Problems 34-39: Solve, for 0 ≤ θ ≤ 2π :

34. sinθ = cosθ 37. sinθ cotθ = 12

35. sin2θ =1+ cosθ 38. cosθ tanθ =1

36. sin2θ =1−cos2θ 39. sinθ tanθ = secθ

G. Graphs of trig functions:

By finding values of sin x when x is a multiple

of π2 , we can get a quick sketch of y = sin x .

The sine is periodic (it repeats every 2π , its

period). sin x never exceeds one, so the

amplitude of sin x is 1, and we get this graph:

To graph y = sin3x , we note that for a given

value of x, say x = a , the value of y is found on the graph of y = sin x three times as far from

the y-axis as a.

Thus all points of the

graph. y = sin3x

are found by moving

each point of y = sin x

graph to 13 its previous distance from the y-

axis, showing the new graph repeats 3 times in

the period of y = sin x , so the period of

y = sin x , so the period of y = sin3x is 2π3 .

Problems 40-46: Sketch each graph and find its

period and amplitude:

40. y = cos x 44. y = −4sin 23x

41. y = cos2x 45. y = sin2 x

42. y = tan x 46. y =1+ cos x

43. y = tan x3

H. Identities:

example: Find a formula for cos2A, given

cos A + B( ) = cosA cosB − sinAsinB .

Substitute A for B:

cos2A =cos A + A( ) = cosA cosA− sin AsinA =cos

2A− sin

2A

Problems 47-49: Use sin2x + cos

2x =1 and the

above to show:

47. cos2A = 2 cos2A −1

48. cos2A =1−2sin2A

49. cos2 x = 121+ cos 2x( )

50. Given cosA = 1secA

, sinAcosA

= tanA, and

sin2A+ cos

2A =1, show that

tan2x +1= sec

2x .

51. Given sin2A = 2sin AcosA , show

8sin 12x cos 1

2x = 4sin x .

Answers:

1. 43

2. 43

3. 45

4. 1225

5. 925

6. 925

7. 1

8. 34

9. –3, 4, 5

10. − 43

11. − 35

12. − 35

13. − 5π2 , − π

2 , 3π

2 , 7π

2

14. −315°,−135°, 45°, 225° 15. − 3

4

16. − 45

17. tan2 A+1= sec2 A

18. 1+cot2 A = csc2A

19. 2

20. π5

21. − π4

22. π 23. 217π

180

24. 270° 25. − 540

π( )° 26. −210° 27. − 3

2

28. 1

29. 22

a 3a

14 30. –1

31. 0

32. 0

33. 0

34. π4 ,

5π4

35. π2 ,

3π2 , π

36. 0 ≤ θ < 2π 37. π

3 , 5π

3

38. π2

39. π2 ,

3π2

40.

P = 2π A =1

41.

P = π A =1

42.

P = π no A

43.

P = 3π no A

44.

P = 3π A = 4

45.

P = π A = 1

2

46.

P = 2π A =1

47. cos2A = cos2A − sin2

A

= cos2 A− (1−cos2 A) = 2cos2 A−1

48. cos2A = cos2 A− sin2 A

= (1− sin2 A) − sin2 A =1−2sin2 A

49. From problem 47,

cos2x = 2cos2x −1. Add 1,

divide by 2.

50. tan2 A+1= sin 2 A

cos2 A+1

= sin 2 A +cos2 Acos2 A

= 1

cos2 A= sec2 A

51. Multiply given by 4, then

let 2A = x

TOPIC 7: LOGARITHMIC and EXPONENTIAL FUNCTIONS

A. Logarithms and exponents:

Exponential form: 23 = 8

Logarithmic form: log28 = 3

Both of the equations above say the same thing.

‘ log28 = 3’ is read ‘log base two of eight equals

three’ and translates ‘the power of 2 which gives

8 is 3.’

Problems 1-4: Write the following information

in both exponential and logarithmic forms:

1. The power of 3, which gives 9 is 2.

2. The power of x, which gives x3 is 3.

3. 10 to the power −2 is 1100

4. 12 is the power of 169 which gives 13.

exponent rules:

(all quantities real)

log rules:

(base any positive real

number except 1)

ab • ac = ab+c

a b

a c = ab−c

ab( )c = abc

ab( )c = acbc ab( )c = a

c

b c

a0 =1 (if a ≠ 0)

a−b = 1

a b

ap

r = a pr = ar( )p

(think of p

r as

power

root)

logab = loga + logb

log ab

= loga − logb

logab = b loga loga a

b = b a(loga b ) = b loga b = logc b

logc a

(base change rule)

Problems 5-25: Use the exponent and log rules

to find the value of x:

5. 43 • 45 = 4x 8. 52( )3 = 5x

6. 3−2 = x 9. log3 3

7 = x

7. 3x

3= 30 10. 6

log 6 x = 8

11. log10 x = log10 4 + log10 2

12. logx 25 = 2 15. 3loga 4 = loga x

13. log2132

= x 16. logx 3

logx 4= log4 3

14. log16 x = 32 17. log3(27• 3−4 ) = x

18. log 2x − 6( )= log 6 − x( )

19. log4 64 = x 23. 27x = 19( )3

20. 54 = 5x 24. 410 = 2

x

21.log3 81

log3 27− log3

8127

= x 25. 3•2x = 4

22. log4 303 = x log4 30

Problems 26-31: Find the value:

26. 210 = 29. 6

10

310=

27. log4 410 = 30. log49 7 =

28. log6 6 = 31. log7 49 =

32. Find log32 , given: log10 3 = .477log10 2 = .301

Problems 33-34: Given log21024 =10 , find:

33. log210245 = 34. log2 1024 =

Problems 35-46: Solve for x in terms of y and z:

35. 3x = 3y • 3z 36. 9y = 3z

3x

1

-1

π

π 1

-1 2π

π 2π

3π

3π 4

-4

1

-1 2π π

2π 1 -1 π

2

15

37. x3 = y 39. 3•2x = 2y

38. 3x = y 40. log x2 = 3log y

41. log x = 2log y − log z

42. 3log x = log y

43. log x = log y + log z

44. log x + log y3 = logz2

45. log7 3 = y; log7 2 = z x = log32

46. y = loga 9; x = loga 3

B. Inverse functions and graphing:

If y = f x( )and y = g x( ) are inverse functions, then an ordered pair (a, b) satisfies y = f x( ) if and only if the ordered pair (b, a) satisfies

y = g x( ). In other words, ‘f and g are inverses of each other’ means f a( ) = b if and only if

g b( ) = a

To find the inverse of a function y = f x( ) 1) Interchange x and y

2) Solve this equation for y in terms of x, so

y = g x( ) 3) Then if g is a function, f and g are inverses

of each other.

The effect on the graph of y = f x( ) when x and y are switched, is to reflect the graph over the

45° line (bisecting quadrants I and III). This reflected graph represents y = g x( ).

example: Find the inverse of

f x( )= x −1 or y = x −1 1) Switch x and y:

x = y −1 (note y ≥1 and x ≥ 0)

2) Solve for y : x2 = y −1, so y = x

2 +1 ( x ≥ 0 is still true)

3) Thus g(x) = x2 +1 (with x ≥ 0) is the

inverse function, and has

this graph:

Note that the f and g graphs

are reflections of each

other in the 45°–line, and that the ordered pair (2,5)

satisfies g and (5,2) satisfies f.

example: Find the

inverse of y = f (x) = 3x

and graph both

functions on one graph:

1) Switch: x = 3y

2) Solve: log3 x = y = g(x) ,

the inverse to get the graph

of g(x) = log3 x ,

reflect the f graph

over the 45°–line:

Problems 47-48: Find the inverse function and

sketch the graphs of both:

47. f x( ) = 3x − 2

48. f (x) = log2(−x) (note that –x must be

positive, which means x must be negative)

Problems 49-56: Sketch the graph:

49. y = x4 53. y = 4

−x

50. y = 4x 54. y = −log4 x

51. y = 4x−1 55. y = 4

x −1 52. y = log4 x 56. y = log4 (x −1)

Answers:

1. 32 = 9, log3 9 = 2

2. x3 = x 3 , logx x

3 = 3

3. 10−2 = 1100 , log10

1100 = −2

4. 16912 =13, log16913= 1

2

5. 8

6. 19

7. 1

8. 6

9. 7

10. 8

11. 8

12. 5

13. –5

14. 64

15. 64

16. any real number > 0

and ≠1 17. –1

18. 4

19. 3

20. 14

21. 13

22. 13

23. –2

24. 20

25. log4− log 3

log 2 (any base; if

base= 2 , x = 2− log2 3)

26. 1024

27. 10

28. 1

29. 210 =1024

30. 12

31. 2

32. 301477

33. 50

34. 5

35. y + z 36. z −2y 37. y

13

38. log3 y

39. y − log2 3

40. y32 = y y

f

l

g

l

f

l

l

f

l

g

16

41. y 2

z

42. y3

43. yz

44. z 4

y 23

45. zy

46. y 2

47. g(x) = 13x + 2

3

48. g(x) = −2x

49.

50.

51. (1,1)

52.

53.

54.

55.

56.

TOPIC 8: MATHEMATICAL MODELING–WORD PROBLEMS

Word Problems:

1. 2 3 of 16 of

34 of a number is 12. What is

the number?

2. On the number line, points P and Q and 2

units apart. Q has coordinate x. What are the

possible coordinates of P?

3. What is the number, which when multiplied

by 32, gives 32• 46? 4. If you square a certain number, you get 9

2.

What is the number?

5. What is the power of 36 that gives 3612 ?

6. Point X is on each of two given intersecting

lines. How many such points X are there?

7. Point Y is on each of two given circles. How

many such points Y?

8. Point Z is on each of a given circle and a

given ellipse. How many such Z?

9. Point R is on the coordinate plane so its

distance from a given point A is less than 4.

Show in a sketch where R could be.

Problems 10-11:

10. If the length of chord AB is x and length of

CB is 16, what is AC?

11. If AC= y and CB= z , how long is AB (in

terms of y and z)?

12. This square is cut into two smaller squares

and two non-square rectangles as shown.

Before being cut, the large square had area

a+ b( )2. The two smaller squares have areas

a2 and b2 . Find the total area of the two non-square rectangles. Do the areas of the 4 parts

add up to the area of the original square?

13. Find x and y:

14. When constructing an equilateral triangle

with an area that is 100 times the area of a

given equilateral triangle, what length should

be used for a side?

Problems 15-16: x and y are numbers, and two

x’s equal three y’s:

15. Which of x or y must be larger?

16. What is the ratio of x to y?

Problems 17-21: A plane has a certain speed in

still air. In still air, it goes 1350 miles in 3 hours:

17. What is its (still air) speed?

18. How long does it take to fly 2000 miles?

19. How far does the plane go in x hours?

20. If the plane flies against a 50 mph

headwind, what is its ground speed?

21. If it has fuel for 7.5 hours of flying time,

how far can it go against this headwind?

Problems 22-25: Georgie and Porgie bake pies.

Georgie can complete 30 pies an hour.

22. How many can he make in one minute?

23. How many can he make in 10 minutes?

24. How many can he make in x minutes?

25. How long does he take to make 200 pies?

A

BC

O

g

f

f

g

1 1

1

-1

1

17 Problems 26-28: Porgie can finish 45 pies an hour:

26. How many can she make in one minute?

27. How many can she make in 20 minutes?

28. How many can she make in x minutes?

Problems 29-32: If they work together, how

many pies can they produce in:

29. 1 minute 31. 80 minutes

30. x minutes 32. 3 hours

Problems 33-41: A nurse needs to mix some

alcohol solutions, given as a percent by weight

of alcohol in water. Thus in a 3% solution, 3%

of the weight would be alcohol. She mixes x

grams of 3% solution, y grams of 10% solution,

and 10 grams of pure water to get a total of 140

grams of a solution which is 8% alcohol:

33. In terms of x, how many grams of alcohol

are in the 3% solution?

34. The y grams of 10% solution would include

how many grams of alcohol?

35. How many grams of solution are in the final

mix (the 8% solution)?

36. Write an expression in terms of x and y for

the total number of grams in the 8% solution

contributed by the three ingredients (the 3%,

10%, and water).

37. Use your last two answers to write a ‘total

grams equation’.

38. How many grams of alcohol are in the 8%.

39. Write an expression in terms of x and y for

the total number of grams of alcohol in the

final solution.

40. Use the last two answers to write a ‘total

grams of alcohol equation’.

41. How many grams of each solution are

needed?

42. Half the square of a number is 18. What is

the number?

43. If the square of twice a number is 81, what

is the number?

44. Given a positive number x. The square of a

positive number y is at least 4 times x. How

small can y be?

45. Twice the square of half of a number is x.

What is the number?

Problems 46-48: Half of x is the same as one-

third of y.

46. Which of x and y is the larger?

47. Write the ratio x: y as the ratio of two integers.

48. How many x’s equal 30 y’s?

Problems 49-50: A gathering has twice as many

women as men. If W is the number of women

and M is the number of men…

49. Which is correct: 2M=W or M=2W

50. Write the ratio WM +W as the ratio of two

integers.

Problems 51-53: If A is increased by 25%, it

equals B.

51. Which is larger, B or the original A?

52. B is what percent of A?

53. A is what percent of B?

Problems 54-56: If C is decreased by 40%, it

equals D.

54. Which is larger, D or the original C?

55. C is what percent of D?

56. D is what percent of C?

Problems 57-58: The length of a rectangle is

increased by 25% and its width is decreased by

40%.

57. Its new area is what percent of its old area?

58. By what percent has the old area increased

or decreased?

Problems 59-61: Your wage is increased by 20%,

then the new amount is cut by 20% (of the new

amount).

59. Will this result in a wage, which is higher

than, lower than, or the same as the original

wage?

60. What percent of the original wage is this

final wage?

61. If the above steps were reversed (20% cut

followed by 20% increase), the final wage

would be what percent of the original wage?

Problems 62-75: Write an equation for each of

the following statement about real numbers and

tell whether it is true or false:

62. The product of the squares of two numbers is

the square of the product of the two numbers.

63. The square of the sum of two numbers is the

sum of the squares of the two numbers

64. The square of the square root of a number is

the square root of the square of the number.

65. The square root of the sum of the squares of

two numbers is the sum of the two numbers.

66. The sum of the absolute vales of two

numbers is the absolute value of the sum of

the two numbers.

18 67. The product of the absolute values of two

numbers is the absolute value of the product of

the two numbers.

68. The negative of the absolute value of a

number is the absolute value of the negative of

the number.

69. The cube root of the square of a number is

the cube of the square root of the number.

70. The reciprocal of the negative of a number

is the negative of the reciprocal of the number.

71. The reciprocal of the sum of two numbers is

the sum of the reciprocals of the two numbers.

72. The reciprocal of the product of two

numbers is the product of the reciprocals of

the two numbers.

73. The reciprocal of the quotient of two

numbers is the quotient of the reciprocals of

the two numbers.

74. The multiple of x which gives xy is the

multiple of y which gives yx.

75. The power of 2, which gives 2xy is the

power of 2, which gives 2yx .

Answers:

1. 144

2. x +2, x −2 3. 46

4. 9

5. 12

6. 1

7. 0, 1, or 2

8. 0, 1, 2, 3, or 4

9. Inside circle of radius

4 centered on A

10. x −16 11. y + z 12. 2ab, yes:

a + b( )2 = a2 +2ab+ b2 13. x = 40

7 ; y = 163

14. 10 times the original side

15. x

16. 32

17. 450 mph

18. 4 49 hours

19. 450x miles

20. 400 mph

21. 3000 miles

22. 12

23. 5

24. x 2

25. 400 min.

26. 34

27. 15

28. 3x 4

29. 54

30. 5x 4

31. 100

32. 225

33. .03x

34. .1y

35. 140

36. x + y +10 37. x + y +10 =140 38. 11.2

39. .03x + .1y 40. .03x + .1y =11.2 41. x = 180

7 ; y = 7307

42. 6, -6

43. 4.5, -4.5

44. 2 x

45. 2x 46. y

47. 2 : 3 48. 45

49. 2M= W

50. 23

51. B 52. 125%

53. 80%

54. C

55. 166 23%

56. 60%

57. 75%

58. 25% decrease

59. lower

60. 96%

61. same (96%)

62. a2b2 = (ab)2 T

63. (a + b)2 = a2 + b2 F

64. a( )2 = a2 T (if all

expressions are real)

65. a2 + b2 = a + b F

66. a + b = a + b F 67. a • b = ab T

68. − a = −a F

69. a23 = a( )3 F

70. 1−a = − 1

a T

71. 1a+b = 1

a+ 1

b F

72. 1ab

= 1a

• 1b T

73. 1

a

1b

= 1ab

T

74. y = x F 75. xy = yx T