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Unit 1

Foundations of Algebra

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Real Number System

A. Real Number System

1. Counting Numbers (Natural Numbers) {1,2,3,4, …}

2. Whole Numbers { 0,1,2,3,4, …} 3. Integers - Negative and Positive Whole Numbers {…-2,-1,0,1,2,..}

4. Rational - Any number that can be written as a fraction 𝑎

where b≠ 0 EX: -2, ½, 5.46, 𝑏

√𝑝𝑒𝑟𝑓𝑒𝑐𝑡 , .̅̅̅̅1̅̅̅̅̅̅2̅̅̅̅̅̅

5. Irrational - A non-repeating, non-terminating decimal EX: 𝜋, √𝑛𝑜𝑛 − 𝑝𝑒𝑟𝑓𝑒𝑐𝑡

6. Real – All Rational and Irrational Numbers

B. Types of Sets and Numbers

Finite- a set that ends

Infinite - a set that doesn’t end or terminate

Absolute Value – the distance a number is from zero

Prime Numbers – exactly 2 factors {2,3,5…}

Composite Numbers- more than 2 factors {4,6,8,9…}

(One and Zero are neither prime/composite)

Even – {0,2,4,6,….}

Odd – {1,3,5,7…}

Inequalities - >, ≥, <, ≤

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Real Number System

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𝟑

Real Number System

Simplify and then identify each of the following as rational or irrational.

16. √𝟐 ∙ 𝟖 17. √𝟒 ∙ 𝟐 18. √𝟓 ∙ √𝟏𝟔 19. 𝝅 ∙ 𝟏

𝝅

20. Given the list of numbers below, determine which number sets they belong to. 3

√5 , , −0.5 , −√4 , 𝜋 2

a) Integers b) Rational c) Irrational

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Properties/Closure

A. Closure –

The round smiley faces are a closed set. No matter what operation is performed on round smiley faces,

another round smiley face will be created. Thus, there are always only round smiley faces in the box.

1) Is the set of whole numbers closed under addition?

- Add any two whole numbers

- Will the sum still be a whole number?

- Will any sum of 2 whole numbers still be a whole number?

If yes – It is CLOSED

Whole Numbers

If no – It is NOT CLOSED * Always think

about a closed circle

2) Is the set of whole numbers closed under subtraction?

3) Is the set of whole numbers closed under multiplication?

4) Is the set of whole numbers closed under division?

5) Is the set {0,1} Closed under

a) addition? b) Subtraction? c) Multiplication d) Division?

6.

7. Under which operations is the set {odd integers} closed?

A set is closed (under an operation) if and only if the operation on two elements of the set

produces another element of the set. If an element outside the set is produced, then the

operation is not closed

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Properties/Closure

B. Properties

Examples:

What property is illustrated by each statement? 1) -42 + 0 = -42 2) (x + 2.5) + 28 = x + (2.5 + 28) 3) 10x + 0 = 10x

4) 4x ·1 = 4x 5) x + (√𝑦 + z) = (x + √𝑦 ) + z 6) (3 + 4) + 5 = (4 + 3) + 5

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Properties/Closure

C. Inverses – Find the additive and multiplicative inverse of each

Example Additive Inverse Multiplicative inverse

-5

½

5.2

-3/4

2 6/7

8

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Properties/Closure

18) For which of the following numbers: 2, 3, 4, 5, 6 is the expression x + 2 closed? 8

19) Which of the following is a rational number? A) 7.2348 … B) 𝜋 C) √15 D) 9

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Properties/Closure

Choose the best answer:

1) Which of the following is an integer, but not a

whole number?

A) 0 B) -11 C) -1/2 D) -5.5

9) Which of the following is not a rational number?

1 A) 6 B) C) 7𝜋 D) 3.5

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2) Which of the following is not a counting number

(natural number)?

A) 0 B) 1 C) 2 D) 3

10) Which of the following is a rational number?

8 A) 7.2348 … B) 𝜋 C) √15 D)

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3) Which is an example of a whole number?

1 A) 0 B) -11 C) 2.5 D)

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11) Which of the following is not an irrational

number?

𝜋 A) 4 B) C) √7 D) √200

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4) Which number is an integer, a whole number, and

a counting (natural) number?

A) 0 B) -1 C) 15 D) 0.5

12) Which of the following is an irrational number?

A) 9.5 B) 3.14 C) 9𝜋 D) −0.5

5) Which of the following is a rational number?

A) 𝜋 B) −11

C) √2 D) −5.5234 …

6) Which of the following is irrational?

1 A) 0 B)

2 C) 2𝜋 D) 3.14

7) Which of the following is an example of a

non-perfect square?

A) √9 B) √81 C) √225 D) √45

8) What is the first counting (natural) number?

A) 0 B) 1 C) -1 D) 0.1

13) √197 lies between which two consecutive

integers?

A) 196 & 197 B) 14 & 15

C) 15 & 16 D) 197 & 198

14) Jessica is asked if 5𝜋 is a rational number. Which of the following is the most logical

response?

A) “No, it is irrational because any multiple

of 𝜋 is irrational.”

B) “Yes, it is rational because 5𝜋 can be written as

a fraction.”

C) “No, it is irrational because 5 is a prime

number.”

D) “Yes, it is rational because 5 and 𝜋 are both

rational.”

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Properties/Closure

15) Which letter on the number line below best represents the value of √6?

A B C D

0 1 2 3 4 5 6

1) Place the following numbers on the number line:

√5 , 3

2 , −0.5 , −√4 , 𝜋

-2 -1 0 1 2 3 4

2) Harry is asked if √100 is a rational number. He answers “No, it is irrational because the square root of any

number is irrational.” Is Harry correct? Explain why he is or is not correct.

Tell whether each description is rational or irrational:

18) Terminating decimals are .

19) Pi is

20) Fractions are .

21) The square roots of perfect squares are .

22) The square roots of all other positive integers are .

23) Decimals that repeat with a pattern are .

24) Change 4

5

to a decimal. Is that rational or irrational?

25) Change 22

7

to a decimal. Is that rational or irrational?

26) Is 3.14 rational or irrational?

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Numerical Expressions/Evaluating Expressions

Warm Up:

1. 5 – 8 2. -5 – 8 3. 5 + -8 4. -5 + 8 5. 5 - -8 6. 72

9 7. (3)(-5)(-2) 8.

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A. Operations with Integers

1. 5 8 2. - 5 8 3. 5 8 4. 5 + -3

5. 4

2

6. (-4)2 7. 0

8 8.

8 0

9. (-4)(-35+7) 10. 3(-6)(2) 11.

(3)

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B. Simplifying Expressions

Order of Operations

Simplify

1. 18 – (4+3) 2. 18 – 4 + 3 3.

4 32

4. (4 3)

2

5. 18 2 2 5

6. 3(5-1) 22 7. 60 3 4

2 45 9 8. 17 (5

2 10) 5 8 9.

15 3(4) 9 6

C. Expression vs. Equation

D. Evaluate

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Numerical Expressions/Evaluating Expressions

1. 50-3x for x=7 2. 2x2 5x 4 a) for x=7 3. 2a

(n 1)d 5

for a= -4

b) for x= -7 n=10

d=3

4. x3 ; x 1 5. 2x

2 ; x 4

What is the value of the expression for x = -5 and y = 2

6) x2 + x - 12 ÷ y2 7) (xy)2 ÷ x2y

What is the value of the expression for a = 3 and b = -4

8) (a) – (b) 9) 3b – a2 10) 2b2 - 7a

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Simplifying Expressions

Warm Up: 1. Which of the following is the set of negative integers greater than -3?

(1) {-4,-5,-6,-7…} (2) {-3,-4,-5,-6…} (3) {-3,-2,-1} (4) {-2,-1}

2. The exact value of the rational number 5/3 can be written as

(1) 1.6 (2)

1.6 (3) 1.66666666667 (4) 1.6666666666

3. Which of the following numbers is rational?

(1) (2) 2 (3)

1.42 (4)

4. Which of the following inequalities is a true statement?

(1) .026<0.25<0.2 (2) 0.2<.026<0.25 (3) 0.2<0.25<0.026 (4) 0.026<0.2<0.25

5. Which of the following is a correct application of the distributive property?

(1) 4(8+0.2) = 4(8)+4(0.2) (3) 8(5+4)=8(5)+4

(2) 6(3-1)=6(-1+3) (4) 8(5)(4)=8(5)+8(4)

6. The additive inverse of 7 is

(1) -7 (2) 0 (3) 1/7 (4) 7

7. Which set of number is closed under subtraction?

(1) counting numbers (2) whole numbers (3) integers (4) odd numbers

8. Which set of number is not closed under multiplication? (1) counting numbers (2) negative integers (3) even integers (4) rational numbers

9. The set of integers is not closed under which operation? (1) addition (2) subtraction (3) multiplication (4) division

A. Vocabulary

4x² 1) Name the variable 2) Name the coefficient

3) Name the exponent

4) Name the base

Monomial/Term – a number, a variable, the product of a number and variable

Examples - 8, 9x, 9xy, -27x², , ,

Binomial – the sum/difference of 2 different terms

Examples - (8x + 9), (3x – 4), , ,

0.4

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Simplifying Expressions

Trinomial - the sum/difference of 3 different terms

Examples - (8x + 3y – 4), , ,

Degree of a Term (Monomial) – the sum of the variable exponents

Degree of the polynomial – the highest degree of the monomial terms

Standard Form – Decending order of exponents

Simplest Form – no like terms

Like Terms – same variable raised to the same power

B. Combining Like Terms

1) 4t + 6t

4) 12x2 (5x

2 )

2) 17y 15y

5) 8xy2 xy

2

3) 11b2 + 4b2

6) 6y – 6y

7) 2f + 7g 6 + 8g

8) 8x + 3 5x 9

9) 5k 6k2 12k + 10

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Simplifying Expressions

Geometry Write an expression in simplified form for the perimeter of each rectangle.

10. 11. 12

C. Distributive Property

1) 3(x + 8) - x 2) -7(5x – 4)+ 7 3) 5(x – 7) + 6x

4) 12( 3 - 1

𝑥) 5) 3(0.4 + 1.1n) 6) 1(7x + 5)

6 2

Simplify each expression.

11) 2.3 (14 + x) + 2x 12) (8 6t) – 10(2t – 3) 13) (6 + d) + 4d 14) (r + 1)

15) 1 3z 12

3

16)

2 2 2

3

3 m

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Geometry Write an expression in simplified form for the area of each rectangle.

17 18. 19 .

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Simplifying Expressions Simplify each expression.

1. 3(c + 5) + 4(4c – 9) 2. –(n – 1) + 8n + 6

3. The profit a company receives is given by the expression 0.15(855p – 315) where p is the number of

products sold. Rewrite this expression using the Distributive Property. What is the profit for 25 products sold.

4. Reasoning Demonstrate why 15x 5

5

3x 5 . Show your work.

Simplify each expression.

5. -3(-x + 4) – 4x + 8 6. 3.5(2n – 7y + 4) 7. 𝟐

5 + t) 8. –(4x - 7) ( 𝟓

Geometry Write an expression in simplified form for the perimeter and area of each rectangle.

8.

9.

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Express each polynomial in standard form and find the degree of each expression. 11. x3 + 3x – 5x5 + 1 12. -6x2 + 2x3 + 5 – 2x 13. 7 – 4x

State the degree of each polynomial

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Translate Expressions

Warm Up: Ditto

x + 5 x – 5 5x 5 x

The sum of x and 5 The difference of x

and 5

The product of 5

and x

The quotient of 5

and x

x plus 5

5 Multiplied by x 5 Divided by x

x Increased by 5 x minus 5

5 More than x x Diminished by 5 Twice x 2x

5 Less than x x squared x2

5 Subtracted from x The cube of a

number x3

A. Translate

1. W more than 3

2. w less than 3

3. w decreased by 2

4. the product of 5r and x

5. twice x, decreased by 10

6. 25, diminished by 4 times n

7. the sum of t and u, divided by 6

8. 100 decreased by twice (x+5)

9. the distance that is 20 meters shorter than x meters

10. a bill for n baseball caps, each costing d dollars

11. a weight that is 40 pounds heavier than p pounds

12. an amount of money that is twice d dollars

13. Brianna paid 17 dollars for batteries and film for her camera. If the batteries cost x dollars, express the cost

of the film in terms of x

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Translate Expressions

x + 5 x - 5

5x 5 x

A. Translate

1. x more than 3 2. w less than 3

3. w decreased by 2 4. the product of 5r and x

5. twice x, decreased by 10 6. 25 subtracted from 4 times n

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Translate Expressions

7. the sum of t and u, divided by 6 8. 100 decreased by twice (x+5)

10. the distance that is 20 meters shorter than x meters

11. One-eighth 𝜋 times the cube of the diameter d

12. a weight that is 40 pounds heavier than p pounds

13. an amount of money that is twice d dollars

14. The square root of the sum of the squares of the altitude a and the base b

15. 5 times the quantity of four less than x

16. four less than five times x

17. 7 less than twice the width, w

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Translate Expressions

Write an algebraic expression for each word phrase.

9. The difference between the square root of x and 𝜋 squared.

10. The sum of x and y decreased by their product.

11. The square of r increased by a quantity that is fifty times the cube of k.

12. Twice 𝜋 divided by the cube of the sum of x and 2.

13. The difference between the square of the hypotenuse h and the square of the altitude a of a triangle

14. Error Analysis A student writes 5y · 3 to model the relationship the sum of 5y and 3. Explain the

error.

15. Error Analysis A student writes the difference between 15 and the product of 5 and y to describe

the expression 5y 15. Explain the error.

1. 7 minus f 2. the sum of 11 and k

3. x multiplied by 6 4. a number t divided by 3

5. one fourth of a number n 6. the product of 2.5 and a number t

7. the quotient of 15 and y 8. a number q tripled

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Translate Expressions

9.

10.

11.

12.

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Translate Expressions

14. A bike rider is traveling at a speed of 15 feet per second. Write an expression

for the distance the rider has traveled after s seconds.

14 a. If the tax that you pay when you purchase an item is 12% of the sale price,

write an expression that gives the tax on the item with a price p.

Write another expression that gives the total price of the item, including tax.

b. Determine the total price, including tax, of an item that costs $75.

15. The cost to rent a hall for school functions is $60 per hour. Write an expression

for the cost of renting the hall for h hours.

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