ALGEBRA
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Transcript of ALGEBRA
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ALGEBRA
CHAPTER 2
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ALGEBRA
2.1 Real No., Sci. Notation & Order
2.2 Real Number Properties
2.3 Solving Equations & Ineq.
2.4 Evaluating Formulas & Fctns.
2.5 Solving Quadratic Equations
2.6 Systems of Equations & Ineq.
2.7 Proportion, Variation, Word Prob.
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2.1 Operations-Irrationals
Expression - collection of numbers & letters with operation signsLike terms - have exactly the same letters and exponentsLike radicals - have exactly same “inside” Multiply radicals - keep the radical sign & multiply the radicandDivide radicals - keep the radical sign & div.
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2.1 Examples - Radicals
=3 - 75 2.
72 D. 34 C. 66 B. 5 A.
35325 =•3135 −
=2
40 5. 20
52 D. 54 C. 102 B. 40 A.
52 54 =•=
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integeran isn
10 and 1between is M 10 x M n
2.1 Scientific Notation
=)10 x (1.4 x )10 x (6.1 7. -1416
(6.1 x 1.4) )10x (10x -1416
A. 854 B. 8540 C. 85.4 D. -854
210 x 8.54
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2.1 Scientific Notation
8. 0.000904 2,260,000
6
-4
10 x 26.2
10 x 04.9 =
-4 – 6 = -10
integeran isn
10 and 1between is M 10 x M n
A. 4.00102
B. 4.001010
C. 4.00109
D. 4.0010-10
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2.1 Order of Operations
Please Excuse My Dear Aunt SallyParens. Expnts. Mult. Div. Add Subt.
=÷++ 5 x 7 14t 2 t x 10t 10. 2
2t t 10 + 22t + 5x 2t10 +t12
A. B. C. D.
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2.2 Real Number Properties
Properties: Commutative, Assoc., Distributive, Identity, Inverse
1. Choose the expression equivalent to the following: 15(13) + 15(10)
A. 15(13+10) B. 15(15)+13(10) C. (15+15)(13+10) D. 30(13)(10)
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2.2 Properties for Solving
To get an equivalent eq. or ineq.: Add, Subtract, Mult., or * Div. both sides by the same non-zero number. *When Div. or Mult. an Ineq. by a negative, reverse the symbol
4. Choose the equiv. to: 4x - 7 =3x + 6
A. 7x-7=6 B. x-7=6 C. 4x-6=3x+1 D. 4x-1= 3x+6
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2.2 Properties for Solving
To get an equivalent eq. or ineq.: Add, Subtract, Mult., or * Div. both sides by the same non-zero number. *When Div. or Mult. an Ineq. by a negative, reverse symbol
5. Choose the equiv. to: 4 - 2x > 8
A. -2x > 4 B. -2x < 4 C. 2x >4 D. -2x < -4
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2.3 Solving Linear Eqs.
1. If 7x - 6 = 3x + 20, then
€
. A x = 54
Subtract 3x4x - 6 = 20
Add 64x = 26
Divide by 4 x = 26/4
Reduce x = 13/2
€
. B x= 52
€
. C x = 134
€
. D x=132
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2.3 Solving Inequalities
A. x < 37 B. x > 2 C. x <-37/25 D. x > 37
Comb. like 12x + 20 > 6x -1 (17 - 7x)Remove ( )12x + 20 > 6x - 17 + 7x
12x + 20 > 13x - 17 Comb. likeSubtract 13x -x + 20 > -17
-x > -37 Subtract 20Divide by -1* x < 37
4. If 20x - 8x + 20 > 6x - (17 - 7x),
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2.3 Checking Solutions
5. For each of the statements below, determine whether -1 is a solution: i. lx-1l = 0
ii. (t-3)(t-6) < 6
iii. y2+3y+17=15
l-1-1l = l-2l = 0
(-1-3)(-1-6)=(-4)(-7) < 6
D. ii onlyC. iii only
B. ii and iii onlyA. i only
(-1)2+3(-1)+17=1-3+17=15
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2.4 Evaluating
3. The formula for finding simple interest (I) on a loan at rate r, after t years is I =Prt. Find the interest paid on a $10,000, 4 year loan if the rate is 8%?
I = 10,000 x 0.08 x 4
A. $32,000 B. $2000 D.$3200C.$200
=.32
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2.4 Evaluating
34 x f(x)given f(-3) Find 4. 2 +−= x
= (-3)2 - 4(-3) + 3
= 9 + 12 + 3
A. 9 B. 6 C. 24 D. 6
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2.5 Quadratic Expressions
A. Factoring Quadratic Expressions Difference of Squares
4x2-9=(2x+3) (2x-3)
D. 3x-2
A=2x, B=3
C. 2x-3B. 2x-9A. 2x+9
1. Which is a linear factor of 4x2 - 9 ?
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2.5 Quadratic Expressions
A. Factoring Quadratic Expressions Trinomial Forms:
?4113x offactor linear a is Which 2. 2 −− x
D. 3x+2
Key number ac=-12Factors of -12 that add to b=-11 : -12,1Rewrite: 3x2 -12x +x -4 = 3x(x-4)+1(x-4)
=(3x+1)(x-4)
C. 3x+1B. 3x-4A. x+4
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2.5 Quadratic Equations
Factoring:
Set=0 0523 2 =−− xxFactor (3x-5) (x+1) = 0
Factors=0 3x-5 = 0 or x+1 = 0
x = 5/3 or x = -1
6
62-2and
6
622 D.
3
5 and 1- C.
5
3 and 1- B.
5
3 and 1 A.
+
4213x :solutions Find.3 2 +=− x
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2.5 Quadratic Formula
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x =−b ± b2 − 4ac
2a
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4. 3Find solutions to x2 +1 = 6
0163 2 =+− xxa= 3, b= -6, c= 1
)3(2
)1)(3(4)6()6( 2 −−±−−=x
6
12366 −±=
3
63
6
626
6
246 ±=
±=
±=x
Solutions to ax2+bx+c=0Are given by:
A.
B.
C.
D.
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2=0
2.6 Solving Systems
System of Equations: 2 eq. and 2 var.
Solution to System: ordered pairs (x,y) that solve both equations
Possible Solutions:
empty set
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0 = 0 x...{ }
one ordered pair (intersecting lines)
many ordered pair (same line)
no ordered pair (parallel lines)
(x,y)
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2.6 Solving Inequalities
4. Which shaded region identifies the portion of the plane which corresponds to x<0 and y>2?
5
-5 5
-5
5
-5 5
-5
5
-5 5
-5
5
-5 5
-5
A. B.
D.C.
We can pick a point from each shaded region and see if it satisfies the given conditions
In A and B we will try (4,-2)
Is x<0? No!
In C we will try (-4,-2)
Is y>2? No!
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2.6 System Example
1. Choose the correct solution set for the system x + 4y = -1
4x + y = 11
Multiply by -4 -16x - 4y = -44x + 4y = -1Recopy Eq. 1
-15x = -45 x = 3
AddDivide
3 +4y = -1, 4y = -4 y = -1
C. A. {(3,-1)}
B. {(3,1)} D. {(x,y)|y=-4x+11}
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2.7 Proportions
Proportions:1. Two machines can complete 5 tasks every 3 days. Let t represent the number of tasks these machines can complete in a 30-day month. Select the correct relationship.
days
tasks =3
5
30
t
C.A. B. D.
For 2 machines
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2.7 Variation
3 Types: direct: y = kx Directly proprtional to
Varies directly asinvs: y = k/x Inversely proportional to
Varies inversely asjoint: y = kxz Varies jointly as
2. The pressure is directly proportional to the temp. If the pressure is 8 lb/sq.in. when temp. is 480 F, what is the pressure when temp. is 120 F?
<-This one
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D. 16 lb per in2
2.7 Variation
Direct Variation: y = kx
2. The pressure is directly proportional to the temp. If the pressure is 8 lb/sq.in. when temp. is 480 F, what is the pressure when temp. is 120 F?
P = k T8 = k (480)
P = k T P = (1/60)(120)=2
k =8/480=1/60
A. 32 lb per in2
B. 4 lb per in2
C. 2 lb per in2
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REMEMBER
MATH IS FUN
AND …
YOU CAN DO IT