ALGEBRA

CHAPTER 2

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.

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.

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 =•=

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

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

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.

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)

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

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

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

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. D x=132

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),

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

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

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

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 ?

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

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

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.

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)

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!

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}

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

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

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

REMEMBER

MATH IS FUN

AND …

YOU CAN DO IT