testing enhancement workshop math98

Testing Enhancement WorkshopMATH98

Department of Mathematics and Computer Science, Coppin State University

Updated on Oct 20, 2014 by Dr. Min A

Outline

• Solve an equation(linear or quadratic).

• Solve a formula.

• Special equations and inequalities.

• Application of equation, inequality,

proportion and geometry.

• Factoring.

• Link to check answers

http://faculty.coppin.edu/pages/MA/

Linear Equation

Form A #1)

#2)

#3)

Solve a formula

Practice form A4)

Solve for b.

Try Form A 22)

Step 1: Collect all terms with the specified variable one side.

Step 2:Combine terms with the specified variable .

Step 3:Divide each side by the factor of the specified variable.

bhA2

1

bhA 2

hb

A

2

Find the perimeter of Geometric Figure

Practice Form A 6) : find the perimeter of

P = 2(L+W)

=2(9x2+9x+19+9x2-6x+11)

=2( 18x2+3x+30)

=36x2+6x +60

9x2+9x+19

9x2-6x+11

Practice Form B10) factor out the GCF.

48m9+84m6+48m2

= 12m2(4m7)+12m2(7m4)+12m2(4)

= 12m2(4m7+7m4+4)

GCF has the least exponent of all terms.

Practice test form B 12) Factor by grouping.

14 –7s –2t+st

= (14 –7s) –(2t – st) grouping

= 7(2 –s) – t(2 –s) factor separately

=(2 –s)(7 –t) factor out GCF

Practice form A 10)

7x2 – 7x – 42

=7 (x2 –x –6)

= 7 (x )(x )

= 7 (x –3 )(x+2)

Try practice form B 13)

8x2 –8x –48

Factor GCF out if there is any.

Factor ax2+bx+c: Factor out GCF

I. Difference of squares a2 −b2= (a+b)(a −b)

Practice form A 14)

25x2 −36

= (5x+6)(5x −6)

Try practice form B 14)

16x2 − 49

Practice form A 12)

x2 − 16xy + 64y2

=(x 8y)(x 8y)

=(x − 8y)(x −8y)

− 8x

− 8x

−16x

II. Factor a perfect square trinomial.a2 + 2ab+b2 = (a +b)(a +b)a2 − 2ab+b2 = (a −b)(a −b)

Practice form A 13) Solve(9y+20)(6y+19)=0

ab=0 a = 0 or b = 0

Practice form A 13) Solve(9y+20)(6y+19)=0

ab=0 a = 0 or b = 0

Practice Form A13) Solve equation by factoring.

(9y+20)(6y+19) = 0

9y+20= 0 or 6y+19 = 0

y = −20/9 or y = −19/6

The solution set is {−20/9, −19/6}.

Try form A) 14

x2 −x=6

x2 −x −6 = 0

(x −3)(x+2) = 0

x −3 = 0 or x+2 = 0

x = 3 or x = −2

The solution set is {3, −2}.

Two rational solutions.

Find numbers not in domain

Practice form A 17) Find all numbers not in the domain of the function.

Idea: denominator can not be equal to zero.

x2+4x−45 = 0

(x+9)(x −5) = 0

x+9 = 0 or x − 5 = 0

x = −9 or x = 5

The domain is {x|x ≠5 and x ≠ −9}.

454

49)(

2

2

xx

xxf

When to cancel?

Practice form A 18) express the rational expression in lowest

terms.

Practice form A 19)

x

x

3 x

x

3 )2)(3(

)2(

ax

ax

2

22

a

ax

2

22

2

x

ax

)7)(3(

)3)(9(

yy

yy

82615

432

xx

x

)25)(43(

43

xx

x

)25(

1

x

Only common factors can be cancelled.

Application of GeometryA16) The length of a rectangle frame is 3 cm more than the width. The

area is 180 square cm. Find the width.

Let x = the width

x + 3 =another dimension

Area = Length × Width

x(x + 3) = 180

x2 + 3x − 180 = 0

(x + 15)(x − 12) = 0

x + 15 = 0 or x − 12 = 0

x = −15 or x = 12

x = 12 reject the negative solution

x + 3 = 15

Check answers:18/13/15/12For factoring .

VIII. Divide rational expressions

Example 5 write in lowest terms.

Practice form A 20)

11

2

x

x

x

x

x

x

x

x )1(

)1(

2

x

23

9988

p

p

p

p

)99(

3)88( 2

p

p

p

p

)1(9

3)1(8 2

p

p

p

p

3

8p

Practice form A 21)

45

5

16 22

xxx

x

)4)(1(

5

)4)(4(

xxxx

x

)1)(4)(4(

)4(5)1(

xxx

xxx

)1)(4)(4(

2052

xxx

xxx

)1)(4)(4(

2042

xxx

xx

A22) Solve for r

P(1 + rt) = A multiply 1+rt both sides

P + Prt = A clear () by distribution

Prt = A − P subtract P both sides

r = A−P divide Pt both sides

Pt

1

AP

rt

Solve a formula

Application of ProportionA24) If 3.5 ounces of oil are to be added to 14

gallons of gasoline, then how many ounces of oil should be added to 29.2 gallons of gasoline?

Idea: mixture of oil and gasoline at the same ratioLet x= # of ounce of oil should be added

3.5(29.2) = 14x cross product 102.2 = 14x

7.3 = x divide 14 both sidesTry Form B 22

2.2914

5.3 x

Example Graph –3 ≤ x < 2.

End points from left to right : interval notation [–3,2).

Graph the inequality.

Inequalities

A25) Solve − 2x ≤ 18

x ≥ −9

[− 9,∞)

[

− 9

2

18

2

2

xreverse when div negative

Graph the solution first;Determine the direction; Always use ( -∞ or ∞) .

Practice form A 27) For the compound inequality, give the solution set in both interval and graph forms.

x≥2 and x≤5

Try Practice form B 24) For the compound inequality, give the solution set in both interval and graph forms.

x≥3 and x≤6

Practice form A 32) simplify the radical . Assume that all variables represent positive real numbers.

12 8x

Practice form A 34) multiply, then simplify the product.

)15)(15(

)1)(1()5)(1()5)(1()5)(5(

1555

15

4

F O I L

Try form B 32) )111)(111(

Practice form A 35) rationalize the denominator

2

7a

2

7a

)2)(2(

)2(7a

2

)2(7a

Practice form B 33) rationalize the denominator 5

6a

Practice form A 38) solve x2 = 9

x = or x =

x = 3 or x = −3

The solution set is {−3, 3}

Try to solve x2 −25=0

9 9

Two rational solutions.