Practice TestUnit 2 (Part 2)

1 If x2 – 6x + 8 = 0 and y = x + 3, then what are the possible values of y ?

x2 – 6x + 8 = 0

(x – 2)(x – 4) = 0

x – 2 = 0 , x – 4 = 0

x = 4x = 2

y = x + 3Let x = 2:

= 2 + 3= 5

y = x + 3Let x = 4:

= 4 + 3= 7

2(2x + 3y)2 – (2x – 3y)2

(2x + 3y)(2x + 3y) – (2x – 3y) (2x – 3y) (4x2 + 6xy + 6xy + 9y2) – (4x2 – 6xy – 6xy + 9y2)

4x2 + 12xy + 9y2 – 4x2 + 12xy – 9y2

(4x2 + 12xy + 9y2) – (4x2 – 12xy + 9y2)

24xy

3 If x2 + y2 = 37 and xy = 24, what is the value of (x – y)2?

(x – y)2

= x2 – xy – xy + y2

= (x – y)(x – y)

= x2 – 2xy + y2 = x2 + y2 – 2xy= 37 – 2xy= 37 – 2(24)= 37 – 48 = –11

x2 + y2 = 37

xy = 24

4 If (–8x + 3)(–4x2 + 4x + 6) = ax3 + bx2 + cx + d for all real values of x, what is the value of c ?

(–8x + 3)(–4x2 + 4x + 6)

–4x2

–8x

4x

3

32x3

6

–32x2 –48x

–12x2 12x 18

32x3 – 44x2 – 36x + 18 (Add matching colors)

c = –36

5 If (y – 5)2 = 0, then find the value of y2 – 2y ?

(y – 5)2 = 0

(y – 5)(y – 5) = 0

y – 5 = 0 , y – 5 = 0y = 5 y = 5

Find y2 – 2y when y = 5(5)2 – 2(5)

25 – 1015

6If x and y are positive integers, then which of

the following must be equal to ?2 2

4 6

4 9

x y

x y

2 2

4 6

4 9

x y

x y

2(2 3 )

(2 3 )(2 3 )

x y

x y x y

2

2 3x y

7

, then reciprocal is y

x

1

r

y

1x

Step 1

Step 2

Step 3

11

r r 1

r r r +1

r 1

y

x

x + y

x x y

x x

If x

ry

8

3

1

x a=

b

If and ab 0,

then

Reciprocal

3ax =

b 3

1

a

3ax =

b

1 a3 = b x a3 = bx

3

1 1

a bx

9 Solve for m.

LCD = 4m22

1 3 2

4 m m

2

2

2 1 3 2

4

4 4

1 1m

m

m

m

2 2 2

2

4 12 8

4

m m m

m m

m2 + 12 = 8m

m2 – 8m + 12 = 0

m2 – 8m + 12 = 0(m – 2)(m – 6) = 0m – 2 = 0 , m – 6 = 0

m = 2 m = 6

10 If , then find the value of x + 4 ?

Find x + 4when x = 7

7 + 4

11

4 3 5x

4 3 5x

2 2

4 3 5x

4x – 3 = 25+3 +3

4x = 28x = 7

11

If x is an integer and , howmany different values of x are possible?

7 2 8x

7 2 8x

+2

22 2

7 2 8x

49 < x – 2 < 64+2 +2

51 < x < 66

First, solveinequality.

11

51 < x < 66Integers between

51 and 66

52 53 54 55 56

57 58 59 60 61

62 63 64 65

Answer: 14 Values

OR

66 – 51 = 15

15 – 1 = 14

If x is an integer and , howmany different values of x are possible?

7 2 8x

11

2 < m < 7

Try this strategyUse an inequality with two numbers

that are close together.

Integers between 2 and 7

3 4 5 6

Answer: 4 Values

OR

7 – 2 = 55 – 1 = 4

How many valuesof m are possible?

10 < m < 45

45 – 10 = 3535 – 1 = 34

12 16 5w

2

216 5w

16 25w –16 –16

9w

2 2

9w

w = 81

13 If and , then what is the value of b ?

8a 3b a

8a

2 2

8a

a = 64

3 64b

4b

2 2

4b

b = 16

–3x + 13 < –14–13–13

–3x < –27

3 2

3

7

3

>x

x > 9

14

If , then which of the following

values could be x ?

10x

x

Strategy: Test each answer by substituting for x.

A. –10

10 < xx

–1010 < –10

–1 < –10NO

B. –5

10 < xx

–510 < –5

–2 < –5NO

15

If , then which of the following

values could be x ?

10x

x

D. 2

10 < xx

210 < 2

5 < 2

NO

E. 5

10 < xx

510 < 5

2 < 5

YES

C. 1

10 < xx

110 < 1

10 < 1

NO

15

If , then which of thefollowing statements must be true?

2x = 1

10

2x

Strategy: Substitute ¼ for x in each answer.

2 1

2 11

1 4

1

4

2 1

41

12

2x x2

1 1

4 16 .25

1

.062

1

5

.2500 .0

4

25

4

6

2

2

xx

2

2.0625

.252

.25

.25.

.0

25

3125

NO

NO

16

If , then which of thefollowing statements must be true?

10

2x

Strategy: Substitute ¼ for x in each answer.

2x x 2

1 1

4 16 .25

1

.062

1

5

.2500 .0

4

25

4

6

2x > 1 2 1

2 11

1 4

1

4

2 1

41

12

NO NO

16

If x > y and y > 0 and xz < 0, then whichof the following must be true about all the values of z?

xz < 0xz = ( – ) negative

y > 0 y = ( + ) positive

x > y x = ( + ) positive

(+)z = ( – )

(+)(–) = ( – )z is negative

z < 0

17

18 If the sum of two integers x and k is less than x, which of the following must be true?

–xx + k < x

–x

k < 0

19 Twice the difference between a certain number and its square root is 15 more than twice the number.Which of the following equations represents the statement above?

A.

B.

C.

D.

E.

2 15 N N = N

2 15 2 N N = N

2 15 =N N + N

2 15 N N + N

2 15 2 N N = N

20 If a number is doubled and then increased by 10, the result is 5 less than the square of the number.Which of the following equations represents the statement above?

A.

B.

C.

D. 2N + 10 = N2 – 5

E. 2N + 10 = N2 + 5

2 10 = 5 N + N

2 10 = 5N + N

2 10 = 5N + N

21 If 2 is subtracted from a number and this difference is tripled, the result is 6 more than the number. Find the number.

Let number = x

3(x – 2) = x + 6

–x –x

‘2 is subtracted from a number’ = x – 2‘The difference is tripled’ = 3(x – 2)

3x – 6 = x + 6

2x – 6 = 6

+6 +62x – 6 = 6

2x = 12x = 6

22 If the sum of two consecutive odd integers is 28, what is the product?

x + x + 2 = 28

–2 –2

Let 1st integer = xLet 2nd integer = x + 2

2x + 2 = 28

2x = 26x = 13

1313+2 = 15

Product = 13 15= 195

Sum = 13 + 15 = 28

23 If a positive integer is doubled and then increased by 10, the result is 5 less than the square of the integer. What is the integer?

2x + 10 = x2 – 5–2x – 10

Let integer = x

–2x – 100 = x2 – 2x – 150 = (x + 3)(x – 5)

–15

–23 –5

Two numbers withProduct of –15 and

Sum of –2

x + 3 = 0 , x – 5 = 0x = –3 x = 5

24 Jon buys one pencil and two pens for $3.50. Lauren buys four pencils and three pens for $5.50. How much would one pencil and one pen cost?

Cost of pencil = A

1A + 2B = 3.50

Cost of pen = B

Jon Lauren 4A + 3B = 5.50

(Multiply Jon by –4)

Jon Lauren

–4A – 8B = –14.004A + 3B = 5.50

– 5B = –8.50B = 1.70

AddEquations

(Pen cost)

24 Jon buys one pencil and two pens for $3.50. Lauren buys four pencils and three pens for $5.50. How much would one pencil and one pen cost?

B = 1.70 (Pen cost) Find A. Use one ofthe original equations.

1A + 2B = 3.501A + 2B = 3.504A + 3B = 5.50

A + 2(1.70) = 3.50A + 3.40 = 3.50A = 0.10

Cost of pencil = A Cost of pen = B

Cost of pencil and pen = A + B = 0.10 + 1.70 = 1.80

25

2

2

1

1

x

y

x

y

If y varies directly as x2, and y = 3 when x = 3, what is the value of y when x is 6?

369

3 y

3639 y

1089 y12y

22

22

1

1

x

y

x

y

31 y

31 x

921 x

?2 y

62 x

3622 x

y varies directly as x2y varies directly as x

26

2

5 100

p=

2

2

1

1

x

y

x

y

Students receive 5 bonus points for every 2 community service projects they perform. If Mark received 100 bonus points, how many projects did he perform?

DirectVariation

Note: As the bonus points increase, the community service projects should increase.

5 p = 2 100

5p = 200

p = 40

27 If it takes 4 men 3 hours each to pave a playground, how many hours will it take 12 men to complete the same task?

x1y1 = x2y2

InverseVariation

Note: Increasing the number of men will decrease the amount of time to complete the task.

4 · 3 = 12 · H2

12 = 12H2

1 = H2

1 Hour

M1H1 = M2H2

28 What is the value of f(x) = 3x + 3x + 30 if x = 3 ?

f(3) = 33 + 3(3) + 30

f(3) = 27 + 9 + 1

f(3) = 37

29 If f(x) = x + 2x, what is the value of f(–2)?f(–2) = –2 + 2–2

2

12

2

12

4

1

4

2

1 4

4

8 1

4 4

7

4

2 1

1 4

30Find the domain for

Note: We can only evaluate the square root of numbers greater than or equal to zero.

x – 5 > 0

Let expression inside radical be > 0.

+5 +5x > 5

( ) 5f x x

31The amount a restaurant owner pays for coffee beans is directly proportional to the number of pounds of coffee she buys. If she buys n pounds of coffee at d dollars per pound, what is the total amount she pays, in dollars, in terms of n and d.

n lb.

d dollars/lb.

total1

lb.$3 dollars/lb.

1 3 = $32

lb.$3 dollars/lb.

2 3 = $63

lb.$3 dollars/lb.

3 3 = $9n

lb.$d dollars/lb.

n d

32 The cost of preparing for a book sale is $30. If each book is sold for $3.00, express the profit as a function of n, where n represents the number of books sold.

BookCost

# of bookssold

PreparationCost Profit

3 1 30 3(1) – 30 = 3–30 = –27

3 2 30 3(2) – 30 = 6–30 = –24

3 3 30 3(3) – 30 = 9–30 = –21

3 4 30 3(4) – 30= 12–30 = –18

32 The cost of preparing for a book sale is $30. If each book is sold for $3.00, express the profit as a function of n, where n represents the number of books sold.

BookCost

# of bookssold

PreparationCost Profit

3 6 30 3(6) – 30= 18–30 = –12

3 10 30 3(10) – 30 = 30–30= 0

3 12 30 3(12) – 30 = 36–30= 6

3 n 30 3n – 30

f(n) = 3n – 30

33 Morgan’s plant grew from 42 centimeters to 57 centimeters in a year. Linda’s plant, which was 59 centimeters at the beginning of the year, grew twice as many centimeters as Morgan’s plant did during the same year. How tall, in centimeters, was Linda’s plant at the end of the year?

Centimeters Morgan’s plant grew = 57 – 42 = 15 cm.

Twice centimeters Morgan’s plant grew = 2(15) = 30 cm.

Height of Linda’s plant = 59 cm + Step 2 = 59 cm + 30 cm = 89 cm.

Step 1

Step 2

Step 3

34 ( ) 3 1h x x If , then h(x) is

I. Always Positive II. Never NegativeIII. Always an Integer

We can only evaluate the square root of numbers greater than or equal to zero.

Note 1

Note 2 0 0

False

Note 3 8 2.83

False

35For all numbers x and y, let xy be defined as xy = xy + y2. What is the value of (31)1 ?

(31)

(31)1

= 3 + 1= 4

= 31 + 12

= 41 + 12= 4 + 1= 5

= 41

36

Which values are not in the domain of

Let denominator = 0

x2 – 25 = 0

Solve equation for x.

+5 +5x = 5

(x – 5)(x + 5) = 0x – 5 = 0 , x + 5 = 0

–5 –5x = –5 {–5,5}

2

10( )

25

xf x

x

37 The sign-up fee at a gym is $50. Members then must pay $25 each month. Express the cost of using the gym as a function of m, where m represents the number of months the member participates.

Sign-upFee

# ofMonths

GymCost

50 1 25(1) 50 + 25(1)

= 75

MonthlyCost

50 2 25(2) 50 + 25(2)

= 100

50 3 25(3) 50 + 25(3)

= 125

50 m 25(m) 50 + 25(m)

f(m) = 50 + 25m

38 Each time Shannon pushes the button on a machine, a bell rings 7 times. Each time she turns the switch on the machine, the bell rings 3 times. During one hour, Shannon caused the bell on the machine to ring 23 times. How many times did she push the button?

Pushed Button

Turned Switch

1# of Rings

There are 3 rings each time a switch is turned. Thus, the number of rings must be a multiple of 3.

7(1) = 7Total rings left

23 – 7 = 16 NO

Pushed Button 2# of Rings7(2) = 14

Total rings left23 – 14 = 9 YES

3 3(3) = 9

39 Which system of inequalities best represents the graph?

A.2

23

3

y x

y x

3 2

23

y x

y x

B.

2 2

3 3

y x

y x

C. D.2

23

3

y x

y x

The dotted line has a negative slope.The 2nd inequality for each answer has a negative slope.The inequality sign should be < or > .

Wrong

39 Which system of inequalities best represents the graph?

3 2

23

y x

y x

B.

2 2

3 3

y x

y x

C. D.2

23

3

y x

y x

The solid line has a positive slope.The 1st inequality for each answer has a positive slope.Inequality sign should be < . The shading is below the line.

Wrong

39 Which system of inequalities best represents the graph?

3 2

23

y x

y x

B. 2 2

3 3

y x

y x

C.

- Answers B and C have different slopes for the solid line.

- Use to get to each point on the line. This will determine the slope and equation of the line.

runrise

2

3 3

2

run

risem

- Find two points on the line.

40 Determine the solution of the system of inequalities.

C. D.B.A.

y < –x – 1

y > 2x – 2–2x + y > –2

Negative Slope

Positive Slope

(Shaded Below)

(Shaded Above)