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)
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