Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find an expression equivalent to the one given.
1)
–4k – 8
11k + 10
1)
A)
–
–4k – 8
11k –10
B)
–4k +8
11k +10
C)
–4k +8
11k –10
D)
4k – 8
–11k +10
2)
–8t –3
t +8
2)
A)
–8t +3
t –8
B)
8t –3
t –8
C)
–(8t –3)
t +8
D)
8t –3
t +8
Answer the question.
3)
Which of the following fractions is equivalent to
1
7+1
14
1
2+1
8
?
3)
A)
8
B)
–35
12
C)
12
35
D)
–12
35
Find an expression equivalent to the one given.
4)
–x –12
x +2
4)
A)
–x –12
– x –2
B)
–
– x +12
x +2
C)
–x –12
x –2
D)
– x +12
x +2
5)
–5x +14
15x –11
5)
A)
5x –14
11 +15x
B)
–5x +14
–(15x –11)
C)
5x +14
–(15x –11)
D)
5x –14
15x –11
Answer the question.
6)
Which of the following complex fractions is equivalent to
6–4
3
–1–2
7
?
6)
A)
–6–4
3
1–2
7
B)
6+4
3
–1+2
7
C)
–6+4
3
–1–2
7
D)
–6+4
3
1+2
7
Find an expression equivalent to the one given.
7)
–15k +3
7k –2
7)
A)
–15k –3
7k –2
B)
–
–15k +3
7k +2
C)
15k +3
–7k –2
D)
–15k –3
7k +2
8)
–15k +7
7k +6
8)
A)
–15k –7
7k +6
B)
–15k –7
7k –6
C)
15k +7
–7k +6
D)
–
–15k +7
7k –6
9)
–2x –2
8x +15
9)
A)
–2x +2
8x –15
B)
–2x +2
8x +15
C)
–
–2x –2
8x –15
D)
2k –2
–8x +15
10)
–x +14
x –11
10)
A)
–x +14
– x +11
B)
–
– x –14
x –11
C)
– x –14
x –11
D)
–x +14
x +11
11)
–7t –9
9t +13
11)
A)
–7t +9
9t –13
B)
7t –9
9t +13
C)
7t –9
9t –13
D)
–(7t –9)
9t +13
12)
–8x +9
14x –9
12)
A)
8x –9
14x –9
B)
–(8x – 9)
14x +9
C)
–8x –9
14x –9
D)
8x –9
9+14x
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Answer the question.
13)
What are two possible LCDs which could be used for the sum –4
x – 5 +4
5– x ?
13)
Provide an appropriate response.
14)
State whether the following situation represents direct variation, inverse variation, or
neither, and explain why: The weight of a roast and the number of servings obtained from
it.
14)
Answer the question.
15)
Explain how to add rational expressions with different denominators.
15)
16)
If 4 is substituted for x in the rational expression x –4
x2–16 , the result is 0
0. Mathematicians
have been known to say “Any number divided by itself is 1.” Does this mean that this
expression is equal to 1 for x =4? Explain why or why not.
16)
17)
How can you tell just by looking at a rational expression, if it is equal to –1?
17)
Provide an appropriate response.
18)
Explain in your own words how to multiply rational expressions. (How would you explain
this to a student in this class who was absent from class?).
18)
19)
When finding a
b÷c
d, what are the restrictions on the variables and why?
19)
Answer the question.
20)
Explain why someone would want to solve A =1
2bh for b.
20)
Provide an appropriate response.
21)
Tell what words you would use in a telephone conversation with a classmate (who missed
class) to explain how to write a rational expression as an equivalent rational expression
with a given denominator.
21)
Answer the question.
22)
For x 4, the rational expression 9(x –4 )
x –4 is equal to 9. Can the same be said for
9x –4
x –4? Explain why or why not.
22)
Provide an appropriate response.
23)
Consider an equation of inverse variation y =k
x. When x increases, does y increase or
decrease?
23)
Answer the question.
24)
Describe, in words, how to simplify a complex fraction by using the method of writing it as
a division problem.
24)
Provide an appropriate response.
25)
Explain in your own words how to divide rational expressions. (How would you explain
this to a student in this class who was absent from class?).
25)
Answer the question.
26)
What property of real numbers justifies the method of multiplying by the LCD of the
parts? Why?
26)
Provide an appropriate response.
27)
State whether the following situation represents direct variation, inverse variation, or
neither, and explain why: A runner’s speed in a race and the time it takes to run the race
27)
28)
Tell what words you would use in a telephone conversation with a classmate (who missed
class) to explain how to find the least common denominator of a group of denominators.
28)
Answer the question.
29)
Why is 6x + 1
x + 1 not equal to 6 ?
29)
30)
Describe, in words, how to simplify a complex fraction by using the method of multiplying
by the LCD of the parts.
30)
Provide an appropriate response.
31)
If (4 x –9 )2 is the LCD of two fractions, is (9 –4x)2 also acceptable as an LCD? Why or
why not?
31)
Answer the question.
32)
Explain the difference between adding rational expressions and solving rational equations.
32)
33)
Without multiplying by the least common denominator and solving, explain why the
rational equation x
x – 3 =3
x – 3 has no solution. (Hint: Examine both numerators and
denominators carefully.)
33)
Provide an appropriate response.
34)
If (5x –7)( 3x –8) is the LCD of two fractions, is ( 7 –5x)( 8–3x) also acceptable as an
LCD? Why or why not?
34)
Answer the question.
35)
If 5 is substituted for x in the rational expression x –5
x2–25 , the result is 0
0. Mathematicians
have been known to say “Any number divided by itself is 1.” Does this mean that this
expression is equal to 1 for x =5? Explain why or why not.
35)
Provide an appropriate response.
36)
Consider an equation of direct variation y = kx. When x increases, does y increase or
decrease?
36)
37)
State whether the following situation represents direct variation, inverse variation, or
neither, and explain why: The cost of mailing a letter in the United States and the distance
that it travels.
37)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Identify the following as an expression or an equation.
38)
4
7x +7=2
9x –9
38)
A)
Expression
B)
Equation
Write the rational expression in lowest terms.
39)
(y + 7)(y – 2)
(y – 2)(y + 3)
39)
A)
y – 7
y – 3
B)
y + 7
y + 3
C)
y + 2
y + 1
D)
2y – 2
2y + 1
Simplify the complex fraction.
40)
x8
2y5
x4
y2
40)
A)
x4
y3
B)
x4
2y7
C)
x4
2y3
D)
x12
2y7
Add or subtract. Write the answer in lowest terms.
41)
5
8– y –6
y –8
41)
A)
11
8– y
B)
30
8– y
C)
1
8– y
D)
–1
8– y
Find the reciprocal.
42)
3
12m2+ 44m
42)
A)
3
12m2– 44m
B)
12m2+ 44m
3
C)
–12m2– 44m
3
D)
3
12m2+ 44m
43)
y + 6
y
43)
A)
y
y + 6
B)
6
C)
–y + 6
y
D)
1
6
Solve the problem.
44)
Two machines are turned on at 8:00 A.M. If one can produce 58 items each hour and the other can
produce 73 items each hour, at what time will they produce a total of 262 items?
44)
A)
11:30 A.M.
B)
11:00 A.M.
C)
10:00 A.M.
D)
9:00 A.M.
Write the expression in lowest terms.
45)
x +2
x –2
45)
A)
–2
B)
–1
C)
1
D)
Already in lowest terms
D
Solve the problem.
46)
The distance D that a spring is stretched by a hanging object varies directly as the weight W of the
object. If a 10–kg object stretches a spring 42 cm, find the distance when the weight is 18–kg.
46)
A)
4.2857 cm
B)
4.2 cm
C)
70 cm
D)
75.6 cm
D
47)
The sum of an integer and its reciprocal is 17/4. Find the integer.
47)
A)
3
B)
4
C)
5
D)
17
B
C
Answer the question.
48)
List all numbers that must be rejected as possible solutions.
x – 2
8–10 x + 12
14 =x
7
48)
A)
8, 14 , 7
B)
There are no numbers which must be rejected.
C)
–8, –14 , –7
D)
8, 14 , 7, 2, –12
10
Perform the indicated operation and write the answer in lowest terms.
49)
7
r+9
r + 3
49)
A)
16r + 21
r(–3– r)
B)
–21r – 16
r(–3– r)
C)
16r + 21
r(r + 3)
D)
–21r – 16
r(r + 3)
Multiply. Write the answer in lowest terms.
50)
2t2– 13t –24
3t2– 4t –7
·3t2+ 11t –42
t2– 2t –48
50)
A)
(2t + 3)
(t + 1)
B)
(2t + 3)
(t – 1)
C)
(2t + 3)(t + 6)
(t + 1)(t – 6)
D)
(2t + 3)(t +8)
(t + 6)(3t –7)
Add or subtract. Write the answer in lowest terms.
51)
5
8x –9+4
9–8 x
51)
A)
–1
8x –9
B)
9
8x –9
C)
–9
8x –9
D)
1
8x –9
Answer the question.
52)
List all numbers that must be rejected as possible solutions.
5
3 x + 10 +1
x=1
17 x– 19
52)
A)
–10
3, 19
17
B)
0, –10
3, 19
17 , 5
C)
–10
3, 19
17 , –5
D)
0, –10
3, 19
17
Solve the problem.
53)
An experienced accountant can prepare a tax return in 14 hours. A novice accountant can do the job
in 22 hours. How long will it take them to do the job working together?
53)
A)
38 1
2 hr
B)
1
308 hr
C)
1
36 hr
D)
85
9 hr
Perform the indicated operation and write the answer in lowest terms.
54)
5
r+7
r – 8
54)
A)
40r – 12
r(8– r)
B)
12r – 40
r(r – 8)
C)
12r – 40
r(8– r)
D)
40r – 12
r(r – 8)
Write the rational expression in lowest terms.
55)
4x + 12
3x2+ 14x + 15
55)
A)
4
3x + 5
B)
4x
3x + 5
C)
4x + 3
3x + 14
D)
4x + 12
3x2+ 14x + 15
Solve the problem.
56)
The NCAA Division 4 women’s champion for the 400–meter dash in 1998 was Sally Sprintalot of
World State University. Her winning time was 4.79 seconds. What was her rate in meters per
second?
56)
A)
85.51 meters per second
B)
1916.00 meters per second
C)
81.51 meters per second
D)
83.51 meters per second
Answer the question.
57)
List all numbers that must be rejected as possible solutions.
11 x + 1
x – 13 =8x + 13
18 x + 7
57)
A)
13 , –7
18 , –1
11 , –13
8
B)
0, 13 , –7
18 , –1
11 , –13
8
C)
13 , –7
18
D)
There are no numbers that must be rejected.
The two triangles below are similar. Find the missing length.
58)
6.3
4.2
58)
A)
x =4.9; y = 3.9
B)
x =5.6; y = 3.9
C)
x =4.9; y = 3.6
D)
x =1.60; y = 3.6
Simplify the complex fraction.
59)
3–2
4+1
8–6
59)
A)
–23
5
B)
29
7
C)
23
9
D)
23
5
D)
60)
t +t
3–3
3+ 3
60)
A)
–7
5t
B)
4
5t
C)
3
5t
D)
7
5t
D)
61)
4–3
2+1
3 – 2
61)
A)
–9
4
B)
– 3
C)
9
4
D)
3
D)
Add or subtract. Write the answer in lowest terms.
62)
9
1– m +5
m –1
62)
A)
45
1– m
B)
–4
1– m
C)
4
1– m
D)
14
1– m
D)
Find all values that make the expression undefined.
63)
x– 8
7
63)
A)
Never undefined
B)
0
C)
–8
D)
8
Perform the indicated operation and write the answer in lowest terms.
64)
18
q – 9 –3
q – 9
64)
A)
21
q – 9
B)
15
q – 9
C)
18(q – 9)
3(q – 9)
D)
15
q
B
Write the rational expression in lowest terms.
65)
y2+ 6y – 16
y2+ 2y – 48
65)
A)
6y – 16
2y – 48
B)
6y – 1
2y – 3
C)
–y2+ 6y – 16
y2+ 2y – 48
D)
y – 2
y – 6
D
Solve the equation.
66)
7x
3–4x – 1
7=2
3
66)
A)
17
37
B)
11
61
C)
11
37
D)
13
37
C
A
The two triangles below are similar. Find the missing length.
67)
9
67)
A)
x =11.25; y =15.75
B)
x =6; y =12
C)
x =20; y =28
D)
x =13.5; y =18
Perform the indicated operation. Write the answer in lowest terms.
68)
x2–3x –18
6x ·x2–6x
x2–12x +36
÷x +3
x +6
68)
A)
x +6
3
B)
x +6
6
C)
x +3
x
D)
x –6
x
Solve for the specified variable.
69)
8x +3
z=9
y for z
69)
A)
z =9– 8xy
3y
B)
z =
–3y
8xy – 9
C)
z =3y
9– 8x
D)
z =
–3y
8x – 9
15
Perform the indicated operation and simplify.
70)
x +6y
x2+ 2xy +y2+x – y
x2+7xy +6y2
70)
A)
2x2+12xy +35y2
(x + y)3
B)
2x2+12xy +35y2
(x +6y)2(x + y)
C)
2x2+12xy +35y2
(x + y)2(x +6y)
D)
2x2+12xy +35y2
(x + y)(x +6y)
Find the least common denominator (LCD).
71)
9
x2– 11x + 30 , 3
x2– 7x + 6
71)
A)
(x + 5)(x + 6)(x – 1)
B)
(x – 5)(x – 6)(x – 1)
C)
(x – 6)(x – 1)
D)
(x – 5)(x – 6)
The two triangles below are similar. Find the missing length.
72)
50 725
48 24
72)
A)
x =14
B)
x =21
C)
x =7
D)
x =10
Multiply. Write the answer in lowest terms.
73)
4(k –6)2
15(k +6) ·5(k +6)2
12(k –6)
73)
A)
(k +6)2
9
B)
(k +6)(k –6)
9
C)
20(k +6)
(k –6)
D)
5(k –6)
(k +6)
16
Perform the indicated operation and write the answer in lowest terms.
74)
x
2–13
7
74)
A)
7x + 26
–26
B)
7x – 26
14
C)
x – 13
14
D)
x – 13
9
Solve the equation.
75)
x + 1
2=x + 2
3
75)
A)
1
6
B)
1
2
C)
3
5
D)
{1}
Write the rational expression in lowest terms.
76)
14k
18
76)
A)
9
7k
B)
7k
9
C)
7
9
D)
–14k
18
Multiply. Write the answer in lowest terms.
77)
3p – 3
p·7p2
6p – 6
77)
A)
18p2+ 36p + 18
7p3
B)
21p3– 21p2
6p2– 6p
C)
7p
2
D)
2
7p
17
Solve for the specified variable.
78)
3
x–4
y+10
z= 1 for x
78)
A)
y =4xz
3xz + 10x – xz
B)
z =10xy
xy – 2y + 4x
C)
x =3yz
yz + 4z –10y
D)
x =3xyz
yz + 4z –10y
Find all values that make the expression undefined.
79)
a– 7
5– a
79)
A)
–5
B)
5, 7
C)
Never undefined
D)
5
D
Find the least common denominator (LCD).
80)
5
x2+ 9x + 18 , 3
–3x – 18
80)
A)
–3(x – 3)(x + 6)
B)
–3(x + 3)(x + 6)
C)
–3(x – 3)(x – 6)
D)
–3(x + 3)(x – 6)
B
Simplify the complex fraction.
81)
1
a+ 1
1
a– 1
81)
A)
1
B)
1 –a2
C)
a
1 –a2
D)
1 + a
1 – a
D
C