Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the expression that is equivalent to the given rational expression.
1)
–x +9
x –14
1)
A)
– x –9
x –14
B)
–
– x –9
x –14
C)
–x +9
– x +14
D)
–x +9
x +14
2)
–2t –5
t +6
2)
A)
–(2t –5)
t +6
B)
–2t +5
t –6
C)
2t –5
t –6
D)
2t –5
t +6
3)
–5x +2
4x –11
3)
A)
–5x +2
–(4x –11)
B)
5x +2
–(4x –11)
C)
5x –2
11 +4x
D)
5x –2
4x –11
4)
–15t –13
15t +13
4)
A)
15t –13
15t +13
B)
15t –13
15t –13
C)
15t –13
–(15t +13)
D)
–15t +13
15t –13
5)
–15k +12
2k +8
5)
A)
–
–15k +12
2k –8
B)
15k +12
–2k +8
C)
–15k –12
2k –8
D)
15k –12
–2k –8
6)
–15x –11
8x +9
6)
A)
–15x +11
8x –9
B)
–15x +11
8x +9
C)
–
–15x –11
8x –9
D)
15k –11
–8x +9
7)
–6k +7
2k +2
7)
A)
6k +7
–(2k +2)
B)
–
–6k – 7
2k –2
C)
6k – 7
–2k +2
D)
–6k +7
2k –2
8)
–6k –10
12k –5
8)
A)
6k +10
–12k –5
B)
6k –10
–12k +5
C)
–6k –10
12k +5
D)
–
–6k +10
12k +5
9)
–6x +13
5x –15
9)
A)
–(6x – 13)
5x +15
B)
6x –13
15 +5x
C)
–6x –13
5x –15
D)
6x –13
5x –15
10)
–x –11
x +9
10)
A)
–x –11
– x –9
B)
–(x –11)
x +9
C)
–
– x +11
x +9
D)
–x –11
x –9
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
11)
Explain the difference between adding rational expressions and solving rational equations.
11)
Answer:
Explanation:
12)
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.)
12)
Answer:
Explanation:
13)
Explain why someone would want to solve A =1
2bh for b.
13)
Answer:
Explanation:
Write four rational expressions equivalent to the given expression.
14)
–8x +10
3x –8
14)
Answer:
–(8x +10)
3x –8, –8x –10
3x –8, 8x +10
–(3x –8), 8x +10
–3x +8
Explanation:
Provide an appropriate response.
15)
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?).
15)
Answer:
Answer:
Explanation:
16)
If 3 is substituted for x in the rational expression x –3
x2–9, the result is 0
0. An often–heard
statement is “Any number divided by itself is 1.” Does this mean that this expression is
equal to 1 for x =3? 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)
18)
Describe, in words, how to simplify a complex fraction by using the method of multiplying
by the LCD of the parts (Method 2).
18)
19)
For x 9, the rational expression 3(x –9 )
x –9 is equal to 3. Can the same be said for
3x –9
x –9? Explain why or why not.
19)
20)
What are two possible LCDs which could be used for the sum –2
x – 6 +4
6– x ?
20)
21)
What property of real numbers justifies the method of multiplying by the LCD of the
parts? Why?
21)
22)
Explain how to add rational expressions with different denominators.
22)
23)
Why is 2x + 6
x + 6 not equal to 2 ?
23)
4
24)
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?).
24)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Simplify the complex fraction.
25)
9+3
x
x
4+1
12
25)
A)
36
x
B)
1
C)
x
36
D)
36
Solve the equation.
26)
x
16 +3
8=x – 4
8
26)
A)
{10 }
B)
{14 }
C)
{11 }
D)
{7 }
Multiply. Write the answer in lowest terms.
27)
3z3
5·20
z2
27)
A)
12z2
z3
B)
z
12
C)
12z
D)
12
z
Rewrite the rational expression with the given denominator.
28)
4
5x = ?
20x2
28)
A)
16x
20x2
B)
4x2
20x2
C)
16
20x2
D)
4
20x2
Add or subtract. Write the answer in lowest terms.
29)
x
x –6+12
x+6–72
x2–36
29)
A)
1
B)
x +24
x2–36
C)
x +24
x +6
D)
x –24
x –6
Identify as an equation or an expression.
30)
–3x +8=4x +5
6
30)
A)
Equation
B)
Expression
Add or subtract. Write the answer in lowest terms.
31)
3
1– y –1
y –1
31)
A)
3
1– y
B)
2
1– y
C)
–2
1– y
D)
4
1– y
Solve the problem.
32)
The sum of an integer and its reciprocal is 82
9. Find the integer.
32)
A)
8
B)
82
C)
9
D)
10
Find all values that make the expression undefined.
33)
8q – 4
q2+9
33)
A)
Never undefined
B)
–3
C)
0
D)
3, –3
A
Write the rational expression in lowest terms.
34)
y2– 5y – 14
y2– 7y – 18
34)
A)
–y2– 5y – 14
y2– 7y – 18
B)
–5y – 7
–7y – 9
C)
y – 7
y – 9
D)
–5y – 14
–7y – 18
C
Solve the formula for the specified variable.
35)
3x +5
z=2
y for z
35)
A)
z =5y
2– 3x
B)
z =2– 3xy
5y
C)
z =
–5y
3x – 2
D)
z =
–5y
3xy – 2
D
7
C
Find the LCD for the fractions in the list.
36)
9
x2– 9x + 20 , 3
–4x + 20
36)
A)
–4(x + 4)(x + 5)
B)
–4(x + 4)(x – 5)
C)
–4(x – 4)(x + 5)
D)
–4(x – 4)(x – 5)
Solve the equation.
37)
x
x –2–5=2
x –2
37)
A)
{2 }
B)
{–2 }
C)
–1
2
D)
Find the numerical value of the rational expression for the given value of x.
38)
7x + 2
7 x2+ 5 x + 2 , x = – 4
38)
A)
15
47
B)
–13
47
C)
–15
47
D)
–1
5
Provide an appropriate response.
39)
True or false? The LCD of the two fractions 1
b and 1
e is be if the greatest common factor of b and
e is not 1.
39)
A)
True
B)
False
Add or subtract. Write the answer in lowest terms.
40)
x
x2– 16
–5
x2+ 5x + 4
40)
A)
x2– 4x + 20
(x – 4)(x + 4)
B)
x2– 4
(x – 4)(x + 4)(x +1)
C)
x2– 4x + 20
(x – 4)(x + 4)(x + 1)
D)
x2+ 4x + 20
(x – 4)(x + 4)(x + 1)
Multiply. Write the answer in lowest terms.
41)
3x2
5·15
x3
41)
A)
45x2
5x3
B)
9x2
x3
C)
9
x
D)
x
9
Simplify the complex fraction.
42)
x5
9y8
x3
y4
42)
A)
x8
9y12
B)
x2
9y4
C)
x2
y4
D)
x2
9y12
Solve the equation.
43)
2x
9+ 5 =1
8
43)
A)
–359
16
B)
–351
16
C)
375
16
D)
9
2
Rewrite the rational expression with the indicated denominator.
44)
9
4m =?
28m
44)
A)
9
28m
B)
63
28m
C)
7
28m
D)
4
28m
Multiply or divide. Write the answer in lowest terms.
45)
4(p – 1)
p÷9(p – 1)
5p2
45)
A)
9
20p
B)
20p
9
C)
20p3– 20p2
9p2– 9p
D)
36p2+ 72p + 36
5p3
Solve the problem.
46)
Tom Quig traveled 270 mi east of St. Louis. For most of the trip he averaged 70 mph, but for one
period of time he was slowed to 20 mph due to a major accident. If the total time of travel was 6 hr,
how many miles did he drive at the reduced speed?
46)
A)
55 mi
B)
70 mi
C)
80 mi
D)
60 mi
Write the rational expression in lowest terms.
47)
z3+8
z3–2z2+4z
47)
A)
z –2
2
B)
z +2
z
C)
z –2
z
D)
z +4
z
Simplify the expression, using only positive exponents in your answer.
48)
x–2–25y–2
9y –45x
48)
A)
x +5y
9x2y
B)
9x –45y2
C)
y +5x
9x2y2
D)
y2+5x
9x2
Solve the equation.
49)
7
m +4–6
m –4=
–56
m2–16
49)
A)
{7}
B)
{–4}
C)
{4}
D)
Add or subtract. Write the answer in lowest terms.
50)
8
2r2– 3rs – 2s2–5
6r2– 13rs + 2s2+5
12r2 + 4rs – s2
50)
A)
43r – 13s
(2r + s)(r – 2s)(6r – s)
B)
63r – 23s
(2r + s)(r – 2s)(6r – s)
C)
43r – 23s
(2r + s)(r – 2s)(6r – s)
D)
63r – 13s
(2r + s)(r – 2s)(6r – s)
Solve the formula for the specified variable.
51)
P =A
1 + rt for r
51)
A)
r =A – P
Pt
B)
r =P – A
1 + t
C)
r =P – 1
At
D)
r = P – tA
Find the LCD for the fractions in the list.
52)
3
25r9s , 3
20r3s4
52)
A)
100r3s4
B)
100r9s
C)
100r12s5
D)
100r9s4
Add or subtract. Write the answer in lowest terms.
53)
8
2x –1+6
1–2 x
53)
A)
–2
2x –1
B)
14
2x –1
C)
2
2x –1
D)
–14
2x –1
54)
9y2
y – 1 +
–9y
y – 1
54)
A)
9y(y + 1)
y – 1
B)
9y
C)
9y
y – 1
D)
0
Multiply or divide as indicated. Write the answer in lowest terms.
55)
9s2+ 6st + t2
2s2– 7st – 4t2·5s2– 21st + 4t2
t2+ 2st – 3s2÷15s2+ 2st – t2
2s2+ 3st + t2
55)
A)
t + s
t – s
B)
(t + 3s)
(t + s)(t – s)
C)
1
D)
(t + 3s)2(5s – t)2
(4s + t)2(t2– s2)
Solve the problem.
56)
A jet plane traveling at a constant speed goes 1200 mi with the wind, then turns around and travels
for 1000 mi against the wind. If the speed of the wind is 50 mph and the total flight took 4 hr, find
the speed of the plane.
56)
A)
550 mph
B)
435 mph
C)
525 mph
D)
605 mph
A
Multiply or divide as indicated. Write the answer in lowest terms.
57)
b3–8a3
4a3+4a2b+ ab2÷4a2+2ab +b2
–a3– ab3
57)
A)
–a2–b3
2a + b
B)
–(b –2a)(a2+b3)
(2a + b)2
C)
b3–a2
2a + b
D)
–(b –2a)
(2a + b)2(a2+b3)
B
Write the rational expression in lowest terms.
58)
a2–36
a2+9a +18
58)
A)
a –6
a +3
B)
a +6
a –3
C)
a +6
a +3
D)
a –6
a –3
A
A
Add or subtract. Write the answer in lowest terms.
59)
1
5– m +4
m –5
59)
A)
4
5– m
B)
5
5– m
C)
3
5– m
D)
–3
5– m
Solve the equation.
60)
y
y2+ 11y +28
+y
y2– 16
=y
y2+ 3y – 28
60)
A)
{–8, 0}
B)
{8 , 0}
C)
{1 , 0}
D)
{–1, 0}
Find the LCD for the fractions in the list.
61)
–9
r2+4r , –11
5r +20 , –10
r2+8r +16
61)
A)
5r(r +4)2
B)
5(r +4)
C)
5(r +4)2
D)
r(r +4)2
Provide an appropriate response.
62)
True or false? If a fraction with denominator x – 2 must be written as an equivalent fraction with
denominator (x – 2 )3, then the original fraction must be multiplied by (x + 2 )2 in both the
numerator and the denominator.
62)
A)
True
B)
False
Solve the problem.
63)
Frank can type a report in 6 hr. James can type it in 5 hr . How long will it take the two of them
working together?
63)
A)
1
11 hr
B)
30 hr
C)
28
11 hr
D)
1
30 hr
Write the rational expression in lowest terms.
64)
a2–n2
n–a
64)
A)
–a+n
B)
–1
C)
a+n
D)
–a–n
Rewrite the rational expression with the indicated denominator.
65)
17(x + 4)
x(x – 2)=?
x3+ 6x2– 16x
65)
A)
17(x + 4)(x + 8)
x3+ 6x2– 16x
B)
17(x + 8)
x3+ 6x2– 16x
C)
17(x + 4)(x – 8)
x3+ 6x2– 16x
D)
(x + 4)(x + 8)
x3+ 6x2– 16x
Solve the problem.
66)
The difference of the reciprocals of two consecutive integers is 1
72 . Find the integers.
66)
A)
9 and 10 or –9 and –10
B)
36 and 37 or –36 and –37
C)
7 and 8 or –7 and –8
D)
8 and 9 or –8 and –9
67)
A water tank can be filled in 5 min and emptied in 10 min. If the drain is accidentally left open
when the tank is being filled, how long does it take to fill the tank?
67)
A)
10 min
B)
31
3 min
C)
1
50 min
D)
1
15 min
Simplify the expression, using only positive exponents in your answer.
68)
m–1+ z–1
m–1– z–1
68)
A)
z + m
m
B)
z – m
z
C)
z + m
z
D)
z + m
z – m
Solve the equation.
69)
–5 x
2x +12 =2 x
4x +24 +6x –4
x +6
69)
A)
1
36
B)
4
9
C)
{16}
D)
–4
9
Find all values that make the expression undefined.
70)
x2– 36
x2+ 8x + 15
70)
A)
6, –6
B)
0
C)
5, –3
D)
–5, –3
Find the LCD for the fractions in the list.
71)
13
k2+ 8k+ 15, 5
k2– 6k– 27 , 12
k2+ 6k+ 9
71)
A)
(k + 5)(k + 3)(k – 9)
B)
(k + 5)(k – 9)(k + 3)
C)
(k + 5)(k + 3)(k – 9)(k + 3)
D)
(k + 5)(k + 3)3(k – 9)(k + 3)
Find all values for which at least one denominator is equal to 0. Write answers using the symbol . Do not solve.
72)
13
14 x – 5 –1
x=1
9x – 10
72)
A)
x 5
14 , 10
9, –13
B)
x 5
14 , 10
9
C)
x 0, 5
14 , 10
9, 13
D)
x 0, 5
14 , 10
9
D
Add or subtract. Write the answer in lowest terms.
73)
–7
20 –3
5x
73)
A)
–10
20 –5x
B)
7x –12
20x
C)
–7x –12
20x
D)
–7x +12
20x
C
Find all values that make the expression undefined.
74)
x– 2
8– x
74)
A)
Never undefined
B)
8
C)
8, 2
D)
–8
B
C
Write the rational expression in lowest terms.
75)
a2–b2–3a +3b
2a +2b –6
75)
A)
b –a
2
B)
a2–b2
6
C)
a +b
6
D)
a –b
2
Multiply or divide. Write the answer in lowest terms.
76)
z2+ 8z + 15
z2+ 10z + 21
÷z2+ 5z
z2– 2z – 63
76)
A)
z
z2+ 10z + 21
B)
z – 9
z2+ 7z
C)
z – 9
z
D)
z – 9
Add or subtract. Write the answer in lowest terms.
77)
–2x + 9
x+
–6x + 4
4x
77)
A)
–14x + 40
4x2
B)
–2x + 40
4x
C)
–14x + 40
4x
D)
–14x – 32
4x
Find all values that make the expression undefined.
78)
9y – 8
y2– 4
78)
A)
2, –2
B)
2
C)
8
9
D)
4
18
79)
b –5
5b –15
79)
A)
15
B)
0
C)
3
D)
–5
Multiply. Write the answer in lowest terms.
80)
4p – 4
p·4p2
5p – 5
80)
A)
20p2+ 40p + 20
4p3
B)
5
16p
C)
16p
5
D)
16p3– 16p2
5p2– 5p
Solve the problem.
81)
The area of the rectangle is represented by y2+ 2y – 35. What is the length?
y + 7
81)
A)
y – 7
B)
y + 7
C)
y + 5
D)
y – 5
Add or subtract. Write the answer in lowest terms.
82)
5
r+9
r + 6
82)
A)
14r + 30
r(–6– r)
B)
–30r – 14
r(r + 6)
C)
–30r – 14
r(–6– r)
D)
14r + 30
r(r + 6)
83)
2
r+5
r – 9
83)
A)
7r – 18
r(9– r)
B)
7r – 18
r(r – 9)
C)
18r – 7
r(r – 9)
D)
18r – 7
r(9– r)
Solve the equation.
84)
1
m – 4 –6
m + 4 =3
m2– 16
84)
A)
{5, –5}
B)
{5}
C)
{–5}
D)
B
Find all values that make the expression undefined.
85)
–3
4z
85)
A)
0
B)
4
C)
Never undefined
D)
0.25
A
Solve the problem.
86)
The winner of the 1998 Peoria 500 (mile) race was Serge Bologna, with an average rate of 175.786
mi per hr. What was his time (to the nearest thousandth of an hour)?
86)
A)
2.919 hr
B)
2.844 hr
C)
12.844 hr
D)
0.786 hr
B
B