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This equation is false no matter what value the variable x might have. Thus, there is no solution. The solution set is { } or ∅.
9. 5 2 3 5( 1)5 2 3 5 55 2 2 5
5 2 2 2 5 25 5
5 5 5 50 0
x xx xx x
x xx x
x x x x
− = + −− = + −− = − +
− + = − + +=
− = −=
Since 0 = 0 is a true statement for every value of x, all real numbers are solutions. The solution set is the set of all real numbers or {x|x is a real number}.
Intermediate Algebra 7th Edition Martin Gay Solutions ManualFull Download: http://testbanklive.com/download/intermediate-algebra-7th-edition-martin-gay-solutions-manual/
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1. Equations with the same solution set are called equivalent equations.
2. A value for the variable in an equation that makes the equation a true statement is called a solution of the equation.
3. By the addition property of equality, y = −3 and y − 7 = −3 − 7 are equivalent equations.
4. By the multiplication property of equality,
2y = −3 and 2 3
2 2
y −= are equivalent equations.
5. 1
53
x − expression
6. 2(x − 3) = 7 equation
7. 5 1 2
9 3 9x x+ = − equation
8. 5 1 2
9 3 9x x+ − − expression
9. The addition property of equality allows us to add the same number to (or subtract the same number from) both sides of an equation and have an equivalent equation. The multiplication property of equality allows us to multiply (or divide) both sides of an equation by the same nonzero number and have an equivalent equation.
10. distributive property
11. to make the calculations less tedious
12. When solving a linear equation and all variable terms subtract out and:
a. you have a true statement, then the equation has all real numbers for which the equation is defined as solutions.
b. you have a false statement, then the equation has no solution.
Exercise Set 2.1
2. 2 182 18
2 29
xx
x
− =− =− −
= −
Check: 2 182( 9) 18
18 18 True
x− =− −
=�
The solution is −9.
4. 25 3025 30 30 30
55
yyy
− = +− − = + −
− =
Check: 25 3025 55 3025 25 True
y− = +− − +− = −
�
The solution is −55.
6. 8.6 6.38.6 8.6 6.3 8.6
2.3
yy
y
− = −− + = − +
=
Check: 8.6 6.32.3 8.6 6.3
6.3 6.3 True
y − = −− −− = −
�
The solution is 2.3.
8. 5 3 11 35 3 11 3
2 142 14
2 27
y yy y
yy
y
− = +− = +
=
=
=
Check: 5 3 11 35(7) 3 11 3(7)
35 3 11 2132 32 True
y y− = +− +− +
=
�
�
The solution is 7.
10. − = −− − = − −
− = −− −=− −
=
10.3 6 2.310.3 6 10.3 2.3 10.3
6 12.66 12.6
6 62.1
xx
xx
x
Check: − = −− −
− −− = −
10.3 6 2.310.3 6(2.1) 2.3
10.3 12.6 2.32.3 2.3 True
x�
�
The solution is 2.1.
Chapter 2: Equations, Inequalities, and Problem Solving ISM: Intermediate Algebra
70. The difference means to subtract. The difference of 8 and a number: 8 − x
72. Two more than three times a number: 3x + 2
74. −3(−4) = 12 not −12; 3( 4) 103 12 10
3 23 2
3 32
3
xx
xx
x
− − =− + =
− = −− −=− −
=
76. 3 7 21 not 7;3
⎛ ⎞+ = + +⎜ ⎟⎝ ⎠
xx x
57
3 35
3 7 33 3
21 521 421 4
4 421
4
x x
x x
x xxx
x
+ =
⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
+ ==
=
=
78. 5x − 3 = 5x − 3 Since the two sides of the equation are identical, the equation is true for any value of x. All real numbers are solutions.
80. 5x − 2 = 5x − 7 Subtracting 2 from a number and subtracting 7 from the same number will not result in equal numbers for any value of x. There is no solution.
82. answers may vary
84. answers may vary
86. 7.6 10 1.1 127.6 1.1 22
y yy y
− − = − +− = − +
From this we see that K = 22.
88. 46 3
6 4 66 3
24 2
x x
x x
x x
+ =
⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
+ =
From this we see that K = 24.
90. answers may vary
ISM: Intermediate Algebra Chapter 2: Equations, Inequalities, and Problem Solving
2. If x = number of passengers at Los Angeles International Airport, in millions, then x + 3.1 = passengers at Chicago’s O’Hare airport, and 2x − 31.9 = passengers at Atlanta’s Hartsfield-Jackson airport.
In words: passengers at Los Angeles
+ passengers at O’Hare
+ passengers at Hartsfield-Jackson
↓ ↓ ↓ ↓ ↓
Translate: x + (x + 3.1) + (2x − 31.9)
Then x + (x + 3.1) + (2x − 31.9) = x + x + 3.1 + 2x − 31.9 = 4x − 28.8.
3. Let x = the first number, then 3x − 8 = the second number, and 5x = the third number. The sum of the three numbers is 118.
(3 8) 5 1183 5 8 118
9 8 1189 126
14
x x xx x x
xxx
+ − + =+ + − =
− ===
The numbers are 14, 3x − 8 = 3(14) − 8 = 34, and 5x = 5(14) = 70.
4. Let x = the original price. Then 0.4x = the discount. The original price, minus the discount, is equal to $270. 0.4 2700.6 270
270450
0.6
x xx
x
− ==
= =
The original price was $450.
5. Let x = width, then 2x − 16 = length. The perimeter is 160 inches. 2( ) 2(2 16) 160
2 4 32 1606 32 160
6 19232
x xx x
xxx
+ − =+ − =
− ===
2x − 16 = 2(32) − 16 = 48 The width is 32 inches and the length is 48 inches.
6. Let x = first odd integer, then x + 2 = second odd integer, and x + 4 = third odd integer. The sum of the integers is 81.
( 2) ( 4) 813 6 81
3 7525
x x xx
xx
+ + + + =+ =
==
x + 2 = 27 x + 4 = 29 The integers are 25, 27, and 29.
Vocabulary, Readiness & Video Check 2.2
1. 130% of a number > the number.
2. 70% of a number < the number.
ISM: Intermediate Algebra Chapter 2: Equations, Inequalities, and Problem Solving
14. The original application asks you to find three numbers. The solution x = 45 only gives you the first number. You need to INTERPRET this result.
Exercise Set 2.2
2. The perimeter is the sum of the lengths of the four sides. ( 5) ( 5) 5 5
4 10x x x x x x x x
x+ − + + − = + + + − −
= −
4. Let x = first odd integer, then x + 2 = second odd integer, and x + 4 = third odd integer.
( 2) ( 4) 2 4 3 6x x x x x x x+ + + + = + + + + = +
6. Find the sum of y quarters worth 25¢ each, 7y dimes worth 10¢ each, and (2y − 1) nickels worth 5¢ each. 25 10(7 ) 5(2 1) 25 70 10 5
105 5y y y y y y
y+ + − = + + −
= −
The total amount is (105y − 5) cents.
8. 4x + 5(3x − 15) = 4x + 15x − 75 = 19x − 75
10. The length of the side denoted by ? is 18 − 10 = 8. Similarly, the length of the unmarked side is (x + 14) − (x + 8) = x + 14 − x − 8 = 6. The perimeter of the floor plan is 18 + (x + 8) + 10 + 6 + 8 + (x + 14) = 2x + 64
Chapter 2: Equations, Inequalities, and Problem Solving ISM: Intermediate Algebra
14. Let x = the first number, then x − 6 = the second number, and 2x = the third number.
( 6) 2 3064 6 306
4 31278
x x xx
xx
+ − + =− =
==
x − 6 = 72 2x = 156 The numbers are 78, 72, and 156.
16. 90% of 70 = 0.90 ⋅ 70 = 63 70 − 63 = 7 7 million acres are not federally owned.
18. 32.2% of 881 = 0.322 ⋅ 881 ≈ 284 Approximately 284 tornadoes occurred in the United States during June 2014.
20. Let x be the number of people employed in the restaurant industry. Then x is 10% of 147 million. x = 0.10(147 million) = 14.7 million There were 14.7 million people employed in the restaurant industry in the U.S. in 2014.
22. From the circle graph, 39% of time is spent on role-specific tasks. 39% of 47 = 0.39 ⋅ 47 ≈ 18.3 An average worker would spend 18.3 hours on role-specific tasks.
24. The percents in the circle graph sum to 100%. 39 2 19 100
3 58 1003 42
14
x xx
xx
+ + + =+ =
==
2x = 2(14) = 28 28% of an average worker’s time at work is spent on e-mail.
26. 3 ( 10) 1805 10 180
5 17034
x x xx
xx
+ + + =+ =
==
3x = 3(34) = 102 x + 10 = 34 + 10 = 44 The angles measure 34°, 44°, and 102°.
38. Let x be the decline in the number of travel agent
jobs (in hundreds). Then x − 17 is the decline in the number of reporter or correspondent jobs and 2x − 21 is the decline in the number of flight attendant jobs.
40. Let x be the number of seats in Gillette Stadium. Then x + 11,200 is the number of seats in AT&T Stadium and x − 3800 is the number of seats at CenturyLink Field.
11 200 3800 213 80011 200 3800 213 800
3 7400 213 8003 206 400
68 800
( , ) ( ) ,, ,
,,
,
x x xx x x
xxx
+ + + − =+ + + − =
+ ===
x + 11,200 = 68,800 + 11,200 = 80,000 x − 3800 = 68,800 − 3800 = 65,000 Gillette Stadium seats 68,800, AT&T Stadium seats 80,000, and CenturyLink Field seats 65,000.
42. Let x be the price of the textbook before tax. 0.09 158.601.09 158.60
145.50
x xxx
+ ==≈
The human anatomy book cost $145.50 before tax.
44. Let x be the population in 2004. This population, decreased by 1.96%, is the 2014 population of 80.9 million.
0 0196 80 90 9804 80 9
82 5
. .
. ..
x xxx
− ==≈
The population of Germany in 2004 was 82.5 million.
46. Let x be the size of the workforce prior to layoffs. 0.15 11,000
73,333xx
=≈
Prior to layoffs, Dana’s workforce was 73,333 people.
48. Let x = measure of complement; then 2x + 30 = measure of angle.
2 30 903 60
20
x xxx
+ + ===
2x + 30 = 2(20) + 30 = 70 The angles measure 20° and 70°.
50. Let x = base angle; then 3x − 10 = third angle. 2 3 10 180
5 10 1805 190
38
x xx
xx
+ − =− =
==
3x − 10 = 3 ⋅ 38 − 10 = 104 The angles measure 38°, 38°, and 104°.
52. Let x = length of side of pentagon, then x + 7 = length of side of square. 5 4( 7)5 4 28
28
x xx xx
= += +=
x + 7 = 28 + 7 = 35 The pentagon has a side length of 28 inches and the square has a side length of 35 inches.
54. Let x = first integer, then x + 1 = second integer, and x + 2 = third integer, and x + 3 = fourth integer. ( 1) ( 3) 110
2 4 1102 106
53
x xx
xx
+ + + =+ =
==
x + 1 = 54 x + 2 = 55 x + 3 = 56 The integers are 53, 54, 55, and 56.
Chapter 2: Equations, Inequalities, and Problem Solving ISM: Intermediate Algebra
56. Let x be the payroll for the Montreal Canadiens. Then x − 5,049,585 was the payroll for the San Jose Sharks.
5 049 585 129 215 7192 5 049 585 129 215 719
2 134 265 30467 132 652
( , , ) , ,, , , ,
, ,, ,
x xx
xx
+ − =− =
==
5 049 585 67 132 652 5 049 58562 083 067
, , , , , ,, ,
x − = −=
The 2014−2015 payroll for the Montreal Canadiens was $67,132,652 and the payroll for the San Jose Sharks was $62,083,067.
58. Let x be the number of passengers at Los Angeles International Airport, in millions. Then x + 3.1 is the number of passengers at Chicago’s O’Hare airport, and 2x − 31.9 is the number of passengers at Atlanta’s Hartsfield-Jackson airport.
3 1 2 31 9 2264 28 8 226
4 254 863 7
( . ) ( . ).
..
x x xx
xx
+ + + − =− =
==
x + 3.1 = 63.7 + 3.1 = 66.8 2x − 31.9 = 2(63.7) − 31.9 = 95.5 The numbers of passengers are: Los Angeles: 63.7 million; Chicago: 66.8 million; Atlanta: 95.5 million
60. ( 2) 2 (2 3) 1106 1 110
6 11118.5
x x x xx
xx
+ + + + − =− =
==
x + 2 = 18.5 + 2 = 20.5 2x = 2(18.5) = 37 2x − 3 = 2(18.5) − 3 = 34 The bases measure 18.5 meters and 37 meters, and the sides measure 20.5 meters and 34 meters.
62. Let x be the energy cost of an LED bulb. Then x + 26 is the energy cost of a CFL bulb, and 6x + 18 is the energy cost of an incandescent bulb.
26 6 18 4768 44 476
8 43254
( ) ( )x x xx
xx
+ + + + =+ =
==
x + 26 = 54 + 26 = 80 6x + 18 = 6(54) + 18 = 342 The energy costs are: LED bulb: $54 CFL bulb: $80 Incandescent bulb: $342
64. Let x be the number of medals won by the Netherlands. Then Canada won x + 1 medals and Norway won x + 2 medals.
1 2 753 3 75
3 7224
( ) ( )x x xx
xx
+ + + + =+ =
==
x + 1 = 24 + 1 = 25 x + 2 = 24 + 2 = 26 In the 2014 winter Olympics, the Netherlands won 24 medals, Canada won 25 medals, and Norway won 26 medals.
66. Let x = height, then 2x + 12 = length. 2( ) 2(2 12) 312
2 4 24 3126 24 312
6 28848
x xx x
xxx
+ + =+ + =
+ ===
2x + 12 = 2(48) + 12 = 108 The height is 48 inches and the length is 108 inches.
76. Let x° be the measure of an angle. Then its complement measures (90 − x)° and its supplement measures (180 − x)°. 180 2(90 ) 50180 180 2 50180 230 2180 230
50
x xx xx xxx
− = − +− = − +− = −+ =
=
The angle measures 50°.
78. y = −80.6x + 2054 y = −80.6(17) + 2054 ≈ 684 The average number of cigarettes smoked by an American adult is predicted to be 684 in 2017.
ISM: Intermediate Algebra Chapter 2: Equations, Inequalities, and Problem Solving
This does not represent the average number of cigarettes smoked by an American smoker, because it is the average for all Americans, both smokers and non-smokers.
82. Let x be the first odd integer. Then x + 2 is the next consecutive odd integer. 7 5( 2) 547 5 10 547 5 642 64
32
x xx xx xxx
= + += + += +==
No such odd integers exist.
84. 60 50 500010 5000
500
R Cx xxx
== +==
50 5000 50(500) 500025,000 500030,000
x + = += +=
500 computer boards must be sold to break even. It costs $30,000 to produce the 500 boards.
86. The company makes a profit if it makes and sells more products than the break-even number.
Section 2.3 Practice Exercises
1.
or
=
=
= =
I PRTI PRT
PR PRI I
T TPR PR
2. 7 2 57 2 7 5 7
2 5 72 5 7
2 27 5
2 2
x yx y x x
y xy x
y x
− =− − = −
− = −− −=− −
= −
3.
or
A P PrtA P = P+ Prt PA P = PrtA P Prt
Pt PtA P A P
r rPt Pt
= +− −−− =
− −= =
4. Let P = 8000, r = 6% = 0.06, t = 4, n = 2.
⋅
⎛ ⎞= +⎜ ⎟⎝ ⎠
⎛ ⎞= +⎜ ⎟⎝ ⎠
=≈≈
2 4
8
1
0.068000 1
2
8000(1.03)8000(1.266770081)10,134.16
ntr
A Pn
A
AAA
Russ will have $10,134.16 in his account.
5. Let d = 190 and r = 7.5.
190 7.5190 7.5
7.5 7.51
253
==
=
=
d rttt
t
They spent 1
253
hours cycling, or 25 hours
20 minutes.
Vocabulary, Readiness & Video Check 2.3
1. 2 55 2
x yy x
+ == −
2. 7 33 7
3 7 or 7 3
x yy xy x y x
− =− = −
= − + = −
3. 5 85 8
a ba b
− == +
4. 7 1010 7
r ss r
+ == −
5. 5 65 6
5 6
j k hj k h
k h j
+ − =+ = +
= − +
Chapter 2: Equations, Inequalities, and Problem Solving ISM: Intermediate Algebra
34. The total area of the ceiling is 18(12) = 216 square feet. Each package can cover up to 50 square feet. Thus, the number of
packages needed is 216
4.32.50
= Therefore,
5 packages must be purchased.
36. Using the formula 1 ,nt
rA P
n⎛ ⎞= +⎜ ⎟⎝ ⎠
we have
⋅⎛ ⎞= +⎜ ⎟⎝ ⎠
=≈≈
2 3
6
0.0554000 1
2
4000(1.0275)4000(1.176768361)4707.07
A
AAA
Yes, the amount is enough.
38. Note that the wall covers 21 ⋅ 8 = 168 square feet. Because we wish to paint three coats, we actually must cover a total of
168 ⋅ 3 = 504 square feet. Since each gallon covers 300 square feet, we need 504
1.68 gallons300
= of paint. 2 gallons should be
purchased.
40. 2
2825 5825 25
825 2533
( )
V r h
hh
hh
= ππ = ππ = π
==
The height is 33 mm.
42. a. 34
3;V r= π
189
2 2
dr = = =
349
34
7293972
( )
( )
V
V
V
= π
= π
= π
The volume is 972π cubic cm.
b. V = 972π ≈ 3053.63 cubic cm
44. a. 2
24 15753 98
( ) ( ).
V r h
VV
= π= π≈
The volume of the cylinder is 753.98 cubic millimeters.
b. 3
3
4
34
43268 08
( )
.
V r
V
V
= π
= π
≈
The volume of the sphere is 268.08 cubic millimeters.
c. V = 753.98 + 268.08 = 1022.06 The volume of the vitamin is 1022.06 cubic millimeters.
46. Note that the radius of the circle is equal to 22,248 + 4000 = 26,248. C = 2πr C = 2π(26,248) C = 52,496π C ≈ 164,921.0479 The “length” of the Clarke belt is approximately 164,921 miles.
ISM: Intermediate Algebra Chapter 2: Equations, Inequalities, and Problem Solving
5. The set {x|x ≥ −0.4} written in interval notation is [−0.4, ∞).
6. The set {x|x < −0.4} written in interval notation is (−∞, −0.4).
7. The set {x|x ≤ −0.4} written in interval notation is (−∞, −0.4].
8. The set {x|x > −0.4} written in interval notation is (−0.4, ∞).
9. The graph of Example 1 is shaded from −∞ to, but not including, −3, as indicated by a parenthesis. To write interval notation, write down what is shaded for the inequality from left to right. A parenthesis is always used with −∞, so from the graph, the interval notation is (−∞, −3).
10. We can add the same number to (or subtract the same number from) both sides of a linear inequality in one variable and have an equivalent inequality; addition property of equality.
11. If you multiply or divide both sides of an inequality by the same nonzero negative number, you must reverse the direction of the inequality symbol.
12. maximum, or less
Exercise Set 2.4
2. {x|x > 5} (5, ∞)
5(
4. {x|x < −0.2} (−∞, −0.2)
–0.2)
6. {x|−7 ≥ x} (−∞, −7]
–7
8. {x|−5 ≤ x ≤ −1} [−5, −1]
–1–5
10. {x|−3 > x ≥ −7} [−7, −3)
–3)
–7
12. 2 13
xx
+ ≤ −≤ −
(−∞, −3]
–3
14. 11 10 55
x xx
< +<
(−∞, 5)
5)
16. 7 1 6 10
x xx
− ≥ −≥
[0, ∞)
0
18. 5
56
6 5 65
5 6 56
x
x
x
≥
⋅ ≥ ⋅
≥
[6, ∞)
6
20. 4 11.22.8
xx
> −> −
(−2.8, ∞)
–2.8(
22. 4 84 8
4 42
xx
x
− ≥− ≤− −
≤ −
(−∞, −2]
–2
24. 8 5 235 15
3
xxx
− ≤− ≤
≥ −
[−3, ∞)
ISM: Intermediate Algebra Chapter 2: Equations, Inequalities, and Problem Solving
70. a. Let x be Holden’s time on his last trial. 6.85 7.04 6.92
74
6.85 7.04 6.924 4(7)
46.85 7.04 6.92 28
20.81 20.81 28 20.817.19
x
x
xx
x
+ + + <
+ + +⎛ ⎞ <⎜ ⎟⎝ ⎠
+ + + <+ − < −
<
The solution is {x|x < 7.19}.
b. A time of 7.19 minutes or less will result in an average time under 7.0 minutes.
72. a. Let x be the number of additional ouces. 98 21 300
21 202 approximately 9.6
xxx
+ ≤≤≤
The solution is {x|x ≤ 9.6}.
b. Since x represents the number of ounces after the first ounce, you can mail at most 1 ounce plus 9 additional ounces, or 10 ounces.
74. a. Let x be the number of additional half-hour intervals parked. 1 0 0 6 4
10 6 406 30
5
. . xxxx
+ ≤+ ≤
≤≤
The solution is {x|x ≤ 5}.
b. Since x represents the number of half hours after the first hour, you can park for at most 1 hour plus 5 additional half hours, or 1 + 2.5 = 3.5 hours total.
76. a. Let n = number of calls made in a given month.
25 13 0.0612 0.06
200
< +<<
nn
n
{n|n > 200}
b. Plan 1 is more economical than Plan 2 when 200 or more calls are made.
78. Given that F ≥ 977, we know the following: 5
( 32)95
(977 32)95
(945)9525
C F
C
C
C
≥ −
≥ −
≥
≥
{C|C ≥ 525} So stibnite melts when the temperature is at least 525°C.
80. a. 11 8 390 5011 8 340
approximately 28.8
..
ttt
− + <− < −
>
2004 + 28.8 = 2032.8 The consumption will be less than 50 billion during the year 2032 and after.
b. answers may vary
82. Consumption of skim milk is decreasing over time; answers may vary.
84. 2024 is 20 years after 2004, so 2024 corresponds to t = 20. s = −0.22t + 27.4 s = −0.22(20) + 27.4 = −4.4 + 27.4 = 23 The average consumption of skim milk is predicted to be 23 pounds per person per year in 2024.
86. answers may vary
88. answers may vary
90. answers may vary
92. x ≥ 0 and x ≤ 7 The integers are 0, 1, 2, 3, 4, 5, 6, 7.
94. x < 6 and x < −5 The integers are −6, −7, −8, ....
96. 3 12 33 12 12 3 12
3 153 15
3 35
xx
xx
x
− =− + = +
=
=
=
Chapter 2: Equations, Inequalities, and Problem Solving ISM: Intermediate Algebra
6. A = {1, 3, 5, 7, 9} and B = {2, 3, 4, 5, 6}. The numbers that are in either set or both sets are {1, 2, 3, 4, 5, 6, 7, 9}. This set is the union, A ∪ B.
7. 8 5 8 1 28 3 3
33
8
+ ≤ − ≥≤ ≥
≤ ≥
x or xx or x
x or x
3 3, ,
8 8x x
⎧ ⎫ ⎛ ⎤≤ −∞⎨ ⎬ ⎜ ⎥⎩ ⎭ ⎝ ⎦
38
{x|x ≥ 3}, [3, ∞)
3 3 3
or 3 , [3, )8 8
x x x⎧ ⎫ ⎛ ⎤≤ ≥ = −∞ ∪ ∞⎨ ⎬ ⎜ ⎥⎩ ⎭ ⎝ ⎦
338
The solution set is 3
, [3, ).8
⎛ ⎤−∞ ∪ ∞⎜ ⎥⎝ ⎦
8. 3 2 8 5 03 6 0
2 0
− − > − >− > − >
< >
x or xx or xx or x
{x|x < 2}, (−∞, 2)
2)
{x|x > 0}, (0, ∞)
0(
{x|x < 2 or x > 0}, (−∞, ∞)
The solution set is (−∞, ∞).
Vocabulary, Readiness & Video Check 2.5
1. Two inequalities joined by the words “and” or “or” are called compound inequalities.
2. The word and means intersection.
3. The word or means union.
4. The symbol ∩ means intersection.
5. The symbol ∪ represents union.
6. The symbol ∅ is the empty set.
7. For an element to be in the intersection of sets A and B, the element must be in set A and in set B.
8. Graph the two intervals, each on its own number line, so you can see their intersection. Graph this intersection on the third number line⎯this intersection is the solution set.
9. For an element to be in the union of sets A and B, the element must be in set A or in set B.
10. Graph the two intervals, each on its own number line, so you can see their union. Graph this union on the third number line⎯this union is the solution set.
Exercise Set 2.5
2. C ∩ D = {4, 5}
4. A ∪ D = {x|x is an even integer or x = 5 or x = 7}
6. A ∩ B = ∅
8. B ∪ D = {x|x is an odd integer or x = 4 or x = 6}
10. B ∩ C = {3, 5}
12. A ∪ C = {x|x is an even integer or x = 3 or x = 5}
14. x ≤ 0 and x ≥ −2 −2 ≤ x ≤ 0 [−2, 0]
–2 0
Chapter 2: Equations, Inequalities, and Problem Solving ISM: Intermediate Algebra
82. From the graph, we see that the number of single-family housing starts were less than 500 or the number of single-family housing completions greater than 1500 are for the years 2004, 2005, 2006, 2009, 2010, and 2011.
84. answers may vary
86. x + 3 < 2x + 1 < 4x + 6 3 2 1 2 1 4 62 5 2
52
25
22
+ < + + < +< − <
> − <
> > −
x x and x xx and x
x and x
x and x
(2, ∞)
88. 7x − 1 ≤ 7 + 5x ≤ 3(1 + 2x) 7 1 7 5 7 5 3 6
2 8 44 4
− ≤ + + ≤ +≤ ≤≤ ≥
x x and x xx and xx and x
{4}
90. 1 + 2x < 3(2 + x) < 1 + 4x 1 2 6 3 6 3 1 4
5 55 5
+ < + + < +− < <
> − >
x x and x xx and x
x and x
(5, ∞)
92. 10 185
10 ( 32) 189
9 9 5 9( 10) ( 32) (18)
5 5 9 5162
18 325
14 64.4
C
F
F
F
F
− ≤ ≤
− ≤ − ≤
⎛ ⎞− ≤ − ≤⎜ ⎟⎝ ⎠
− ≤ − ≤
≤ ≤
14° ≤ F ≤ 64.4°
94. Let x be Wendy’s grade on the final exam. 1
80 (2 80 90 82 75) 896
480 2 327 534153 2 207
76.5 103.576.5 100
x
xx
xx
≤ + + + + ≤
≤ + ≤≤ ≤≤ ≤≤ ≤
If Wendy scores between 76.5 and 100 inclusive on her final exam, she will receive a B in the course.
Section 2.6 Practice Exercises
1. |q| = 13 q = 13 or q = −13 The solution set is {−13, 13}.
2. |2x − 3| = 5 2 3 5 or 2 3 5
2 8 or 2 24 or 1
x xx xx x
− = − = −= = −= = −
The solution set is {−1, 4}.
3. 1 155
x + =
1 15 or 1 155 5
14 or 165 5
70 or 80
x x
x x
x x
+ = + = −
= = −
= = −
The solutions are −80 and 70.
4. 3 8 14
3 6
x
x
+ ==
3 6 or 3 62 or 2
x xx x
= = −= = −
The solutions are −2 and 2.
5. |z| = 0 The solution is 0.
ISM: Intermediate Algebra Chapter 2: Equations, Inequalities, and Problem Solving
The absolute value of any expression is never negative, so no solution exists. The solution set is ∅.
70. From the circle graph, mozzarella cheese had the highest U.S. production in 2014.
72. In 2014, cream cheese accounted for 7.6% of the total cheese production. 7.6% of 11,201,000,000 is 0.076(11,201,000,000) = 851,276,000 Therefore, 851,276,000 pounds of cream cheese was produced in the U.S. in 2014.
74. answers may vary
76. no solution
78. Since absolute value is never negative, the solution set is ∅.
80. All numbers whose distance from 0 is 2 units is written as |x| = 2.
82. answers may vary
84. |x − 7| = 2
86. answers may vary
88. |2x − 1| = 4
Chapter 2: Equations, Inequalities, and Problem Solving ISM: Intermediate Algebra
6. The left side of the inequality is an absolute value, which must be nonnegative⎯it must be 0 or positive. Therefore, there is no value of x that can make the value of this absolute value be less than the negative value on the right side of the inequality.
7. The solution set involves “or” and “or” means “union.”
Exercise Set 2.7
2. |x| < 6 −6 < x < 6 The solution set is (−6, 6).
( (
–6 6
4. |y − 7| ≤ 5 5 7 52 12
yy
− ≤ − ≤≤ ≤
The solution set is [2, 12].
2 12
6. |x + 4| < 6 6 4 6
10 2xx
− < + <− < <
The solution set is (−10, 2). ( (
–10 2
8. |5x − 3| ≤ 18 18 5 3 1815 5 21
213
5
xx
x
− ≤ − ≤− ≤ ≤
− ≤ ≤
The solution set is 21
3, .5
⎡ ⎤−⎢ ⎥⎣ ⎦
215
–3
10. 6 7
1
x
x
+ ≤≤
−1 ≤ x ≤ 1 The solution set is [−1, 1].
–1 1
12. |8x − 3| < −2 The absolute value of an expression is never negative, so no solution exists. The solution set is ∅.
0
14. 2 7 3
2 4
z
z
+ − < −+ <
4 2 44 2 2 2 4 2
6 2
zzz
− < + <− − < + − < −
− < <
The solution set is (−6, 2). ( (
–6 2
16. |y| ≥ 4 y ≤ −4 or y ≥ 4 The solution set is (−∞, −4] ∪ [4, ∞).
4–4
18. |x − 9| ≥ 2 9 2 or 9 2
7 or 11x x
x x− ≤ − − ≥
≤ ≥
The solution set is (−∞, 7] ∪ [11, ∞).
117
20. 1 3
4
x
x
− >>
x < −4 or x > 4 The solution set is (−∞, −4) ∪ (4, ∞).
((
–4 4
22. |4x − 11| > −1 An absolute value is always greater than a negative number. Thus, the answer is (−∞, ∞).
0
24. 10 3 1 2
10 3 1
x
x
+ + >+ >
10 3 1 or 10 3 13 11 or 3 9
11or 3
3
x xx x
x x
+ < − + >< − > −
< − > −
The solution set is 11
, ( 3, ).3
⎛ ⎞−∞ − ∪ − ∞⎜ ⎟⎝ ⎠
((
–3−113
Chapter 2: Equations, Inequalities, and Problem Solving ISM: Intermediate Algebra
89. Let x = number of tourists for France, then x + 9 = number of tourists for United States, and x + 44 = number of tourists for China.
( 9) ( 44) 3323 53 332
3 27993
x x xx
xx
+ + + + =+ =
==
x + 9 = 102 x + 44 = 137 China is predicted to have 137 million tourists, whereas the United States is predicted to have 102 million and France, 93 million.
90. d = rt or d
rt
=
11:00 a.m. to 1:15 p.m. is 2.25 hours. 130
582.25
r = ≈
His average speed was 58 mph.
91. 3box 8 5 3 120 in ,V lwh= = ⋅ ⋅ = while
2 2 3cyl 3 6 54 170 inV r h= π = π⋅ ⋅ = π ≈
Therefore, the cylinder holds more ice cream.
Chapter 2: Equations, Inequalities, and Problem Solving ISM: Intermediate Algebra
24. Let x be the number of new vehicles sold by Ford in 2010. The number of new vehicles sold is increased by 29.1%, or by 0.291x.
0 291 2 480 9421 291 2 480 942
1 922 000
. , ,
. , ,, ,
x xxx
+ ==≈
Ford sold approximately 1,922,000 new vehicles in 2010.
25. Recall that C = 2πr. Here C = 78.5. 78.5 2
78.5 39.25
2
r
r
= π
= =π π
Also recall that 2.A r= π 2 2 239.25 39.25 39.25
4913.14
A⎛ ⎞= π = ≈ ≈⎜ ⎟π π⎝ ⎠
The area of the pen is about 491 square feet. Each dog requires at least 60 square feet of
space, and 491
8.18.60
≈ At most 8 dogs could be
kept in the pen.
26. Let x be the number of people employed as registered nurses in 2012. The number of people employed in this field in 2022 is x increased by 19%.
0 19 3 240 0001 19 3 240 000
2 723 000
. , ,
. , ,, ,
x xxx
+ ==≈
In 2012, there were 2,723,000 registered nurses employed.
27. Use 1ntr
A Pn
⎛ ⎞= +⎜ ⎟⎝ ⎠
where P = 2500,
r = 3.5% = 0.035, t = 10, and n = 4. 410
40
0.0352500 1
4
2500(1.00875)$3542.27
A
AA
⋅⎛ ⎞= +⎜ ⎟⎝ ⎠
=≈
28. Let x be the amount of money international travelers spend in New York. Then x + 4 is the amount of money international travelers spend in California and 2x − 1 is the amount of money international travelers spend in Florida.
( 4) (2 1) 394 3 39
4 369
x x xx
xx
+ + + − =+ =
==
x + 4 = 9 + 4 = 13
2x − 1 = 2(9) − 1 = 18 − 1 = 17 International travelers spend $9 billion in New York, $13 billion in California, and $17 billion in Florida.
Chapter 2 Cumulative Review
1. a. {101, 102, 103, ...}
b. {2, 3, 4, 5}
2. a. {−2, −1, 0, 1, 2, 3, 4}
b. {4}
3. a. |3| = 3
b. 1 1
7 7− =
c. −|2.7| = −2.7
d. −|−8| = −8
e. |0| = 0
4. a. The opposite of 2
3 is
2.
3−
b. The opposite of −9 is 9.
c. The opposite of 1.5 is −1.5.
5. a. −3 + (−11) = −14
b. 3 + (−7) = −4
c. −10 + 15 = 5
d. −8.3 + (−1.9) = −10.2
e. 1 1 1 2 1
4 2 4 4 4− + = − + =
f. 2 3 14 9 5
3 7 21 21 21− + = − + = −
6. a. −2 − (−10) = −2 + 10 = 8
b. 1.7 − 8.9 = −7.2
c. 1 1 2 1 3
2 4 4 4 4− − = − − = −
Chapter 2: Equations, Inequalities, and Problem Solving ISM: Intermediate Algebra