GUIDED PRACTICE
✓
Concept Check ✓
Vocabulary Check
Skill Check
✓
1. Sample answers:
x < º2 or x > 4
(a compound
inequality because
it consists of two
inequalities
connected by or);
º12 ≤ x ≤ 9 (a
compound
inequality because
it consists of two
inequalities
connected by and).
3 APPLY
1. Write a compound inequality and explain why it is a compound inequality.
See margin.
2. Write a compound inequality that
⫺5 ⫺4 ⫺3 ⫺2 ⫺1
describes the graph at the right.
0
ASSIGNMENT GUIDE
BASIC
1
º4 ≤ x < 0
WRITING INEQUALITIES Write an inequality that represents the statement
and graph the inequality. 3, 4. See margin for graphs.
Day 1: pp. 349–351 Exs. 12–38
even, 39, 40, 45, 50, 62, 65,
Quiz 1 Exs. 1–15
3. x is less than 5 and greater than 2.
2<x<5
Solve the inequality.
AVERAGE
5. 7 < 4 + x < 8 3 < x < 4
4. x is greater than 3 or less than –1.
x > 3 or x < º1
Day 1: pp. 349–351 Exs. 12–38
even, 39, 40, 45, 50, 62, 65,
Quiz 1 Exs. 1–15
6. º1 < 2x + 3 ≤ 13 º2 < x ≤ 5
7. 6 < 4x º 2 ≤ 14 2 < x ≤ 4
ADVANCED
8. 4x º 1 > 7 or 5x º 1 < º6
x > 2 or x < º1
10. 4 ≤ º8 º x < 7
º15 < x ≤ º12
Day 1: pp. 349–351 Exs. 12–38
even, 39, 40, 45, 46–48,
50, 62, 65, Quiz 1 Exs. 1–15
9. 2x + 3 < º1 or 3x º 5 > º2
x < º2 or x > 1
11. In Example 6, write an inequality that describes the situation if you live
BLOCK SCHEDULE WITH 6.2
two miles from school and your friend lives one mile from the same school.
pp. 349–351 Exs. 12–38 even,
39, 40, 45, 50, 62, 65, Quiz 1
Exs. 1–15
1≤d≤3
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 802.
23. x is less than or equal
to º5 or greater than
or equal to 10.
24. x is greater than or
equal to º13 and less
than or equal to º3.
WRITING INEQUALITIES Write an inequality that represents the statement
and graph the inequality. 12–17. See margin for graphs.
12. x is less than 8 and greater than 3.
13. x is greater than 7 or less than 5.
3<x<8
x > 7 or x < 5
14. x is less than 5 and is at least 0.
15. x is less than 4 and is at least º9.
0≤x<5
º9 ≤ x < 4
16. x is less than º2 and is at least º4. 17. x is greater than º6 and less than º1.
º4 ≤ x < º2
º6 < x < º1
SOLVING INEQUALITIES Solve the inequality. Write a sentence that
describes the solution.
18. 6 < x º 6 ≤ 8 x is greater than 12
19. º5 < x º 3 < 6
x is greater than º2 and less than 9.
and less than or equal to 14.
20. º4 < 2 + x < 1
21. 8 ≤ 2x + 6 ≤ 18 x is greater than
x is greater than º6 and less than º1.
or equal to 1 and less than or equal to 6.
22. 6 + 2x > 20 or 8 + x ≤ 0
23. º3x º 7 ≥ 8 or º2x º 11 ≤ º31
See margin.
x is greater than 7 or less than or equal to º8.
24. º4 ≤ º3x º 13 ≤ 26
25. º13 ≤ 5 º 2x < 9 x is greater than
See margin.
º2 and less than or equal to 9.
GRAPHING INEQUALITIES Solve the inequality and graph the solution.
26–31. See margin for graphs.
26. 3 ≤ 2x < 7 }3} ≤ x < }7}
27. º25 < 5x < º20 º5 < x < º4
2
STUDENT HELP
HOMEWORK HELP
Example 1:
Example 2:
Example 3:
Example 4:
Example 5:
Example 6:
Exs. 12–17
Exs. 36–38
Exs. 18–35
Exs. 18–35
Exs. 18–35
Exs. 41–44
2
1
32. º4 < 4x º 8 < 12; x = 1
1 < x < 5; not a solution
34. º2x ≥ 6 or 2x + 1 > 5; x = 0
x ≤ º3 or x > 2; not a solution
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 14, 20, 24, 30, 34. See also
the Daily Homework Quiz:
• Blackline Master (Chapter 6
Resource Book, p. 51)
•
Transparency (p. 48)
2
29. º4 ≤ 9x º 1 < 5 º}3} ≤ x < }3}
30. 2x + 7 < 3 or 5x + 5 ≥ 10
31. 3x + 8 > 17 or 2x + 5 ≤ 7
x < º2 or x ≥ 1
x > 3 or x ≤ 1
CHECKING SOLUTIONS Solve the inequality and graph the solution. Then
check graphically whether the given x-value is a solution by graphing the
x-value on the same number line. 32–35. See margin for graphs.
28. º5 < 3x + 4 < 19 º3 < x < 5
EXERCISE LEVELS
Level A: Easier
12–17
Level B: More Difficult
18–38, 49–69
Level C: Most Difficult
39–48
COMMON ERROR
EXERCISES 15 AND 16 Some
students may have difficulty representing the expression at least.
Have them choose several points
from their graphs and verify that
they satisfy both conditions of the
inequality. Help students compare
and contrast at least, at most, not
more than, and not less than.
33. º1 < 7x º 15 ≤ 20; x = 5
2 < x ≤ 5; solution
35. 7 ≤ º2x + 21 < 31; x = 0
º5 < x ≤ 7; solution
6.3 Solving Compound Inequalities
12–17. See the next page.
26–35. See Additional Answers
beginning on page AA1.
349
3.
4.
⫺1
0
1
2
3
4
5
⫺2 ⫺1
0
1
2
3
4
349
FOCUS ON
APPLICATIONS
x > a and x < b ; x > b and x < a ; no
APPLICATION NOTE
EXERCISES 41–44 You may
want to discuss the logarithmic
scale used in the diagram. Explain
that the planets are so far apart that
it is otherwise difficult to show
them together in one picture.
13.
17.
2
4
6
8
1
2
3
4
5
⫺1
RE
The painting above
is Still Life with Apples by
Paul Cézanne. Another of
Cézanne’s paintings, Still Life
with Curtain, Pitcher and
Bowl of Fruit, sold at auction
for $60.5 million in 1999.
0
1
2
10 12
6
3
4
5
⫺10 ⫺8 ⫺6 ⫺4 ⫺2
0
2
7
⫺9
15.
16.
0
4
⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
⫺6 ⫺5 ⫺4 ⫺3 ⫺2 ⫺1
0
PRICES OF FINE ART In 1958, the painting Still Life with Apples by Paul
Cézanne sold at auction for $252,000. The painting was auctioned again in
1993 and sold for $28.6 million. Write a compound inequality that represents
the different values that the painting probably was worth between 1958 and
252,000 ≤ x ≤ 28,600,000
1993.
37.
TELEVISION ADVERTISING In 1967 a 60-second TV commercial during
the first Super Bowl cost $85,000. In 1998 advertisers paid $2.6 million for
60 seconds of commercial time (two 30-second spots). Write a compound
inequality that represents the different prices that 60 seconds of commercial
time during the Super Bowl probably cost between 1967 and 1998.
FINE ART
3
12.
14.
L
AL I
FE
MATHEMATICAL REASONING
To check students’ understanding
of and compound inequalities, have
them rewrite the inequalities
a < x < b and a > x > b each as
two simple inequalities joined by
and, with x on the left side of each
inequality. Ask them if for any
two real numbers a and b both
compound inequalities can be true.
36.
40. A solution is a number
that is on the graph of
either inequality.
Since every point that
is not on the graph of
one inequality is on
the graph of the other,
every real number is a
solution.
85,000 ≤ x ≤ 2,600,000
38.
ANTELOPES The table gives
the weights of some adult antelopes.
The eland is the largest antelope in
Africa. The royal antelope is the
smallest of all. Write a compound
inequality that represents the
different weights of these adult
antelopes. 7 ≤ x ≤ 2000
Antelope
Weight (lb)
Eland
2000
Kudu
700
Nyala
280
Springbok
95
Royal
7
39–40. Sample answers are given.
39. LOGICAL REASONING Explain why the inequality 3 < x < 1 has no
solution. A solution would be a number on the graph of both inequalities. Since the
graphs do not overlap, there are no such numbers.
40. LOGICAL REASONING Explain why the inequality x < 2 or x > 1 has every
real number as a solution.
SCIENCE
CONNECTION
Use the diagram of distances in our solar system.
41. Which compound inequality
best describes the distance d
(in miles) between the Sun and
any of the nine planets? A
7
10
A. 10 < d < 10
B. 106 < d < 1011
C. 108 < d < 1011
42. Write an inequality to describe
Jupiter
Neptune
an estimate of the distance d
(in miles) between the Sun
and Mercury. 107 < d < 108
Mercury
43. Write an inequality to describe
ADDITIONAL PRACTICE
AND RETEACHING
44. The Moon’s distance d
For Lesson 6.3:
• Practice Levels A, B, and C
(Chapter 6 Resource Book, p. 39)
• Reteaching with Practice
(Chapter 6 Resource Book, p. 42)
•
See Lesson 6.3 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
350
(in miles) from Earth varies
from about 220,000 miles to
about 250,000 miles. Write
an inequality to represent
this fact. 220,000 < d < 250,000
350
Chapter 6 Solving and Graphing Linear Inequalities
Uranus
10
an estimate of the distance d
(in miles) between the Sun and
Saturn. 108 < d < 109
Mars
00 mi
00,000,0
100,0
mi
0,000,000
10,00
00 mi
0,000,0
1,00
mi
000,000
100,
i
00,000 m
10,0
0,000 mi
1,00
,000 mi
100
0 0 mi
10,0
0 0 mi
1,0
mi
100
mi
Venus
Pluto
Saturn
Earth
Moon
Not drawn to scale
1011
1010
109
108
107
106
105
104
103
102
101
Sun
Test
Preparation
45. MULTI-STEP PROBLEM You have a friend Kiko who lives one mile from a
pool. Another friend D’evon lives 0.5 mile from Kiko. Kiko walks from her
home to the pool, where she meets D’evon. They both walk from the pool to
D’evon’s home and then to Kiko’s home.
The pool is on
this circle.
a. Write an inequality that describes
1
the values, d, of the possible
distances that D’evon could live
from the pool. }1} ≤ d ≤ 1}1}
2
1
2
2
D’evon lives
values, D, of the possible distances
somewhere on
this circle.
that Kiko could have walked.
(Assume that each part of the walk followed a straight path.) 2 ≤ d ≤ 3
★ Challenge
GEOMETRY CONNECTION Write a compound inequality that must be satisfied
by the length of the side labeled x. Use the fact that the sum of the lengths
of any two sides of a triangle is greater than the length of the third side.
46.
47.
x
1
4
34
x
2 < x < 10
3
4
1
4
4}} < x < 10 }}
–1 < x < 3; x is greater than –1
and less than 3.
3. –2x – 3 > 5 or 3x + 2 ≥ –1
Solve the inequality. Graph the
solution. Is the indicated value of
x a solution?
4. 3 ≤ 2x – 7 < 11; x = 9
x
1
72
6
x ≤ –4 or x ≥ 1
Solve the inequality. Write a sentence that describes the solution.
2. –11 < –3x – 2 < 1
x < –4 or x ≥ –1; x is less than
–4 or x is at least –1.
11.75
5.5
EXTRA CHALLENGE
www.mcdougallittell.com
48.
DAILY HOMEWORK QUIZ
Transparency Available
1. Write an inequality that represents the statement “x is at
most –4 or at least 1.”
Kiko’s home
b. Write an inequality that describes the
4 ASSESS
6.25 < x < 17.25
5 ≤ x < 9; no
MIXED REVIEW
5
0
50. 6.5a when a = 4 26
51. m º 20 when m = 30
10
x
52. ᎏᎏ when x = 30 2
53. 5x when x = 3.3 16.5 54. 4.2p when p = 4.1
15
17.22
SOLVING EQUATIONS Solve the equation. (Review 3.1, 3.2 for 6.4)
1
55. x + 17 = 9 º8 56. 8 = x + 2ᎏᎏ
2
1
59. ᎏᎏx = º6 º12 60. º3x = º27
2
1
5}} 57. x º 4 = 12 16 58. x º (º9) = 15 6
2
3
9 61. 4x = º28 º7 62. ºᎏᎏx = 21 º28
4
x
1
5
64. x º 4 = 16.7
65. ᎏᎏ = ºᎏᎏ 3
66. ᎏᎏx = º25 º30
º6
2
6
20.7
63. x + 3.2 = 11
7.8
67.
ICE SKATING An ice skating rink charges $4.75 for admission and skate
rental. If you bring your own skates, the admission is $3.25. You can buy a
pair of ice skates for $45. How many times must you go ice skating to justify
buying your own skates? (Review 3.4) at least 30 times
HIKING You are hiking a six-mile trail at a constant rate in Topanga
Canyon. You begin at 10 A.M. At noon you are two miles from the end of
the trail. (Review 5.5)
68. Write a linear equation that gives the distance d (in miles) from the end of the
trail in terms of time t. Let t represent the number of hours since 10 A.M.
69. Find the distance you are from the end of the trail at 11 A.M.
4 mi
d = 6 º 2t
6.3 Solving Compound Inequalities
4
6
8
10
12
5. 5x – 3 ≤ 6 or 7 < 4x – 9; x = 1.9
EVALUATING EXPRESSIONS Evaluate the expression. (Review 1.1)
49. x + 5 when x = 2 7
2
9
x ≤ 1.8 or x > 4; no
1.8
⫺1
0
1
2
3
4
5
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 6.3 are available in
blackline format in the Chapter 6
Resource Book, p. 47 and at
www.mcdougallittell.com.
ADDITIONAL TEST
PREPARATION
1. WRITING Write two different
sentences that describe the
compound inequality –4 ≤ x ≤ 6.
Sample sentences: x is greater
than or equal to –4 and less than
or equal to 6; x is at least –4 and
at most 6; x is between –4 and 6,
inclusive; x is not less than –4
and is not greater than 6.
351
351