328 Chapter 8 Inequalities
STATE
STANDARDS
MA.6.A.3.2
S
Writing and Graphing Inequalities
8.1
How can you use a number line to represent
solutions of an inequality?
Work with a partner.
a. Consider the statement “Your friend
is more than 3 minutes late.”
Circle each number that makes the
statement true.
3 2 1 0 1 2 3 4 5 6
●
Write four other numbers that make the statement true.
b. Consider the statement “The temperature
is at most 2 degrees.”
●
Can the temperature be exactly 2 degrees? Explain.
●
Circle each number that makes the statement true.
5 4 3 2 1 0 1 2 3 4
●
Write four other numbers that make the statement true.
c. Consider the statement “You need at least
4 pieces of paper for your math homew
ork.”
●
Can you have exactly 4 pieces of paper? Explain.
●
Circle each number that makes the
statement true.
3 2 1 0 1 2 3 4 5 6
●
Write four other numbers that make the statement true.
d. Consider the statement “After playing a video game
for 20 minutes, you have fewer than 6 points.”
●
Circle each number that makes the
statement true.
2 1 0 1 2 3 4 5 6 7
●
Write four other numbers that make the statement true.
ACTIVITY: Understanding Inequality Statements
1
1
45
6
m
e
6
7
English
Spanish
Section 8.1 Writing and Graphing Inequalities 329
Work with a partner.
a. Consider the statement “x is a number such that x
<
2.”
●
Can the number be exactly 2? Explain.
●
Circle each number that makes the statement true.
5 4 3 2 1 0 1 2 3 4
●
Write four other numbers that make the statement true.
b. Consider the statement “x is a number such that x ≥
1.”
●
Can the number be exactly 1? Explain.
●
Circle each number that makes the statement true.
5 4 3 2 1 0 1 2 3 4
●
Write four other numbers that make the statement true.
ACTIVITY: Understanding Inequality Symbols
2
2
Work with a partner.
a. Which number line shows x
>
0? Which number line shows x ≥ 0?
Explain your reasoning.
8
6
4
2 0
86
4
2
8
6
4
2 0
86
4
2
b. Write the smallest positive number you can think of that is still a
solution of the inequality x
>
0. Explain your reasoning.
ACTIVITY: How Close Can You Come to 0?
3
3
4. IN YOUR OWN WORDS How can you use a number line to represent
solutions of an inequality?
5. Write an inequality. Graph all solutions of your inequality on a
number line.
Use what you learned about graphing inequalities to complete
Exercises 17–20 on page 333.
English
Spanish
330 Chapter 8 Inequalities
Lesson
8.1
An inequality is a mathematical sentence that compares expressions.
It contains the symbols <, >, ≤, or ≥. To write an inequality, look for the
following phrases to determine where to place the inequality symbol.
Key Vocabulary
inequality, p. 330
solution of an
inequality, p. 331
solution set, p. 331
graph of an
inequality, p. 332
EXAMPLE
Writing Inequalities
1
1
Write the word sentence as an inequality.
a. A number c is less than 4.
A number c is less than 4.
c < 4
An inequality is c
<
4.
b. A number k plus 5 is greater than or equal to 8.
A number k plus 5 is greater than or equal to 8.
k 5 ≥ 8
An inequality is k 5 ≥ 8.
c. Four times a number q is at most 16.
Four times a number q is at most 16.
4q ≤ 16
An inequality is 4q ≤ 16.
Write the word sentence as an inequality.
1. A number n is greater than 1.
2. Twice a number p is fewer than 7.
3. A number w minus 3 is less than or equal to 10.
4. A number z divided by 2 is at least 6.
Exercises 5–10
Lesson Tutorials
Inequality Symbols
Symbol <>
≤ ≥
Key
Phrases
● is less
than
● is fewer
than
● is greater
than
● is more
than
● is less than or
equal to
● is at most
● is no more than
● is greater than
or equal to
● is at least
● is no less than
English
Spanish
Section 8.1 Writing and Graphing Inequalities 331
A solution of an inequality is a value that makes the inequality true. An
inequality can have more than one solution. The set of all solutions of an
inequality is called the solution set.
Reading
The symbol ≤
/
means
“is not less than or
equal to.”
EXAMPLE
Checking Solutions
2
2
Tell whether the given value is a solution of the inequality.
a. x 1
>
7; x 8
x 1
>
7 Write the inequality.
8 1
>
?
7 Substitute 8 for
x
.
9
>
7 ✓ Add. 9 is greater than 7.
So, 8 is a solution of the inequality.
b. 7y
<
27; y 4
7y
<
27
Write the inequality.
7(4)
<
?
27 Substitute 4 for
y
.
28 <
/
27
✗
Multiply. 28 is
not
less than 27.
So, 4 is not a solution of the inequality.
c.
z
3
≥ 5; z 15
z
3
≥ 5 Write the inequality.
15
3
≥
?
5 Substitute 15 for
z
.
5 ≥ 5 ✓ Divide. 5 is greater than or equal to 5.
So, 15 is a solution of the inequality.
Tell whether 3 is a solution of the inequality.
5. b 4 < 6 6. 9 n ≥ 6 7. 18 x ≤ 10
Exercises 11–16
Value of x
x 3
≤
7
Is the inequality true?
3
3 3 ≤
?
7
6 ≤
7
✓
yes
4
4 3 ≤
?
7
7 ≤
7
✓
yes
5
5 3 ≤
?
7
8 ≤
/
7
✗
no
English
Spanish
1
3
4
2 0
15
4
2
5
3
Use an open circle because
2 is
not
a solution.
Test a number to the left of 2.
g
0 is
not
a solution.
Test a number to the right of 2.
g
3 is a solution.
332 Chapter 8 Inequalities
The graph of an inequality shows all of the solutions of the inequality
on a number line. An open circle
is used when a number is not a
solution. A closed circle
is used when a number is a solution. An arrow
to the left or right shows that the graph continues in that direction.
EXAMPLE
Graphing an Inequality
3
3
Graph g > 2.
1
3
4
2 0
15
4
2
5
3
Shade the number line on the side
where you found the solution.
EXAMPLE
Real-Life Application
4
4
The NASA Solar Probe can withstand temperatures up to and
including 2600°F. Write and graph an inequality that represents
the temperatures the probe can withstand.
Words temperatures up to and including 2600°F
Variable Let t be the temperatures the probe can withstand.
Inequality t ≤ 2600
An inequality is t ≤ 2600.
2000
2200
3000
2800
2400
2600
Graph the inequality on a number line.
8. a < 4 9. f ≤ 7 10. n > 0 11. p ≥ 3
Write and graph an inequality for the situation.
12. A cruise ship can carry at most 3500 passengers.
13. A board game is designed for ages 12 and up.
Exercises 25–36
English
Spanish
Section 8.1 Writing and Graphing Inequalities 333
Exercises
8.1
9
+(-6)=3
3
+(-3)=
4
+(-9)=
9
+(-1)=
1. VOCABULARY How are “greater than” and “greater than or equal to” similar?
How are they different?
2. DIFFERENT WORDS, SAME QUESTION Which is different? Write “both” inequalities.
3. WRITING Explain how the graph of x ≤ 6 is differ
ent from the graph of x < 6.
4. WRITING Are the graphs of x ≤ 5 and 5 ≥ x the same or differ
ent? Explain.
Write the word sentence as an inequality.
5. A number k is less than 10. 6. A number a is more than 6.
7. A number z is fewer than
3
4
. 8. A number b is at least 3.
9. One plus a number y is no more than 13.
10. A number x divided by 3 is at most 5.
Tell whether the given value is a solution of the inequality.
11. x 1 ≤ 7; x 6 12. y 5 < 13; y 17
13. 3z > 6; z 3 14.
b
2
≥ 6; b 10
15. c 2.5 < 4.3; c 1.8 16. a ≤ 0; a 5
Match each inequality with its graph.
17. x ≥ 2 18. x < 2 19. x > 2 20. x ≤ 2
A.
1
3
2
0
123
B.
1
3
2
0
123
C.
1
3
2
0
123
D.
1
3
2
0
123
A number n is at most 3. A number n is at least 3.
A number n is less than or equal to 3. A number n is no more than 3.
1
1
2
2
Help with Homework
English
Spanish
334 Chapter 8 Inequalities
Write an inequality and a word sentence that represent the graph.
21.
1
3
2
0
123
22.
1
4
2
0
123
23.
2
6
4
0
246
24.
5
5
0
10
15 20 25
Graph the inequality on a number line.
25. a
>
4 26. n ≥ 8 27. 3 ≥ x 28. y
<
1
2
29. x
<
2
9
30. 3 ≥ c 31. m
>
5 32. b ≥ 0
33. f
<
1.5 34. t ≥
1
2
35. p
>
1.6 36. z ≤
7
3
ERROR ANALYSIS Describe and correct the error in graphing the inequality.
37.
38.
39. FISHING In the Suwannee River, you are
allo
wed to catch at most 3 striped bass.
Each striped bass must be no less than
18 inches in length.
a. Write and graph an inequality to
represent the number of striped
bass you are allowed to catch.
b. Write and graph an inequality to
represent the length of each striped
bass you are allowed to catch.
40. LOW SODIUM For a food to be labeled lo
w sodium,
there must be no more than 140 milligrams of sodium
per serving.
a. Write and graph an inequality to represent the
amount of sodium in a low sodium serving.
b. Write and graph an inequality to represent the
amount of sodium in a serving that does not qualify
as low sodium.
c. Does the food represented by the label qualify as a low
sodium food? Explain.
x
≥
1
1
0
2
3
✗
x
>
1
1
0
2
3
✗
3
3
39
.
18 in.
English
Spanish
Section 8.1 Writing and Graphing Inequalities 335
Solve the equation. Check your solution.
47. x 3 12 48. x 6 8 49. 16 x 44 50. 7.6 x 6.5
51. MULTIPLE CHOICE A stack of boards is 24 inches high. Each board is
3
8
of an
inch thick. How many boards are in the stack?
A
1
6
B
1
9
C 9 D 64
41. SHOPPING You have $33. You want to buy a necklace
and one other item from the list.
a. Write an inequality to represent the situation.
b. Can the other item be a T-shirt? Explain.
c. Can the other item be a book? Explain.
Determine whether the statement is sometimes, always, or never true.
Explain your reasoning.
42. A number that is a solution of the inequality x > 5 is also a solution of the
inequality x ≥ 5.
43. A number that is a solution of the inequality 5 ≤ x is also a solution of the
inequality x > 5.
44. BUS RIDE A bus ride costs $1.50. A 30-day bus pass costs $36. Write an
inequality to r
epr
esent the number of bus rides you would need to take
for the bus pass to be a better deal.
45. MOVIE THEATER Fifty people are seated in a movie theater. The maximum
capacity of the theater is 425 people.
Write an inequality to represent the
number of additional people who can still be seated.
46.
The map shows the elevations above sea level for an area of land.
a. Graph the possible elevations of A. Write
the set of elevations as two inequalities.
b. Graph the possible elevations of C. How
can you write this set of elevations as a
single inequality? Explain.
c. What is the elevation of B ? Explain.
0–100 ft
100–200 ft
200–300 ft
300–400 ft
400–500 ft
A
B
C
Key
Item
T-shirt
Book
DVD
Necklace
$ 15
$ 20
$ 13
$ 16
Price (with tax)
Ex
pl
ai
n.
en
t
is
so
English
Spanish
SECTION 7.2
SECTION 2.5
