2 Solving Linear Inequalities
2 Solving Linear Inequalities
2.1
2.2
2.3
2.4
2.5
2.6
Writing and Graphing Inequalities
Solving Inequalities Using Addition or Subtraction
Solving Inequalities Using Multiplication or Division
Solving Multi-Step Inequalities
Solving Compound Inequalities
Solving Absolute Value Inequalities
Camel Physiology (p. 91)
Mountain Plant Life (p. 85)
SEE the Big Idea
Digital Camera (p. 70)
Microwave Electricity (p. 64)
Natural Arch (p. 59)
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Maintaining Mathematical Proficiency
Graphing Numbers on a Number Line
(6.NS.C.6c)
Example 1 Graph each number.
a.
a.
3
3
b.
?5
b.
?5
?5
?4
?3
?2
?1
0
1
2
3
4
5
?2
?1
0
1
2
3
4
5
Example 2 Graph each number.
?5
?4
¨O4¨O
a.
?3
Example 2 Graph
each
¨O number.
b. ¨O ?2
a. ¨O 4 ¨O
The absolute value of a positive number is positive.
Graph the number.
1. 6
2.
4. 2 + ¨O ?2 ¨O
?5
?4
?3
?2
¨O2¨O
3.
?1 1 ?0¨O ?4 1
¨O
5.
¨O ?2 ¨O
b.
2
3
4
5
¨O ?1 ¨O
6. ?5 + ¨O 3 ¨O
The absolute value of a negative number is positive.
Comparing Real Numbers
(6.NS.C.7a)
Example 3
0
Complete
the statement ?1
?5 ?4 ?3 ?2 ?1
?5
with
,
or 4=.
1
2
3
5
Graph the number.
?1is to the right of ?5. So, ?1 > ?5.
2. ¨O 2 ¨O
1. 6
3.
4. 2 + ¨O ?2 ¨O
5. 1 ? ¨O ?4 ¨O
Example 4 Evaluate 15 ¡Â (?3).
¨O ?1 ¨O
6. ?5 + ¨O 3 ¨O
Comparing Real Numbers
Example 3 Complete the statement ?1
?5 with , or =.
15 ¡Â (?3) = ?5
Graph ¨C 5.
Multiply or divide.
10. ?3 (8)
?7
13. ?24 ¡Â (?6)
?6
?5
?4
Graph ¨C 1.
?3
?2
?1
0
? ?1 > ?5.
?1 is to the right of ?5. So,
1
2
3
? (?7)
11. ?7 (?9)
12. 4
14. ?16 ¡Â 2
15. 12 ¡Â (?3)
Complete
the statement with
16. 6 8
17. 36,
6 or =.
?
18. ?3(?4)
7. 2
9
8. ?6
5
9. ?12
?4
19. ABSTRACT REASONING Summarize the rules for (a) adding integers, (b) subtracting integers,
¨O ?8 ¨O integers.
¨O 8 ¨O Give an example12.of each.
10. (c)
?7multiplying
?13 integers, and (d)
11.dividing
?10
¨O ?18 ¨O
13. ABSTRACT REASONING A number a is to the left of a number b on the number line.
How do the numbers ?a and ?b compare?
Dynamic Solutions available at
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Mathematical
Practices
Mathematically proficient students use technology tools to
explore concepts.
Using a Graphing Calculator
Core Concept
Solving an Inequality in One Variable
You can use a graphing calculator to solve an inequality.
1.
Enter the inequality into a graphing calculator.
2.
Graph the inequality.
3.
Use the graph to write the solution.
Using a Graphing Calculator
Use a graphing calculator to solve (a) 2x ? 1 < x + 2 and (b) 2x ? 1 ¡Ü x + 2.
SOLUTION
a.
Enter the inequality 2x ? 1 < x + 2 into a graphing calculator. Press graph.
Y1=2X-1
¡Ý
¡ñ
is less
than
¡ñ
is greater
than
¡ñ
is less than or
equal to
¡ñ
is greater than
or equal to
¡ñ
is fewer
than
¡ñ
is more
than
¡ñ
is at most
¡ñ
is at least
¡ñ
is no more than
¡ñ
is no less than
Writing Inequalities
Write each sentence as an inequality.
a. A number w minus 3.5 is less than or equal to ?2.
b. Three is less than a number n plus 5.
c. Zero is greater than or equal to twice a number x plus 1.
SOLUTION
a. A number w minus 3.5 is less than or equal to ?2.
w ? 3.5
¡Ü
?2
An inequality is w ? 3.5 ¡Ü ?2.
READING
b. Three is less than a number n plus 5.
The inequality 3 < n + 5 is
the same as n + 5 > 3.
3
n+5
<
An inequality is 3 < n + 5.
c. Zero is greater than or equal to twice a number x plus 1.
¡Ý
0
2x + 1
An inequality is 0 ¡Ý 2x + 1.
Monitoring Progress
Help in English and Spanish at
Write the sentence as an inequality.
1. A number b is fewer than 30.4.
7
2. ?¡ª
10 is at least twice a number k minus 4.
54
Chapter 2
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Solving Linear Inequalities
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