Chapter 3 Section 7

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3-7 Solving Absolute-Value Inequalities
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Warm Up
California Standards
Lesson Presentation
3-7 Solving Absolute-Value Inequalities
Warm Up
Solve each inequality and graph the solution.
1. x + 7 < 4 x < –3 –5 –4 –3 –2 –1 0 1 2 3 4 5
2. 14x ≥ 28
3. 5 + 2x > 1
x≥2
x > –2
–5 –4 –3 –2 –1
0
1
2
3
4
5
–5 –4 –3 –2 –1
0
1
2
3
4
5
3-7 Solving Absolute-Value Inequalities
California
Standards
3.0 Students solve equations and inequalities
involving absolute values.
Also covered:
5.0
3-7 Solving Absolute-Value Inequalities
When an inequality contains an absolute-value
expression, it can be written as a compound
inequality. The inequality |x| < 5 describes all
real numbers whose distance from 0 is less
than 5 units. The solutions are all numbers
between –5 and 5, so |x|< 5 can be rewritten
as –5 < x < 5, or as x > –5 AND x < 5.
3-7 Solving Absolute-Value Inequalities
3-7 Solving Absolute-Value Inequalities
Additional Example 1A: Solving Absolute-Value
Inequalities Involving <
Solve the inequality and graph the solutions.
|x|– 3 < –1
|x|– 3 < –1
+3 +3
|x| < 2
Since 3 is subtracted from |x|, add 3
to both sides to undo the
subtraction.
x > –2 AND x < 2
Write as a compound inequality.
The solution set is {x: –2 < x < 2}.
2 units
–2
–1
2 units
0
1
2
3-7 Solving Absolute-Value Inequalities
Additional Example 1B: Solving Absolute-Value
Inequalities Involving <
Solve the inequality and graph the solutions.
|x – 1| ≤ 2
x – 1 ≥ –2 AND x – 1 ≤ 2 Write as a compound inequality.
+1 +1
+1 +1 Solve each inequality.
x ≥ –1AND
–3
–2
–1
0
1
x ≤ 3 Write as a compound inequality.
The solution set is
{x: –1 ≤ x ≤ 3}.
2
3
3-7 Solving Absolute-Value Inequalities
Helpful Hint
Just as you do when solving absolute-value
equations, you first isolate the absolute-value
expression when solving absolute-value
inequalities.
3-7 Solving Absolute-Value Inequalities
Check It Out! Example 1a
Solve the inequality and graph the solutions.
2|x| ≤ 6
2|x| ≤ 6
2
2
|x| ≤ 3
x ≥ –3 AND x ≤ 3
3 units
–3
–2
–1
Since x is multiplied by 2, divide both
sides by 2 to undo the
multiplication.
Write as a compound inequality.
The solution set is {x: –3 ≤ x ≤ 3}.
3 units
0
1
2
3
3-7 Solving Absolute-Value Inequalities
Check It Out! Example 1b
Solve each inequality and graph the solutions.
|x + 3|– 4.5 ≤ 7.5
|x + 3|– 4.5 ≤ 7.5
+ 4.5 +4.5
|x + 3| ≤ 12
x + 3 ≥ –12 AND x + 3 ≤ 12
–3
–3
–3 –3
x ≥ –15 AND
x≤9
–20 –15 –10
–5
0
5
10
15
Since 4.5 is subtracted from
|x + 3|, add 4.5 to both
sides to undo the
subtraction.
Write as a compound
inequality.
The solution set is
{x: –15 ≤ x ≤ 9}.
3-7 Solving Absolute-Value Inequalities
The inequality |x| > 5 describes all real numbers
whose distance from 0 is greater than 5 units. The
solutions are all numbers less than –5 or greater
than 5. The inequality |x| > 5 can be rewritten as
the compound inequality x < –5 OR x > 5.
3-7 Solving Absolute-Value Inequalities
3-7 Solving Absolute-Value Inequalities
Additional Example 2A: Solving Absolute-Value
Inequalities Involving >
Solve the inequality and graph the solutions.
|x| + 14 ≥ 19
|x| + 14 ≥ 19
– 14 –14
|x|
≥ 5
x ≤ –5 OR x ≥ 5
Since 14 is added to |x|, subtract 14
from both sides to undo the addition.
Write as a compound inequality. The
solution set is {x: x ≤ –5 OR x ≥ 5}.
5 units 5 units
–10 –8 –6 –4 –2
0
2
4
6
8 10
3-7 Solving Absolute-Value Inequalities
Additional Example 2B: Solving Absolute-Value
Inequalities Involving >
Solve the inequality and graph the solutions.
3 + |x + 2| > 5
Since 3 is added to |x + 2|,
subtract 3 from both sides to
undo the addition.
3 + |x + 2| > 5
–3
–3
|x + 2| > 2
Write as a compound inequality.
x + 2 < –2 OR x + 2 > 2
Solve each inequality.
–2 –2
–2 –2
Write as a compound inequality.
x
< –4 OR x
>0
The solution set is
{x: x < –4 or x > 0}.
–10 –8 –6 –4 –2
0
2
4
6
8 10
3-7 Solving Absolute-Value Inequalities
Check It Out! Example 2a
Solve each inequality and graph the solutions.
|x| + 10 ≥ 12
|x| + 10 ≥ 12
– 10 –10
|x|
≥
Since 10 is added to |x|, subtract 10
from both sides to undo the
addition.
2
x ≤ –2 OR x ≥ 2
Write as a compound inequality. The
solution set is {x: x ≤ –2 or x ≥ 2}.
2 units 2 units
–5 –4 –3 –2 –1
0
1
2
3
4
5
3-7 Solving Absolute-Value Inequalities
Check It Out! Example 2b
Solve the inequality and graph the solutions.
Since is added to |x + 2 |, subtract
from both sides to undo the addition.
Write as a compound inequality.
Solve each inequality.
OR
x ≤ –6
x≥1
Write as a compound
inequality. The solution
set is {x: x ≤ –6 or x ≥ 1}
3-7 Solving Absolute-Value Inequalities
Check It Out! Example 2b Continued
Solve the inequality and graph the solutions.
–7 –6 –5 –4 –3 –2 –1 0
1
2
3
3-7 Solving Absolute-Value Inequalities
Additional Example 3: Application
A pediatrician recommends that a baby’s
bath water be 95°F, but it is acceptable for
the temperature to vary from this amount by
as much as 3°F. Write and solve an absolutevalue inequality to find the range of
acceptable temperatures. Graph the
solutions.
Let t represent the actual water temperature.
The difference between t and the ideal
temperature is at most 3°F.
t – 95
≤
3
3-7 Solving Absolute-Value Inequalities
Additional Example 3 Continued
t – 95
≤
3
|t – 95| ≤ 3
t – 95 ≥ –3 AND t – 95 ≤
3
+95 +95
+95 +95
t
≥ 92 AND t
≤ 98
90
92
94
96
98
Solve the two
inequalities.
100
The range of acceptable temperature is 92 ≤ t ≤ 98.
3-7 Solving Absolute-Value Inequalities
Check It Out! Example 3
A dry-chemical fire extinguisher should be
pressurized to 125 psi, but it is acceptable
for the pressure to differ from this value by
at most 75 psi. Write and solve an absolutevalue inequality to find the range of
acceptable pressures. Graph the solution.
Let p represent the desired pressure.
The difference between p and the ideal
pressure is at most 75 psi.
p – 125
≤
75
3-7 Solving Absolute-Value Inequalities
Check It Out! Example 3 Continued
p – 125
≤
75
|p – 125| ≤ 75
p – 125 ≥ –75 AND p – 125 ≤ 75
+125 +125
+125 +125
p
≥
50 AND p
≤ 200
25
50
Solve the two
inequalities.
75 100 125 150 175 200 225
The range of pressure is 50 ≤ p ≤ 200.
3-7 Solving Absolute-Value Inequalities
When solving an absolute-value inequality, you may
get a statement that is true for all values of the
variable. In this case, all real numbers are solutions
of the original inequality. If you get a false
statement when solving an absolute-value
inequality, the original inequality has no solutions.
Its solution set is ø.
3-7 Solving Absolute-Value Inequalities
Additional Example 4A: Special Cases of AbsoluteValue Inequalities
Solve the inequality.
|x + 4|– 5 > – 8
|x + 4|– 5 > – 8
+5
+5
|x + 4|
>
–3
Add 5 to both sides.
Absolute-value expressions
are always nonnegative.
Therefore, the statement is
true for all real numbers.
The solution set is all real numbers.
3-7 Solving Absolute-Value Inequalities
Additional Example 4B: Special Cases of AbsoluteValue Inequalities
Solve the inequality.
|x – 2| + 9 < 7
|x – 2| + 9 < 7
–9 –9
|x – 2|
< –2
Subtract 9 from both sides.
Absolute-value expressions
are always nonnegative.
Therefore, the statement is
false for all values of x.
The inequality has no solutions. The solution set
is ø.
3-7 Solving Absolute-Value Inequalities
Remember!
An absolute value represents a distance, and
distance cannot be less than 0.
3-7 Solving Absolute-Value Inequalities
Check It Out! Example 4a
Solve the inequality.
|x| – 9 ≥ –11
|x| – 9 ≥ –11
+9 ≥ +9
|x|
≥ –2
Add 9 to both sides.
Absolute-value expressions
are always nonnegative.
Therefore, the statement is
true for all real numbers.
The solution set is all real numbers.
3-7 Solving Absolute-Value Inequalities
Check It Out! Example 4b
Solve the inequality.
4|x – 3.5| ≤ –8
4|x – 3.5| ≤ –8
4
4
|x – 3.5| ≤ –2
Divide both sides by 4.
Absolute-value expressions
are always nonnegative.
Therefore, the statement is
false for all values of x.
The inequality has no solutions. The solution set
is ø.
3-7 Solving Absolute-Value Inequalities
Lesson Quiz: Part I
Solve each inequality and graph the solutions.
1. 3|x| > 15
–10
–5
x < –5 or x > 5
0
2. |x + 3| + 1 < 3
–6
–5
–4
5
10
–5 < x < –1
–3
–2
–1
0
3. A number, n, is no more than 7 units away
from 5. Write and solve an inequality to show
the range of possible values for n.
|n– 5| ≤ 7; –2 ≤ n ≤ 12
3-7 Solving Absolute-Value Inequalities
Lesson Quiz: Part II
Solve each inequality.
4. |3x| + 1 < 1 ø
5. |x + 2| – 3 ≥ – 6
all real numbers

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