Solve compound inequalities in one variable involving absolute-value expressions.

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Objectives
Solve compound inequalities in one variable
involving absolute-value expressions.
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Find all numbers whose absolute value is less
than 5.
Absolute value inequality
Compound inequality
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To Solve Absolute-Value Inequalities To Solve Absolute-Value Inequalities
1. Perform inverse operations to isolate the absolute value bars.
2.
3. Solve the compound inequality.
4. Graph the solution set.
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Additional Example 1A Solving Absolute-Value
Inequalities Involving lt
Solve the inequality and graph the solutions.
x 3 lt 1
Since 3 is subtracted from x, add 3 to both
sides to undo the subtraction.
Write as a compound inequality.
x gt 2 AND x lt 2
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Additional Example 1B Solving Absolute-Value
Inequalities Involving lt
Solve the inequality and graph the solutions.
x 1 2
Write as a compound inequality.
x 1 2 AND x 1 2
Solve each inequality.
Write as a compound inequality.
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Check It Out! Example 1a
Solve the inequality and graph the solutions.
2x 6
Since x is multiplied by 2, divide both sides by
2 to undo the multiplication.
x 3
Write as a compound inequality.
x 3 AND x 3
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Check It Out! Example 1b
Solve each inequality and graph the solutions.
x 3 4.5 7.5
Since 4.5 is subtracted from x 3, add 4.5 to
both sides to undo the subtraction.
Write as a compound inequality.
x 3 12 AND x 3 12
Subtract 3 from both sides of each inequality.
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Find all numbers whose absolute value is greater
than 5.
Absolute value inequality
Compound inequality
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Additional Example 2A Solving Absolute-Value
Inequalities Involving gt
Solve the inequality and graph the solutions.
x 14 19
Since 14 is added to x, subtract 14 from both
sides to undo the addition.
x 5
x 5 OR x 5
Write as a compound inequality.
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Additional Example 2B Solving Absolute-Value
Inequalities Involving gt
Solve the inequality and graph the solutions.
3 x 2 gt 5
Since 3 is added to x 2, subtract 3 from both
sides to undo the addition.
Write as a compound inequality. Solve each
inequality.
Write as a compound inequality.
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Check It Out! Example 2a
Solve each inequality and graph the solutions.
x 10 12
x 10 12
Since 10 is added to x, subtract 10 from both
sides to undo the addition.
x 2 OR x 2
Write as a compound inequality.
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Check It Out! Example 2b
Solve the inequality and graph the solutions.
Write as a compound inequality. Solve each
inequality.
Write as a compound inequality.
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Check It Out! Example 2b Continued
Solve the inequality and graph the solutions.
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Homework

• Sec. 2-7 Practice B Wksht (1-8) Sec. 2-7
  Practice A Wksht (1-8)

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Additional Example 3 Application
A pediatrician recommends that a babys bath
water be 95F, but it is acceptable for the
temperature to vary from this amount by as much
as 3F. Write and solve an absolute-value
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 3F.
t 95 3
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Additional Example 3 Continued
t 95 3
t 95 3
Solve the two inequalities.
t 95 3 AND t 95 3
The range of acceptable temperature is 92 t
98.
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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 absolute-value
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
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Check It Out! Example 3 Continued
p 125 75
p 125 75
Solve the two inequalities.
p 125 75 AND p 125 75
The range of pressure is 50 p 200.
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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.
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Additional Example 4A Special Cases of
Absolute-Value Inequalities
Solve the inequality.
x 4 5 gt 8
Add 5 to both sides.
Absolute-value expressions are always
nonnegative. Therefore, the statement is true for
all real numbers.
All real numbers are solutions.
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Additional Example 4B Special Cases of
Absolute-Value Inequalities
Solve the inequality.
x 2 9 lt 7
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.
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Check It Out! Example 4a
Solve the inequality.
x 9 11
Add 9 to both sides.
Absolute-value expressions are always
nonnegative. Therefore, the statement is true for
all real numbers.
All real numbers are solutions.
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Check It Out! Example 4b
Solve the inequality.
4x 3.5 8
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.
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Homework

• Sec. 2-7 Practice C Problem Solving Worksheets