Warm Up

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Warm Up
Lesson Presentation
Lesson Quiz
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Warm Up Solve each inequality and graph the
solution. 1. x 7
lt 4 2.
14x 28
3. 5 2x gt 1
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Sunshine State Standards
MA.912.A.3.6 Solve and graph the solutions of
absolute valueinequalities with one variable.
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Objective

Solve inequalities in one variable involving
absolute-value expressions.
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When an inequality contains an absolute-value
expression, it can be written as a compound
inequality. The inequality x lt 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 lt 5 can be rewritten as 5 lt x lt 5,
or as x gt 5 AND x lt 5.
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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.
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.
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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
x 3 AND x 3
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Check It Out! Example 1b
Solve each inequality and graph the solutions.
Isolate the absolute value function.
x 3 4.5 7.5
Since 4.5 is subtracted from x 3, add 4.5 to
both sides to undo the subtraction.
x 3 12 AND x 3 12
Write as a compound inequality.
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The inequality x gt 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 gt 5 can be
rewritten as the compound inequality x lt 5 OR x
gt 5.
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Additional Example 2A Solving Absolute-Value
Inequalities Involving gt
Solve the inequality and graph the solutions.
x 14 19
Isolate the absolute value function.
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. The solution set
is x x 5 OR x 5.
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Additional Example 2B Solving Absolute-Value
Inequalities Involving gt
Solve the inequality and graph the solutions.
Isolate the absolute value function.
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
Isolate the absolute value function.
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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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.
The solution set is all real numbers.
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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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Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
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Lesson Quiz Part I
Solve each inequality and graph the solutions.
1. 3x gt 15
x lt 5 OR x gt 5
2. x 3 1 lt 3
5 lt x lt 1
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
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Lesson Quiz Part II
Solve each inequality.
no solutions
4. 3x 1 lt 1
5. x 2 3 6
all real numbers
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Lesson Quiz for Student Response Systems
1. Solve the inequality.
2x 10
A. x lt 5 OR x gt 5
C. x lt 5
B. x lt 5 OR x gt 5
D. no solution
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Lesson Quiz for Student Response Systems
2. Solve the inequality.
2x 10
A. x lt 5 OR x gt 5
C. x lt 5
B. x lt 5 OR x gt 5
D. no solution
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Lesson Quiz for Student Response Systems
3. Solve the inequality.
2x 3 9
A. x lt 3 OR x gt 3
C. x lt 6 OR x gt 6
B. x lt 3 OR x gt 3
D. x lt 6 OR x gt 6
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Lesson Quiz for Student Response Systems
4. Graph the inequality.
6x 24
A.
10
8
6
4
2
0
2
4
6
8
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B.
10
8
6
4
2
0
2
4
6
8
10
C.
10
8
6
4
2
0
2
4
6
8
10
D.
10
8
6
4
2
0
2
4
6
8
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