Solving AbsoluteValue

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Solving Absolute-Value Equations and Inequalities
2-8
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
2
Warm Up Solve.
1. y 7 lt 11
y lt 18
2. 4m 12
m 3
3. 5 2x 17
x 6
Use interval notation to indicate the graphed
numbers.
4.
(-2, 3
(-?, 1
5.
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Objectives
Solve compound inequalities. Write and solve
absolute-value equations and inequalities.
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Vocabulary
disjunction conjunction absolute-value
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A compound statement is made up of more than one
equation or inequality. A disjunction is a
compound statement that uses the word or.
Disjunction x 3 OR x gt 2 Set builder
notation xx 3 U x gt 2
A disjunction is true if and only if at least one
of its parts is true.
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A conjunction is a compound statement that uses
the word and.
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Example 1A Solving Compound Inequalities
Solve the compound inequality. Then graph the
solution set.
6y lt 24 OR y 5 3
Solve both inequalities for y.
6y lt 24
y 5 3
or
y lt 4
y 2
The solution set is all points that satisfy yy
lt 4 or y 2.
(8, 4) U 2, 8)
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Example 1B Solving Compound Inequalities
Solve the compound inequality. Then graph the
solution set.
Solve both inequalities for c.
The solution set is the set of points that
satisfy both c 4 and c lt 0.
4, 0)
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Example 1C Solving Compound Inequalities
Solve the compound inequality. Then graph the
solution set.
x 5 lt 2 OR 2x 10
Solve both inequalities for x.
x 5 lt 2 or 2x 10
x lt 3 x 5
The solution set is the set of all points that
satisfy xx lt 3 or x 5.
(8, 3) U 5, 8)
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Check It Out! Example 1a
Solve the compound inequality. Then graph the
solution set.
x 2 lt 1 OR 5x 30
Solve both inequalities for x.
x 2 lt 1
5x 30
or
x 6
x lt 3
The solution set is all points that satisfy xx
lt 3 U x 6.
(8, 3) U 6, 8)
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Check It Out! Example 1b
Solve the compound inequality. Then graph the
solution set.
2x 6 AND x gt 4
Solve both inequalities for x.
2x 6 and x gt 4
x 3 x lt 4
3, 4)
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Check It Out! Example 1c
Solve the compound inequality. Then graph the
solution set.
x 5 lt 12 OR 6x 12
Solve both inequalities for x.
x 5 lt 12 or 6x 12
x lt 17 x 2
Because every point that satisfies x lt 2 also
satisfies x lt 2, the solution set is xx lt 17.
(-8, 17)
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Check It Out! Example 1d
Solve the compound inequality. Then graph the
solution set.
3x lt 12 AND x 4 12
Solve both inequalities for x.
3x lt 12 and x 4 12
x lt 4 x 8
The solution set is the set of points that
satisfy both x4 lt x 8.
(4, 8
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Recall that the absolute value of a number x,
written x, is the distance from x to zero on
the number line. Because absolute value
represents distance without regard to direction,
the absolute value of any real number is
nonnegative.
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Absolute-value equations and inequalities can be
represented by compound statements. Consider the
equation x 3.
The solutions of x 3 are the two points that
are 3 units from zero. The solution is a
disjunction x 3 or x 3.
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The solutions of x lt 3 are the points that are
less than 3 units from zero. The solution is a
conjunction 3 lt x lt 3.
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The solutions of x gt 3 are the points that are
more than 3 units from zero. The solution is a
disjunction x lt 3 or x gt 3.
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Helpful Hint
Think Greator inequalities involving gt or
symbols are disjunctions. Think Less thand
inequalities involving lt or symbols are
conjunctions.
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Note The symbol can replace lt, and the rules
still apply. The symbol can replace gt, and the
rules still apply.
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Example 2A Solving Absolute-Value Equations
Solve the equation.
This can be read as the distance from k to 3 is
10.
3 k 10
Rewrite the absolute value as a disjunction.
3 k 10 or 3 k 10
Add 3 to both sides of each equation.
k 13 or k 7
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Example 2B Solving Absolute-Value Equations
Solve the equation.
Isolate the absolute-value expression.
Rewrite the absolute value as a disjunction.
Multiply both sides of each equation by 4.
x 16 or x 16
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Check It Out! Example 2a
Solve the equation.
This can be read as the distance from x to 9 is
4.
x 9 13
Rewrite the absolute value as a disjunction.
x 9 13 or x 9 13
Subtract 9 from both sides of each equation.
x 4 or x 22
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Check It Out! Example 2b
Solve the equation.
6x 8 22
Isolate the absolute-value expression.
6x 30
Rewrite the absolute value as a disjunction.
6x 30 or 6x 30
Divide both sides of each equation by 6.
x 5 or x 5
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You can solve absolute-value inequalities using
the same methods that are used to solve an
absolute-value equation.
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Example 3A Solving Absolute-Value Inequalities
with Disjunctions
Solve the inequality. Then graph the solution.
4q 2 10
Rewrite the absolute value as a disjunction.
4q 2 10 or 4q 2 10
Subtract 2 from both sides of each inequality.
4q 8 or 4q 12
Divide both sides of each inequality by 4 and
reverse the inequality symbols.
q 2 or q 3
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Example 3A Continued
qq 2 or q 3
(8, 2 U 3, 8)
To check, you can test a point in each of the
three region.
4(0) 2 10 2 10 x
4(3) 2 10 14 10 ?
4(4) 2 10 14 10 ?
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Example 3B Solving Absolute-Value Inequalities
with Disjunctions
Solve the inequality. Then graph the solution.
0.5r 3 3
Isolate the absolute value as a disjunction.
0.5r 0
Rewrite the absolute value as a disjunction.
0.5r 0 or 0.5r 0
Divide both sides of each inequality by 0.5.
r 0 or r 0
The solution is all real numbers, R.
(8, 8)
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Check It Out! Example 3a
Solve the inequality. Then graph the solution.
4x 8 gt 12
Rewrite the absolute value as a disjunction.
4x 8 gt 12 or 4x 8 lt 12
Add 8 to both sides of each inequality.
4x gt 20 or 4x lt 4
Divide both sides of each inequality by 4.
x gt 5 or x lt 1
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Check It Out! Example 3a Continued
xx lt 1 or x gt 5
(8, 1) U (5, 8)
To check, you can test a point in each of the
three region.
4(0) 8 gt 12 8 gt 12 x
4(2) 8 gt 12 16 gt 12 ?
4(6) 8 gt 12 32 gt 12 ?
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Check It Out! Example 3b
Solve the inequality. Then graph the solution.
3x 36 gt 12
Isolate the absolute value as a disjunction.
3x gt 24
Rewrite the absolute value as a disjunction.
3x gt 24 or 3x lt 24
Divide both sides of each inequality by 3.
x gt 8 or x lt 8
The solution is all real numbers, R.
(8, 8)
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Example 4A Solving Absolute-Value Inequalities
with Conjunctions
Solve the compound inequality. Then graph the
solution set.
Multiply both sides by 3.
2x 7 3
Rewrite the absolute value as a conjunction.
2x 7 3 and 2x 7 3
Subtract 7 from both sides of each inequality.
2x 4 and 2x 10
Divide both sides of each inequality by 2.
x 2 and x 5
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Example 4A Continued
The solution set is x5 x 2.
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Example 4B Solving Absolute-Value Inequalities
with Conjunctions
Solve the compound inequality. Then graph the
solution set.
Multiply both sides by 2, and reverse the
inequality symbol.
p 2 6
Rewrite the absolute value as a conjunction.
p 2 6 and p 2 6
Add 2 to both sides of each inequality.
p 4 and p 8
Because no real number satisfies both p 4
andp 8, there is no solution. The solution set
is ø.
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Check It Out! Example 4a
Solve the compound inequality. Then graph the
solution set.
Multiply both sides by 2.
x 5 8
Rewrite the absolute value as a conjunction.
x 5 8 and x 5 8
Add 5 to both sides of each inequality.
x 13 and x 3
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Check It Out! Example 4
The solution set is x3 x 13.
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Check It Out! Example 4b
Solve the compound inequality. Then graph the
solution set.
2x 5 gt 10
Divide both sides by 2, and reverse the
inequality symbol.
x 5 lt 5
Rewrite the absolute value as a conjunction.
x 5 lt 5 and x 5 gt 5
Subtract 5 from both sides of each inequality.
x lt 10 and x gt 0
Because no real number satisfies both x lt 10 and
x gt 0, there is no solution. The solution set is
ø.