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198 Chapter 2 Solving Linear Equations
2.7 Solve Absolute Value Inequalities
Learning Objectives
By the end of this section, you will be able to:
Solve absolute value equations
Solve absolute value inequalities with “less than”
Solve absolute value inequalities with “greater than”
Solve applications with absolute value
Be Prepared!
Before you get started, take this readiness quiz.
1. Evaluate: −7.
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If you missed this problem, review Example 1.12.
2. Fill in <, >, or = for each of the following pairs of numbers.
( )
ⓐ −8___− −8 ⓑ 12___− −12 ⓒ −6___−6 ⓓ − −15 ___− −15
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If you missed this problem, review Example 1.12.
14−2 ( )
3. Simplify: 8−34−1 .
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If you missed this problem, review Example 1.13.
Solve Absolute Value Equations
As we prepare to solve absolute value equations, we review our definition of absolute value.
Absolute Value
The absolute value of a number is its distance from zero on the number line.
The absolute value of a number n is written as n and n ≥ 0 for all numbers.
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Absolute values are always greater than or equal to zero.
Welearnedthatbothanumberanditsoppositearethesamedistancefromzeroonthenumberline.Sincetheyhavethe
same distance from zero, they have the same absolute value. For example:
−5 is 5 units away from 0, so −5 = 5.
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5 is 5 units away from 0, so 5 = 5.
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Figure 2.6 illustrates this idea.
Figure 2.6 The numbers 5 and −5 are both five
units away from zero.
Fortheequation |x| = 5, wearelookingforallnumbersthatmakethisatruestatement.Wearelookingforthenumbers
whosedistancefromzerois5.Wejustsawthatboth5and −5 arefiveunitsfromzeroonthenumberline.Theyarethe
solutions to the equation.
If |x| = 5
then x = −5 or x = 5
Thesolutioncanbesimplifiedtoasinglestatementbywriting x = ±5. Thisisread,“xisequaltopositiveornegative5”.
We can generalize this to the following property for absolute value equations.
This OpenStax book is available for free at http://cnx.org/content/col12119/1.3
Chapter 2 Solving Linear Equations 199
Absolute Value Equations
For any algebraic expression, u, and any positive real number, a,
if u =a
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then u = −a or u = a
Remember that an absolute value cannot be a negative number.
EXAMPLE 2.68
Solve: ⓐ x = 8 ⓑ y = −6 ⓒ z = 0
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Solution
ⓐ
|x| = 8
Write the equivalent equations. x = −8or x = 8
x = ±8
ⓑ
y = −6
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No solution
Since an absolute value is always positive, there are no solutions to this equation.
ⓒ
z = 0
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Write the equivalent equations. z = −0orz = 0
Since−0 = 0, z = 0
Both equations tell us that z = 0 and so there is only one solution.
TRY IT : : 2.135 Solve: ⓐ x = 2 ⓑ y = −4 ⓒ z = 0
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TRY IT : : 2.136 Solve: ⓐ x = 11 ⓑ y = −5 ⓒ z = 0
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Tosolveanabsolutevalueequation,wefirstisolatetheabsolutevalueexpressionusingthesameproceduresweusedto
solve linear equations. Once we isolate the absolute value expression we rewrite it as the two equivalent equations.
EXAMPLE 2.69 HOW TO SOLVE ABSOLUTE VALUE EQUATIONS
Solve 5x−4 −3 = 8.
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Solution
200 Chapter 2 Solving Linear Equations
TRY IT : : 2.137
Solve: 3x − 5 − 1 = 6.
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TRY IT : : 2.138
Solve: 4x − 3 − 5 = 2.
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The steps for solving an absolute value equation are summarized here.
HOW TO : : SOLVE ABSOLUTE VALUE EQUATIONS.
Step 1. Isolate the absolute value expression.
Step 2. Write the equivalent equations.
Step 3. Solve each equation.
Step 4. Check each solution.
EXAMPLE 2.70
Solve 2|x − 7| + 5 = 9.
Solution
2|x − 7| + 5 = 9
Isolate the absolute value expression. 2|x − 7| = 4
|x − 7| = 2
Write the equivalent equations. x −7 = −2 or x−7 = 2
Solve each equation. x = 5 or x = 9
This OpenStax book is available for free at http://cnx.org/content/col12119/1.3
Chapter 2 Solving Linear Equations 201
Check:
TRY IT : : 2.139 Solve: 3 x − 4 − 4 = 8.
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TRY IT : : 2.140 Solve: 2 x − 5 + 3 = 9.
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Remember, an absolute value is always positive!
EXAMPLE 2.71
Solve: 2x −4 +11 = 3.
|3 |
Solution
2
x −4 +11 = 3
|3 |
2
Isolate the absolute value term. x −4 = −8
|3 |
An absolute value cannot be negative. No solution
TRY IT : : 2.141
Solve: 3x −5 +9 = 4.
|4 |
TRY IT : : 2.142
Solve: 5x +3 +8 = 6.
|6 |
Some of our absolute value equations could be of the form u = v where u and v are algebraic expressions. For
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example, |x − 3| = 2x + 1.
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Howwouldwesolve them? If two algebraic expressions are equal in absolute value, then they are either equal to each
other or negatives of each other. The property for absolute value equations says that for any algebraic expression, u, and
a positive real number, a, if u = a, then u = −a or u = a.
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This tell us that
if u = v
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then u = −v or u = v
This leads us to the following property for equations with two absolute values.
Equations with Two Absolute Values
For any algebraic expressions, u and v,
if u = v
| | | |
then u = −v or u = v
When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.
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