Absolute Value Equations and Inequalities (EQ4)

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Section 1.4 Absolute Value Equations and Inequalities (EQ4)

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Objectives

Solve linear equations involving an absolute value. Solve linear inequalities involving absolute values and express the answers graphically and using interval notation.

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Subsection 1.4.1 Activities

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Remark 1.4.1.

An absolute value, written

| x |,

is the non-negative value of

x .

x

is a positive number, then

| x | = x .

x

is a negative number, then

| x | = - x .

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Activity 1.4.2.

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Let’s consider how to solve an equation when an absolute value is involved.

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(a)

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Which values are solutions to the absolute value equation

| x | = 2 ?

$x = 2$
$x = 0$
$x = - 1$
$x = - 2$

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(b)

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Which values are solutions to the absolute value equation

| x - 7 | = 2 ?

$x = 9$
$x = 7$
$x = 5$
$x = - 9$

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(c)

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Which values are solutions to the absolute value equation

3 | x - 7 | + 5 = 11 ?

It may be helpful to rewrite the equation to isolate the absolute value.

$x = 7$
$x = - 9$
$x = 5$
$x = 9$

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Activity 1.4.3.

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Absolute value represents the distance a value is from 0 on the number line. So,

| x - 7 | = 2

means that the expression

x - 7

2

units away from

0 .

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(a)

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What values on the number line could

x - 7

equal?

$x = - 7$
$x = - 2$
$x = 0$
$x = 2$
$x = 7$

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(b)

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This gives us two separate equations to solve. What are those two equations?

$x - 7 = - 7$
$x - 7 = - 2$
$x - 7 = 0$
$x - 7 = 2$
$x - 7 = 7$

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(c)

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Solve each equation for

x .

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Remark 1.4.4.

When solving an absolute value equation, begin by isolating the absolute value expression. Then rewrite the equation into two linear equations and solve. If

c > 0,

| a x + b | = c

becomes the following two equations

a x + b = c and a x + b = - c

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Activity 1.4.5.

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Solve the following absolute value equations.

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(a)

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| 3 x + 4 | = 10

${- 2, 2}$
${- \frac{14}{3}, 2}$
${- 10, 10}$
No solution

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(b)

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3 | x - 7 | + 5 = 11

${- 2, 2}$
${- 9, 9}$
${5, 9}$
No solution

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(c)

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2 | x + 1 | + 8 = 4

${- 4, 4}$
${- 6, 6}$
${5, 7}$
No solution

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Remark 1.4.6.

Since the absolute value represents a distance, it is always a positive number. Whenever you encounter an isolated absolute value equation equal to a negative value, there will be no solution.

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