Absolute Value Equations Absolute Value Inequalities

1.8

Absolute Value Equations and Inequalities

Absolute Value Equations Absolute Value Inequalities

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Distance

is 3.

Distance is greater than 3.

Distance is less than 3.

Distance

is 3.

Distance

Distance is

is greater

less than 3.

than 3.

?3

0

3

By definition, the equation |x| = 3 can be

solved by finding real numbers at a distance

of three units from 0. Two numbers satisfy

this equation, 3 and ? 3.

So the solution set is {-3,3}.

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Properties of Absolute Value 1. For b > 0, a = b if and only if a = b or a = -b. 2. a = b if and only if a = b or a = -b.

For any positive number b: 3. a < b if and only if - b < a < b. 4. a > b if and only if a < -b or a > b.

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Example 1 SOLVING ABSOLUTE VALUE EQUATIONS

Solve a. 5 - 3x = 12

Solution For the given expression 5 ? 3x to have absolute value 12, it must represent either 12 or ?12 . This requires applying Property 1, with a = 5 ? 3x and b = 12.

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Example 1 SOLVING ABSOLUTE VALUE EQUATIONS

Solve

a. 5 - 3x = 12

Solution

5 - 3x = 12

5 - 3x = 12 or -3x = 7 or x = - 7 or 3

5 - 3x = -12 Property 1

-3x =-17 Subtract 5.

x = 17 3

Divide by ? 3.

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Example 1 SOLVING ABSOLUTE VALUE EQUATIONS

Solve

a. 5 - 3x = 12

Solution x = - 7 or 3

x = 17 3

Divide by ?3.

Check the solutions by substituting them in

the original absolute value equation. The

{ } solution set is - 7,17 . 33

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Example 1

Solve

SOLVING ABSOLUTE VALUE EQUATIONS

b. 4x - 3 = x + 6 Solution 4x - 3 = x + 6

4x - 3 = x + 6 or 4x - 3 =-(x + 6) Property 2

3x = 9 or 4x - 3 =-x - 6

x = 3 or 5x = -3

{ } The solutionset is - 3,3 . x = - 3

5

5

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Example 2

Solve

SOLVING ABSOLUTE VALUE INEQUALITIES

a. 2x + 1 < 7

Solution

Use Property 3, replacing a with 2x + 1 and b with 7. 2x + 1 < 7

-7 < 2x + 1< 7 -8 < 2x < 6 -4< x ................
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