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Solving Absolute Value Inequalities

Solving an Absolute Value Inequality of the Form | x| < a

Example 1

Solve: -6|4x| > -72

Solution

Step 1 Isolate the absolute value.

Divide both sides by -6 and reverse the direction of the inequality symbol.

Simplify.

Step 2 Make the substitution w = 4x.

Step 3 Use the Absolute Value Principle to solve for w.

Step 4 Replace w with 4x.

Replace w with 4x.

Step 5 Solve for x.

Divide all three parts by 4.

-6|4x| > -72

 

|4x| < 12

|w| < 12

-12 < w < 12

 

-12 < 4x < 12

 

-3 < x < 3

Notes:

1) When multiplying or dividing both sides of an inequality by a negative number, reverse the direction of the inequality.

2) The compound inequality -3 < x < 3 is read “-3 is less than x” and “x is less than 3”.

Step 6 Check the answer.

We leave the check to you.

So, the solution is -3 < x < 3.

 

Example 2

Solve: -4 + 5|2x - 4| < 36

Solution

Step 1 Isolate the absolute value.

Add 4 to both sides.

Divide both sides by 5.

Step 2 Make the substitution w = 2x - 4.

Step 3 Use the Absolute Value Principle to solve for w.

Step 4 Replace w with 2x - 4.

Step 5 Solve for x.

Add 4 to all three parts.

Divide each part by 2.

 -4 + 5|2x - 4| < 36

 

5|2x - 4| < 40

|2x - 4| < 8

|w| < 8

-8 < w < 8

-8 < 2x - 4 < 8

 

4 < 2x < 12

-2 < x < 6

Step 6 Check the answer.

We leave the check to you.

So, the solution is -2 < x < 6.

Note:

-8 < 2x - 4 < 8 is a compound inequality since it contains two inequality symbols.

A compound inequality is solved when the variable has been isolated in the middle part.

For example, -2 < x < 6 is solved.

 

Remember, absolute value always represents a nonnegative number.

Therefore, some absolute value inequalities have no solution, as the next example shows.

 

Example 3

Solve: |x| + 6 < 4

Solution

Step 1 Isolate the absolute value.

 Subtract 6 from both sides.

 |x| + 6 < 4

 

|x| < -2

 

For any value of x, we know that |x| is a nonnegative number. Thus, |x| cannot be less than -2.

Therefore, |x| + 6 < 4 has no solution.

 
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