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 Number of inequalities to solve: 23456789
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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 â€œ-3 is less than xâ€ and â€œx is less than 3â€.

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 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.