Absolute Value Inequalities

Absolute Value Inequalities

Absolute value measures distance from 0, so inequalities with (|x|) are really questions about how far a number can be from 0.

1) Identify the type

• If the inequality looks like (|x| < a) or (|x| \le a), it means the value is close to 0.
• If it looks like (|x| > a) or (|x| \ge a), it means the value is far from 0.

2) Rewrite without absolute value

For a positive number (a):

• (|x| < a) becomes (-a < x < a)
• (|x| \le a) becomes (-a \le x \le a)
• (|x| > a) becomes (x < -a) or (x > a)
• (|x| \ge a) becomes (x \le -a) or (x \ge a)

3) Solve carefully

If there is a more complicated expression inside the absolute value, isolate it first. Then solve the resulting inequality or inequalities using ordinary algebra.

4) Check your answer

Test a number from each part of the solution set. Make sure it satisfies the original inequality, especially when the answer is a union of two intervals.

5) Simplify the final result

Write the answer in interval notation or as a clear inequality, and combine overlapping parts if needed. If the inequality has no solution, say so clearly.