Absolute Value Inequality Calculator
Solve absolute value inequalities by turning them into inside intervals or outside intervals, then write the answer in compound inequality and interval notation form.
Less than means inside, greater than means outside
The easiest way to understand absolute value inequalities is to think about distance. A statement like |x| < 4 means x is less than 4 units from zero, so the answer is inside the interval from -4 to 4. A statement like |x| > 4 means x is more than 4 units from zero, so the answer is outside that interval.
OpenStax explains this same idea in its section on solving absolute value inequalities with less-than and greater-than cases. The calculator above uses that rule after first isolating the absolute value expression.
Worked example: less-than absolute value inequality
This is an inside solution. Every value between -2 and 5 makes the expression 2x - 3 stay within 7 units of zero.
Worked example: greater-than absolute value inequality
This answer has two separate pieces, so interval notation uses the union symbol. Values between -5 and 3 are too close to -1, so they do not satisfy the inequality.
Absolute value inequality rules
| Form | Meaning | Pattern |
|---|---|---|
| |u| < k | u is within k units of 0 | -k < u < k |
| |u| ≤ k | u is within or exactly k units from 0 | -k ≤ u ≤ k |
| |u| > k | u is farther than k units from 0 | u < -k or u > k |
| |u| ≥ k | u is at least k units from 0 | u ≤ -k or u ≥ k |
Common mistakes with absolute value inequalities
The most common mistake is using the wrong connector. Less-than absolute value inequalities become AND statements because the answer is between two endpoints. Greater-than absolute value inequalities become OR statements because the answer is outside two endpoints. Another mistake is forgetting to isolate the absolute value before applying the rule.
Common questions
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First isolate the absolute value expression. Then decide whether the inequality is a less-than type or a greater-than type. Less-than absolute value inequalities become inside compound inequalities, such as -k < expression < k. Greater-than absolute value inequalities become outside compound inequalities, such as expression < -k or expression > k. Finally, solve the resulting inequality and write the answer in interval notation.
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Less than means the expression must stay within a certain distance from zero. For example, |x| < 5 means x must be greater than -5 and less than 5. Both conditions must be true at the same time, so the answer is an AND inequality: -5 < x < 5.
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Greater than means the expression must be outside a distance from zero. For example, |x| > 5 means x can be less than -5 or greater than 5. The answer has two separated regions, so it uses OR and interval notation with the union symbol.
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Yes. If the isolated form is |expression| < 0, there is no solution because absolute value cannot be negative and cannot be less than zero. If the isolated form is |expression| ≤ -3, there is also no solution. The calculator checks these cases before solving.
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Yes. If the isolated form is |expression| ≥ 0, every real number works because absolute value is always nonnegative. Also, any inequality like |expression| > -2 is true for all real numbers because every absolute value is greater than a negative number.
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For inside solutions, write one interval between the two endpoints, such as [-2, 5]. For outside solutions, write two intervals joined by the union symbol, such as (-∞, -2) ∪ (5, ∞). Use brackets for included endpoints and parentheses for excluded endpoints.
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The biggest mistake is treating every absolute value inequality like an equation with two separate equalities. Inequalities need direction and connector logic. Less-than cases stay between two endpoints; greater-than cases split into two outside regions. The sign and connector matter as much as the numeric endpoints.
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Yes. Testing values is very useful, especially for greater-than cases. Choose one value inside the final interval and one value outside it. Substitute them into the original inequality. The inside value should make the inequality true, and the outside value should make it false.