Math
Absolute Value Inequalities Calculator
An absolute value inequality asks how far ax + b is from 0.
The absolute value inequalities calculator rewrites |ax + b| < c as the compound 'and' inequality -c < ax + b < c, and |ax + b| > c as the 'or' inequality ax + b < -c or ax + b > c, then solves each part exactly. You get the solution as an inequality, in interval notation, in set-builder notation, and on a number line, including the special cases where c is negative or zero.
|ax + b|. Use 1 if the inside is just x + b.
Use 0 if there is no constant inside.
The number the absolute value is compared with. Negative and zero are handled.
Try an example
Result
Solution
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- Interval notation
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- Compound inequality
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- Set-builder notation
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- Critical points
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- Type of solution set
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- Number line
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the absolute value inequalities calculator solves
|u| < c means u is within c units of 0, so u is trapped between -c and c. |u| > c means u is more than c units from 0, so u is below -c or above c. Replace u with ax + b, solve each piece, and remember to flip the inequality signs if you divide by a negative a.
Special cases
| Inequality | Solution | Why |
|---|---|---|
| |ax + b| < c with c < 0 | no solution | an absolute value is never negative |
| |ax + b| > c with c < 0 | all real numbers | every absolute value beats a negative number |
| |ax + b| < 0 | no solution | nothing is below 0 |
| |ax + b| ≤ 0 | the single point x = -b/a | only 0 is ≤ 0 |
| |ax + b| > 0 | all reals except x = -b/a | only the zero is excluded |
| |ax + b| ≥ 0 | all real numbers | always true |
Example: |2x - 3| < 5
- -5 < 2x - 3 < 5
- Add 3: -2 < 2x < 8
- Divide by 2: -1 < x < 4
- Writing a range with brackets and parentheses, where brackets include the endpoint. (-1, 4); open dots at -1 and 4 with the segment between them shaded.
How to use it
- Enter a, b, the inequality sign, and c, or type the inequality such as |x + 4| >= 2.
- Read the solution as an inequality, in interval notation, and on the number line.
- Open the steps to see the compound inequality and each algebra move, including sign flips.
How to read the answer
A 'less than' inequality gives one interval between the two critical points; a 'greater than' inequality gives two rays pointing away from them. Open dots and round brackets mean the endpoint is excluded (< or >); filled dots and square brackets mean it is included (≤ or ≥). ∅ means no solution and (-∞, ∞) means every real number.
Common mistakes and edge cases
- Writing -c > ax + b > c for a 'greater than' inequality: that chain is impossible. Use 'or' with two separate inequalities.
- Forgetting to flip the inequality signs when dividing by a negative The number multiplying a variable, like the 3 in 3x..
- Answering |ax + b| < -3 with an interval: a negative right side gives no solution.
- Dropping the How far a number sits from zero, so the answer is never negative. without considering both the positive and negative cases.
Worked examples
Less than: one interval
|2x - 3| < 5 → -1 < x < 4
Solution
-1 < x < 4
Greater than or equal: two rays
|x + 4| ≥ 2 → x ≤ -6 or x ≥ -2
Solution
x ≤ -6 or x ≥ -2
Negative coefficient flips the signs
|-3x + 1| ≤ 7 → -2 ≤ x ≤ 8/3
Solution
-2 ≤ x ≤ 2.666667
No solution
|x - 2| < -1 has no solution
Solution
no solution
All real numbers
|5x + 1| > -4 is always true
Solution
x is any real number
Right side zero
|x - 3| ≤ 0 only at x = 3
Solution
x = 3
Quadratic inside is rejected
|x^2 - 1| < 3 is outside this calculator's scope
Solution
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Frequently asked questions
How do I solve an absolute value inequality?+
Isolate the absolute value, then split it: |u| < c becomes -c < u < c, and |u| > c becomes u < -c or u > c. Solve the resulting linear inequalities and write the answer as an interval or on a number line.
When does an absolute value inequality have no solution?+
When it asks an absolute value to be less than a negative number (or less than 0). Since |u| ≥ 0 always, nothing works.
When is the answer all real numbers?+
When the inequality asks an absolute value to be greater than a negative number, or ≥ 0. Every real number satisfies it.
What is the difference between 'and' and 'or' solutions?+
'Less than' inequalities give 'and' (a bounded interval between two points). 'Greater than' inequalities give 'or' (two rays heading away from the points).
Why do the signs flip?+
Dividing or multiplying an inequality by a negative number reverses its direction. If a < 0 in |ax + b|, the final division by a flips every sign.
How do I graph the solution on a number line?+
Mark the critical points with open dots for < or > and filled dots for ≤ or ≥, then shade between them for 'and' or outward from them for 'or'. The calculator draws this for you.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
References
- OpenStax, Intermediate Algebra 2e, Section 2.7 “Solve Absolute Value Inequalities” — The 'and'/'or' rewrite rules and number-line graphs.
- OpenStax, College Algebra 2e, Section 2.7 “Linear Inequalities and Absolute Value Inequalities”
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Last updated: September 4, 2026