Absolute Value Inequalities Calculator
Solve absolute value inequalities of the form |ax + b| < c, > c, ≤ c, or ≥ c instantly.
Enter coefficients a and b, select the inequality operator, and input constant c to find the complete solution set.
Absolute Value Inequalities Calculator
Solve absolute value inequalities of the form |ax + b| < c, > c, ≤ c, or ≥ c instantly.
About the Absolute Value Inequalities Calculator
An absolute value inequality is a mathematical statement that uses an inequality sign (<, >, ≤, or ≥) together with an absolute value expression. The absolute value |ax + b| measures the distance of the expression ax + b from zero on the number line. Asking when that distance satisfies an inequality condition defines a set of values for x, known as the solution set or solution interval.
This calculator handles inequalities of the standard form |ax + b| ○ c, where a is the non-zero coefficient of x, b is the constant inside the absolute value, c is the constant on the right side, and ○ is one of the four inequality operators. The structure of the solution depends on which operator you use and on the relationship between c and zero.
For less-than inequalities (< or ≤): |ax + b| < c is equivalent to −c < ax + b < c, which produces a bounded solution interval. The solution set is the range of x values where ax + b stays within distance c of zero. When c ≤ 0, a strict less-than inequality has no solution because absolute values cannot be less than a non-positive number.
For greater-than inequalities (> or ≥): |ax + b| > c is equivalent to ax + b > c or ax + b < −c, producing two separate unbounded rays. The solution set is the union of all x values where ax + b is at least c units away from zero in either direction. When c < 0, every real number satisfies the inequality because absolute values are always ≥ 0 > c. When c = 0 and the operator is >, the solution is all real numbers except the single point x = −b/a.
Absolute value inequalities model tolerance and range requirements in countless real-world settings. In manufacturing, the acceptable range of a machine part dimension is expressed as |measurement − target| ≤ tolerance. In science, confidence intervals express the uncertainty of a measured value as |measured − true| ≤ error. Speed limits use absolute value logic: if you must stay within 5 mph of 60 mph, then |speed − 60| ≤ 5. Financial portfolios set acceptable risk bands using the same structure.
The most common mistake when solving these inequalities by hand is applying the wrong rule for the two cases. Remember: for less-than, you get a single interval (compound inequality with AND); for greater-than, you get a union of two rays (compound inequality with OR). Never write a less-than solution as two separate rays — that represents a completely different set. Always pay attention to whether the operator is strict (<, >) or non-strict (≤, ≥) when writing the endpoints of your solution.
Absolute Value Inequality Examples
Examples showing bounded and unbounded solution sets for each inequality type.
| Inequality | Solution | Explanation |
|---|---|---|
| |x| < 5 | −5 < x < 5 | Less-than: distance from 0 is less than 5, so x lies in the bounded interval (−5, 5). |
| |2x − 4| ≤ 6 | −1 ≤ x ≤ 5 | Solve −6 ≤ 2x − 4 ≤ 6. Add 4: −2 ≤ 2x ≤ 10. Divide by 2: −1 ≤ x ≤ 5. |
| |x + 1| > 2 | x < −3 or x > 1 | Greater-than: x + 1 > 2 gives x > 1; x + 1 < −2 gives x < −3. |
| |3x − 6| ≥ 9 | x ≤ −1 or x ≥ 5 | 3x − 6 ≥ 9 gives x ≥ 5; 3x − 6 ≤ −9 gives x ≤ −1. |
How to Use the Absolute Value Inequalities Calculator
- Enter coefficient a — the multiplier of x inside the absolute value. It must be a non-zero real number.
- Enter constant b — the number added to ax inside the absolute value. Any real number, including zero.
- Select the inequality operator from the dropdown: < (less than), ≤ (less than or equal), > (greater than), or ≥ (greater than or equal).
- Enter constant c — the right-hand side value. Its sign and the operator determine whether you get a bounded interval, two rays, no solution, or all real numbers.
- Click 'Calculate Solution' to see the complete solution set with the correct interval or union notation.
Absolute Value Inequalities FAQ
What is the difference between a less-than and greater-than absolute value inequality?
A less-than inequality |ax + b| < c produces a single bounded interval — a range of x values between two endpoints. A greater-than inequality |ax + b| > c produces two separate unbounded rays — x values that are either smaller than a lower bound or larger than an upper bound.
When does an absolute value inequality have no solution?
A strict less-than inequality |ax + b| < c has no solution when c ≤ 0, because absolute values are always ≥ 0 and cannot be less than zero or a negative number. A non-strict inequality |ax + b| ≤ c has no solution when c < 0 for the same reason.
When is the solution all real numbers?
A greater-than inequality |ax + b| > c has all real numbers as its solution when c < 0, because absolute values are always ≥ 0 and every real value already satisfies the condition. A non-strict |ax + b| ≥ c also yields all real numbers when c ≤ 0 for the same reason, since |ax + b| is always non-negative.
How does the sign of a affect the solution?
The sign of a flips the inequality direction when you divide both sides to isolate x. The calculator handles this automatically, so you can enter any non-zero a, whether positive or negative, and get the correct orientation of the solution interval.
Can I use this calculator for inequalities with fractions or decimals?
Yes. Enter decimal values for a, b, and c. The calculator works with any real number inputs and returns exact decimal results for the solution endpoints.
How do I write the solution in interval notation?
For a bounded solution like −1 ≤ x ≤ 5, the interval notation is [−1, 5]. For an unbounded solution like x < −3 or x > 1, the interval notation is (−∞, −3) ∪ (1, ∞). Square brackets indicate the endpoint is included; parentheses indicate it is excluded.