Absolute Value Inequality Calculator
Absolute value inequalities describe a range of numbers rather than one or two exact answers, and the shape of that range depends entirely on which direction the inequality points. A "less than" inequality closes in on a single connected interval, while a "greater than" inequality splits apart into two separate rays. This calculator covers five situations: |ax + b| less than a value, |ax + b| greater than a value, double/compound inequalities like m < |ax + b| < n, comparing two absolute value expressions directly, and converting any solution range into interval notation.
|ax + b| > c → ax + b > c or ax + b < −c (two separate rays, an "or" statement)
Pick the exact inequality shape you have in the tool below — each is explained with formulas and interval notation further down the page.
What an Absolute Value Inequality Means
Since |x| measures distance from zero, an inequality like |x| < 3 is really asking: "which numbers are within 3 units of zero?" Geometrically, that's every point between −3 and 3 on the number line. An inequality like |x| > 3, on the other hand, asks for numbers farther than 3 units from zero — which means the solution has to split into two separate pieces, one to the left of −3 and one to the right of 3.
- |expression| < c (or ≤ c) always produces one connected interval — an "and" condition
- |expression| > c (or ≥ c) always produces two separate rays — an "or" condition
- If c is negative, "less than" has no solution and "greater than" is true for all real numbers
- ≤ and ≥ include their boundary points; < and > do not
Solving |ax + b| < c ("and")
Example: Solve |x − 2| < 5
−5 < x − 2 < 5 → add 2 to all three parts → −3 < x < 7
Rewriting the absolute value as a compound "and" inequality lets you solve for x across all three parts simultaneously, the same way you'd solve any linear inequality — whatever operation you perform on one part, you perform on all three, including flipping both inequality signs if you multiply or divide by a negative.
Solving |ax + b| > c ("or")
Example: Solve |x + 1| > 4
x + 1 > 4 → x > 3
x + 1 < −4 → x < −5
Solution: x < −5 or x > 3
Because both branches can never overlap, the solution to a "greater than" absolute value inequality is always reported as two separate pieces joined by "or," not a single range. Graphed on a number line, this looks like two rays pointing away from each other, with a gap in between.
Double/Compound Inequalities
Example: Solve 2 < |x − 3| < 7
2 < x−3 < 7 → 5 < x < 10
−7 < x−3 < −2 → −4 < x < 1
Solution: −4 < x < 1 or 5 < x < 10
A double inequality like this asks for values whose absolute value is neither too small nor too large — it combines a "greater than" style split with a "less than" style bound on each branch, producing two separate finite intervals instead of one interval or two infinite rays.
Comparing Two Absolute Values
Example: Solve |2x − 1| ≤ |x + 4|
A=B: 2x−1 = x+4 → x = 5
A=−B: 2x−1 = −(x+4) → 3x = −3 → x = −1
Solution: −1 ≤ x ≤ 5
Comparing two absolute value expressions is equivalent to comparing their squares, since squaring removes the absolute value bars entirely while preserving the inequality's truth for any real numbers. That turns the problem into a quadratic-style sign analysis: find the two points where the expressions are equal or opposite, and those points become the boundaries of the solution.
Interval Notation
Example: −3 < x ≤ 7 in interval notation is (−3, 7]
Example: x < 4 in interval notation is (−∞, 4)
Interval notation is a compact way to write a solution set without inequality symbols. A "less than" solution (one connected range) is written as a single interval, while a "greater than" solution (two separate rays) is written as a union of two intervals joined with the ∪ symbol, such as (−∞, −5) ∪ (3, ∞).
The Trap: Flipping the Wrong Sign
Solving |x + 1| > 4 as x+1 > 4 or x+1 > −4 (using the same direction twice)
Correct: the second branch must flip direction — x+1 < −4, giving x < −5
On the negative branch of a "greater than" inequality, the sign must flip to "less than" — this is easy to forget because it feels asymmetric compared to the positive branch. A quick sanity check: pick a number from your claimed solution set and substitute it into the original inequality; if it doesn't work, the sign was likely applied incorrectly on one branch.
Real-Life Uses
- Manufacturing tolerances — a part is acceptable only if it's within a stated tolerance of the target measurement, an "and"-style absolute value inequality
- Safety and alert thresholds — systems that trigger an alarm only when a reading strays too far from normal use "greater than" absolute value logic
- Signal and error bounds — engineering and statistics frequently bound acceptable error as |actual − expected| ≤ tolerance
- Scheduling and range windows — "within 2 hours of the appointment time" or "more than 5 miles from the depot" are everyday absolute value inequality statements
- Algebra, precalculus, and standardized test prep — compound and double absolute value inequalities are a recurring topic on the SAT, ACT, and in college algebra courses
Glossary
- Compound Inequality
- An inequality that combines two conditions with "and" (producing one connected interval) or "or" (producing two separate intervals).
- Interval Notation
- A shorthand way of writing a solution range using brackets and parentheses instead of inequality symbols.
- Open Interval
- A range that excludes its endpoints, written with round parentheses, such as (2, 9).
- Closed Interval
- A range that includes its endpoints, written with square brackets, such as [2, 9].
- Union
- The combination of two separate solution intervals, written with the ∪ symbol, used when a solution is made of disconnected pieces.
Frequently Asked Questions
Q: What is an absolute value inequality?
A: An inequality comparing an absolute value expression to a number using <, ≤, >, or ≥, which describes a range of values rather than exact solutions.
Q: How do you solve |ax + b| < c?
A: Rewrite as −c < ax + b < c and solve across all three parts for one continuous interval.
Q: How do you solve |ax + b| > c?
A: Split into ax + b > c or ax + b < −c, producing two separate ranges.
Q: What is the difference between "and" and "or" in absolute value inequalities?
A: "Less than" inequalities become "and" statements (one interval); "greater than" inequalities become "or" statements (two disconnected intervals).
Q: How do you write an absolute value inequality solution in interval notation?
A: Use square brackets for included endpoints (≤, ≥) and round parentheses for excluded or infinite endpoints (<, >). For example, −2 ≤ x < 5 becomes [−2, 5).
Q: What is a common mistake when solving absolute value inequalities?
A: Forgetting to flip the inequality direction on the negative branch of a "greater than" inequality — the correct second branch uses < −c, not > −c.