Absolute Value Inequality Solver

Free Online Absolute Value Inequality Solver

Our Absolute Value Inequality Solver solves inequalities of the form |ax + b| op c, where op is one of <, ≤, >, ≥, or =. It provides step-by-step solutions with AND/OR logic explanations and interval notation. Whether you are solving |x + 3| < 5, |2x - 1| ≤ 7, or |x - 2| > 3, this calculator handles all cases including special edge cases like negative right-hand sides and zero.

What Is an Absolute Value Inequality?

An absolute value inequality is an inequality that contains an absolute value expression. The absolute value of a number represents its distance from zero on the number line, always resulting in a non-negative value. When solving absolute value inequalities, you must consider both the positive and negative possibilities of the expression inside the absolute value bars, which leads to either an AND (compound) or OR (union of intervals) solution.

In mathematical notation, for any real numbers $$a$$, $$b$$, and positive constant $$c$$:

• $$|ax + b| < c$$ means $$-c < ax + b < c$$ (AND logic)
• $$|ax + b| ≤ c$$ means $$-c ≤ ax + b ≤ c$$ (AND logic)
• $$|ax + b| > c$$ means $$ax + b < -c$$ or $$ax + b > c$$ (OR logic)
• $$|ax + b| ≥ c$$ means $$ax + b ≤ -c$$ or $$ax + b ≥ c$$ (OR logic)

How to Solve Absolute Value Inequalities

Follow these steps to solve any absolute value inequality:

1. Isolate the absolute value expression on one side of the inequality.
2. Check the right-hand side: If c < 0 and the inequality is < or ≤, there is no solution. If c < 0 and the inequality is > or ≥, all real numbers are solutions.
3. Apply the correct logic:
   • For < or ≤ (AND logic): Rewrite as $$-c < ax + b < c$$ (or with ≤).
   • For > or ≥ (OR logic): Rewrite as $$ax + b < -c$$ or $$ax + b > c$$ (or with ≤/≥).
4. Solve the resulting compound inequality by isolating $$x$$ in each part.
5. Express the solution in interval notation.

Examples

Example 1: AND Logic (|x + 3| < 5)

Solve $$|x + 3| < 5$$.

1. The absolute value is already isolated: $$|x + 3| < 5$$.
2. Since $$5 \ge 0$$ and the operator is <, use AND logic: $$-5 < x + 3 < 5$$.
3. Subtract 3 from all parts: $$-5 - 3 < x < 5 - 3$$.
4. Simplify: $$-8 < x < 2$$.
5. Interval notation: $$(-8, 2)$$.

Example 2: OR Logic (|x - 2| > 3)

Solve $$|x - 2| > 3$$.

1. The absolute value is already isolated: $$|x - 2| > 3$$.
2. Since $$3 \ge 0$$ and the operator is >, use OR logic: $$x - 2 < -3$$ or $$x - 2 > 3$$.
3. Add 2 to both sides of each inequality: $$x < -1$$ or $$x > 5$$.
4. Interval notation: $$(-\infty, -1) \cup (5, \infty)$$.

Example 3: With Coefficient (|2x - 1| ≤ 7)

Solve $$|2x - 1| ≤ 7$$.

1. Absolute value is isolated: $$|2x - 1| ≤ 7$$.
2. Use AND logic (because ≤): $$-7 ≤ 2x - 1 ≤ 7$$.
3. Add 1: $$-6 ≤ 2x ≤ 8$$.
4. Divide by 2: $$-3 ≤ x ≤ 4$$.
5. Interval notation: $$[-3, 4]$$.

AND vs OR Logic

The key to solving absolute value inequalities is understanding when to use AND versus OR logic:

A helpful mnemonic: "Less thAND" — when the absolute value is less than a number, use AND logic. "GreatOR" — when the absolute value is greater than a number, use OR logic.

Special Cases

• Negative right-hand side with < or ≤: No solution. Example: $$|x + 1| < -3$$ has no solution because absolute value is always ≥ 0.
• Negative right-hand side with > or ≥: All real numbers. Example: $$|x + 1| > -3$$ is always true.
• Zero right-hand side with <: No solution. $$|x| < 0$$ is impossible.
• Zero right-hand side with ≤: Only where the expression equals zero. $$|x| ≤ 0$$ means $$x = 0$$.
• Zero right-hand side with >: All real numbers except the zero point.
• Zero right-hand side with ≥: All real numbers.

Applications of Absolute Value Inequalities

• Error Tolerance: Manufacturing tolerances like $$|x - 10| ≤ 0.05$$ ensure parts are within specification.
• Distance Problems: "Within 5 units of 3" translates to $$|x - 3| < 5$$.
• Range Constraints: Expressing acceptable ranges in engineering and science.
• Temperature Ranges: "The temperature should not deviate more than 2 degrees from 70°F" means $$|T - 70| ≤ 2$$.

Using the Calculator

Enter your inequality in the input field using the format |expression|operator value. Supported operators are < (less than), <= (less than or equal), > (greater than), >= (greater than or equal), and = (equal). The calculator automatically detects whether to use AND or OR logic based on the operator and displays the solution in interval notation with full step-by-step reasoning.

For related absolute value topics, check the absolute value calculator for single-number evaluation and the absolute value equation solver for equations with discrete solutions. The absolute difference calculator explores the distance interpretation of absolute values, and the domain and range calculator helps visualize the complete set of possible inputs and outputs for absolute value functions.