📍 Algebra Solver

Absolute Value Inequality Calculator (|ax + b| < c)

Q: How do you solve an absolute value inequality?

First isolate the absolute value expression. For a less-than inequality, write a compound inequality between the two boundary values. For a greater-than inequality, create two inequalities joined by OR, then solve for the variable.

Q: What is the difference between less than and greater than absolute value inequalities?

A less-than absolute value inequality gives values inside the boundary points, while a greater-than absolute value inequality gives values outside the boundary points. For example, |x| 3 gives x 3.

Solve absolute value inequalities step-by-step using positive and negative cases with interval notation and number line solutions.

Absolute Value Inequality

Enter values to calculate instant result

Enter Absolute Value Inequality ℹ️ Info

💡 Try:

⚡ Quick Tools: 👥 2×2 System of Equations Calculator 🧊 3×3 System of Equations Calculator 🔄 Substitution Method Calculator

Verified Solution

---

📋

Calculation Reference Table

Benchmark values and quick reference intervals

Top 10 Reference Values

📖

How to Use & Educational Guide

In-depth guide for Absolute Value Inequality Calculator

Educational Overview

What Is an Absolute Value Inequality?

An absolute value inequality is an inequality that contains an absolute value expression, such as $|ax + b| < c$ or $|ax + b| \ge c$. Absolute value represents the distance from zero on a number line, so the solution can be a range of values rather than a single number. To solve an absolute value inequality, first isolate the absolute value expression and then apply the appropriate inside or outside interval rule.

Absolute Value Inequality Rules

The inequality symbol determines whether the solution lies between two boundary values or outside them.

$|X| < c \implies -c < X < c$

$|X| \le c \implies -c \le X \le c$

$|X| > c \implies X < -c \quad \text{OR} \quad X > c$

$|X| \ge c \implies X \le -c \quad \text{OR} \quad X \ge c$

Quick rule: Less than means inside the two boundary points, while greater than means outside the two boundary points. Inclusive signs ($\le$, $\ge$) include the boundary values.

How to Solve an Absolute Value Inequality Step-by-Step?

Use these steps to solve most single-variable absolute value inequalities:

Step 1 (Isolate the Absolute Value): Move constants and coefficients so that the absolute value expression is alone on one side of the inequality.
Step 2 (Check the Right Side): Remember that an absolute value is always non-negative. For example, $|X| < -3$ has no solution, while $|X| > -3$ is true for every real number.
Step 3 (Apply the Correct Rule): For less-than inequalities, create a compound inequality. For greater-than inequalities, create two separate inequalities joined by OR.
Step 4 (Solve for x): Use normal algebraic operations to isolate the variable. If you multiply or divide by a negative number, reverse the inequality sign.
Step 5 (Write the Solution): Express the answer as an inequality, interval notation, or number line depending on the required format.

How to Graph an Absolute Value Inequality?

Absolute value inequalities can be represented visually on a number line. Use an open circle when the endpoint is not included ($<$ or $>$), and a closed circle when the endpoint is included ($\le$ or $\ge$).

Inside interval: For $|x| < c$ or $|x| \le c$, shade between the two boundary points.
Outside interval: For $|x| > c$ or $|x| \ge c$, shade to the left and right of the boundary points.
Open endpoint: Use an open circle when the boundary value is excluded.
Closed endpoint: Use a closed circle when the boundary value is included.

Absolute Value Inequality Examples

Example 1: Solve $|2x - 3| < 5$

• Step 1: Apply the less-than rule: $-5 < 2x - 3 < 5$.

• Step 2: Add 3 to all three parts: $-2 < 2x < 8$.

• Step 3: Divide all parts by 2: $\mathbf{-1 < x < 4}$.

• Step 4: Convert to interval notation: $(-1, 4)$.

👉 Solution: $-1 < x < 4$

Example 2: Solve $|x + 4| \ge 7$

• Step 1: Apply the greater-than-or-equal rule: $x + 4 \le -7$ OR $x + 4 \ge 7$.

• Step 2: Solve the first branch: $x \le -11$.

• Step 3: Solve the second branch: $x \ge 3$.

• Step 4: Write the union: $(-\infty, -11] \cup [3, \infty)$.

👉 Solution: $x \le -11$ OR $x \ge 3$

Example 3: Solve $3|x - 2| + 4 \le 16$

• Step 1: Subtract 4 from both sides: $3|x - 2| \le 12$.

• Step 2: Divide by 3: $|x - 2| \le 4$.

• Step 3: Apply the less-than-or-equal rule: $-4 \le x - 2 \le 4$.

• Step 4: Add 2 to all three parts: $\mathbf{-2 \le x \le 6}$.

• Step 5: Interval notation: $[-2, 6]$.

👉 Solution: $-2 \le x \le 6$

Example 4: Solve $|x - 1| > 3$

• Step 1: Apply the greater-than rule: $x - 1 < -3$ OR $x - 1 > 3$.

• Step 2: Solve both branches: $x < -2$ OR $x > 4$.

• Step 3: Write the interval solution: $(-\infty, -2) \cup (4, \infty)$.

👉 Solution: $x < -2$ OR $x > 4$

Absolute Value Inequalities with Negative Constants

Since an absolute value cannot be negative, inequalities with a negative number on the right side can often be solved immediately.

• $|x + 2| < -4$ → No Solution

• $|x + 2| \le -4$ → No Solution

• $|x + 2| > -4$ → All Real Numbers

• $|x + 2| \ge -4$ → All Real Numbers

Real-World Applications of Absolute Value Inequalities

🏭 Manufacturing Tolerances

Manufacturers use absolute value inequalities to describe acceptable measurement ranges. For example, if a part should measure 50 mm with a tolerance of 0.2 mm, the condition can be written as $|x - 50| \le 0.2$, giving $49.8 \le x \le 50.2$.

🌡️ Temperature & Measurement Limits

Absolute value inequalities can describe how far a measured value may differ from a target. For example, a temperature within 3 degrees of 20°C satisfies $|T - 20| \le 3$, so the acceptable range is $17 \le T \le 23$.

Common Mistakes to Avoid

Using the Wrong Rule: Remember that less-than inequalities produce an inside interval, while greater-than inequalities produce outside intervals.
Forgetting the OR: Greater-than absolute value inequalities must be split into two branches joined by OR.
Ignoring Negative Constants: An absolute value can never be less than a negative number. Check the right side before doing algebra.
Forgetting to Reverse the Sign: When solving either branch, reverse the inequality whenever you multiply or divide by a negative number.
Using the Wrong Endpoint: Use parentheses for excluded endpoints and square brackets for included endpoints in interval notation.

Mathematical accuracy verified by AptCalc Engine

Frequently Asked Questions (FAQs)

Answers to common questions about this calculator

What is an absolute value inequality?

An absolute value inequality is an inequality containing an absolute value expression, such as |ax + b| < c or |ax + b| ≥ c. Its solution is usually a range of values rather than one exact number.

How do you solve an absolute value inequality?

First isolate the absolute value expression. For a less-than inequality, write a compound inequality between the two boundary values. For a greater-than inequality, create two inequalities joined by OR, then solve for the variable.

What is the difference between less than and greater than absolute value inequalities?

A less-than absolute value inequality gives values inside the boundary points, while a greater-than absolute value inequality gives values outside the boundary points. For example, |x| < 3 gives -3 < x < 3, while |x| > 3 gives x < -3 or x > 3.

When do you flip the inequality sign?

Flip the inequality sign whenever you multiply or divide both sides of an inequality by a negative number.

What happens if the number on the right side is negative?

Because absolute values are always non-negative, |x| < a negative number has no solution. An inequality such as |x| > a negative number is true for all real numbers.