Free Absolute Value Inequalities Calculator

What Is an Absolute Value Inequality?

An absolute value inequality involves an expression such as |ax + b| related by <, ≤, >, or ≥ to a constant. Geometrically it describes the distance of ax + b from zero on the number line. This absolute value inequalities solver walks you through each stage of the solution, from isolating the absolute value to splitting the condition into two cases. Because the tool also outputs the result in compact interval form, it serves both as an inequality solver with steps and as an interval notation calculator.

With the coefficients a, b, and c entered for |ax + b| ⋈ c, the calculator instantly decides which rule to apply and displays the transformed inequality. Whether you are dealing with strict inequalities (<, >) or non‑strict ones (≤, ≥), the online solver keeps the steps clear and the algebra correct.

How the Solver Works

The logic behind solving |ax + b| ⋈ c depends on the comparison symbol:

For < or ≤ (conjunction)

If the inequality is |ax + b| < c and c > 0, then

$$-c < ax + b < c$$

If |ax + b| ≤ c, the “<” is simply replaced by “≤”. The calculator then isolates x by performing the same operation on all three parts.

For > or ≥ (disjunction)

If |ax + b| > c and c ≥ 0, then

$$ax + b < -c \quad \text{or} \quad ax + b > c$$

Similarly, |ax + b| ≥ c leads to “≤” and “≥”. Each branch is solved separately, and the union forms the final answer.

The solver also handles special cases:

• When c is negative, |ax + b| < c has no solution while |ax + b| > c is true for all real x.
• When c is zero, the inequality reduces to a simple equality check (e.g., |ax + b| < 0 is impossible).

Step‑by‑Step Example

Suppose you want to solve |2x – 3| ≤ 5. The |ax+b| inequality calculator shows:

1. Because the symbol is “≤”, it applies the conjunction rule: –5 ≤ 2x – 3 ≤ 5.
2. Add 3 to all parts: –2 ≤ 2x ≤ 8.
3. Divide by 2: –1 ≤ x ≤ 4.

The solution in interval notation is [–1, 4]. Every arithmetic step is listed, making this truly a solve absolute value inequalities tool that builds understanding.

Why Use an Online Inequality Solver with Steps?

Manual solving can lead to sign errors, especially when splitting an inequality that involves a negative coefficient or a fraction. An absolute value inequality solver online automates the process while displaying each transformation. It also doubles as an interval notation calculator because it converts the final inequality into brackets and parentheses automatically. Students can verify their homework step by step, and teachers can use it to generate quick examples.

Key Formulas

For the standard form |ax + b| ⋈ c:

• If ⋈ is < or ≤:

$$|ax + b| \leq c \iff -c \leq ax + b \leq c \quad (\text{provided } c \geq 0)$$

• If ⋈ is > or ≥:

$$|ax + b| \geq c \iff ax + b \leq -c \quad \text{or} \quad ax + b \geq c \quad (\text{provided } c \geq 0)$$

These equivalences are the engine behind every calculation. By keeping them visible, the |ax+b| inequality calculator reinforces the mathematical concept while delivering fast, reliable answers.