Inequality Solver

📘 Guide

What this Inequality Solver does

An inequality is like an equation, except instead of asking “what value makes both sides equal,” it asks “what values make the left side smaller or larger than the right side.” This calculator solves one‑variable linear inequalities (the most common type you see in school, homework, and day‑to‑day reasoning): things like 2x + 5 < 17, -3x ≥ 12, or 4 - x ≤ 9. It also supports compound inequalities where the variable is trapped between two bounds, like 1 < x ≤ 6.

Your result is shown in a few useful formats: interval notation (like (-∞, 6]), set‑builder notation (like { x | x ≤ 6 }), and a quick plain‑English interpretation. The calculator also prints step‑by‑step working, including the only “gotcha” that causes most mistakes: when you multiply or divide by a negative number, the inequality flips.

Formula breakdown (linear inequality)

Most linear inequalities can be simplified into the standard form:

Ax + B < C (or ≤, >, ≥)

The solving strategy is basically the same as solving a linear equation: get all x terms on one side and all constant terms on the other, then isolate x.

If you start with a general inequality:

a(x) + b < c(x) + d

move the variable terms together and constants together:

Subtract c(x) from both sides → (a - c)x + b < d
Subtract b from both sides → (a - c)x < (d - b)
Divide both sides by (a - c) → x < (d - b)/(a - c)

Important: if (a - c) is negative, dividing by it flips the sign:

(a - c)x < k becomes x > k/(a - c) when (a - c) < 0.

How compound inequalities work

A compound inequality traps x between two bounds:

L < x ≤ U

You can think of it as two inequalities at the same time:

L < x (x is bigger than L)
x ≤ U (x is at most U)

So the solution is the overlap: all x values that satisfy both. In interval notation, the example above becomes (L, U]. Parentheses mean “not included,” brackets mean “included.”

Examples (with explanations)

Example 1: Solve 2x + 5 < 17

Subtract 5 from both sides: 2x < 12
Divide by 2: x < 6
Interval notation: (-∞, 6)

Example 2: Solve -3x ≥ 12

Divide by -3 (negative!) → flip the sign: x ≤ -4
Interval notation: (-∞, -4]

Example 3: Solve 4 - x ≤ 9

Subtract 4: -x ≤ 5
Multiply by -1 (negative!) → flip: x ≥ -5
Interval notation: [-5, ∞)

Example 4 (compound): Solve 1 < x ≤ 6

This already isolates x. Solution is (1, 6]
In set-builder form: { x | 1 < x ≤ 6 }

How it works (what the calculator does behind the scenes)

To solve a linear inequality, the calculator turns each side into a simple form:

(coefficient)·x + (constant)

For example, 3x - 7 becomes “coefficient = 3” and “constant = -7.” Then it subtracts the right side from the left side to combine terms:

(aL - aR)x < (bR - bL)

Finally it isolates x by dividing by the remaining coefficient. If that coefficient is negative, the solver flips the direction automatically and formats the final answer as an interval and as a set.

Common mistakes this page helps you avoid

Forgetting to flip the inequality when dividing or multiplying by a negative number.
Sign errors when moving terms across the inequality.
Mixing up brackets vs parentheses in interval notation.
Confusing “and” vs “or” when interpreting compound inequalities.

FAQ (quick, clear answers)

What types of inequalities can this solver handle?
This page is optimized for one‑variable linear inequalities and the most common compound inequalities (bounds around x). If you paste a more complex expression (like absolute-value inequalities or quadratics), the result may be incomplete — use it as a quick checker, then verify manually.
Why do inequalities “flip” when dividing by a negative?
Multiplying by a negative reverses order on the number line. A simple way to remember: if 2 < 5, multiplying both sides by -1 gives -2 > -5. The direction must flip to stay true.
What does (-∞, 6] mean?
It means “all numbers less than or equal to 6.” Infinity always uses parentheses because it’s not a real number you can include.
What’s the difference between < and ≤?
< means “strictly less than” (endpoint not included), while ≤ means “less than or equal to” (endpoint included).
How do I write compound inequalities in interval notation?
Use parentheses for strict bounds and brackets for inclusive bounds. For example: 2 < x < 9 → (2, 9), while 2 ≤ x < 9 → [2, 9).
How can I sanity-check my answer?
Pick a test number inside your solution interval and plug it into the original inequality. Then pick a test number outside the interval. The inside number should make the inequality true, and the outside number should make it false.
Can an inequality have “no solution” or “all real numbers”?
Yes. If the x terms cancel out, you might end up with something like 5 < 2 (never true → no solution) or 5 < 9 (always true → all real numbers). The solver detects this case and prints it clearly.

Educational note: This tool is designed for quick checking and learning. For graded work, always show your steps.

Enter an inequality