Why do you flip the inequality sign when dividing by a negative?

Because multiplying or dividing by a negative reverses the order of numbers on the number line. For example, 2 > 1 is true, but multiplying both sides by −1 gives −2 and −1, and −2 < −1. So the inequality sign must flip to keep the statement true.

What is interval notation for an inequality?

Interval notation expresses the solution set using brackets and parentheses. A square bracket [ or ] means the endpoint is included (≤ or ≥). A parenthesis ( or ) means the endpoint is excluded ( ). Infinity always uses a parenthesis. Example: x > 3 is written (3, ∞).

How do you solve a compound inequality?

For 'and' inequalities like 2 5, solve each separately and the solution is their union: (−∞, 2) ∪ (5, ∞).

Inequality Calculator – Solve Linear Inequalities Step by Step

Enter a linear inequality in the form ax + b > c (or <, ≥, ≤). Supports positive and negative coefficients.

a (x coefficient)

x +

b (constant, left)

Sign

c (right side)

Quick examples:

Inequality Calculator FAQ

Common questions about solving and graphing inequalities.

How do you solve a linear inequality?

Solve a linear inequality just like an equation — isolate x using addition, subtraction, multiplication, or division. The critical rule: if you multiply or divide by a negative number, flip the inequality sign.

Example: 2x + 3 > 7 → 2x > 4 → x > 2

Why do you flip the sign when dividing by a negative?

Multiplying or dividing by a negative number reverses the order on the number line. For example, 3 > 1 is true. Multiply both sides by −1: −3 and −1 — but now −3 < −1. The relationship flipped, so the sign must flip too to keep the statement true.

What is interval notation?

Interval notation uses brackets and parentheses to show the solution set:

( ) — endpoint excluded (strict inequality < or >)
[ ] — endpoint included (≤ or ≥)
Infinity always uses ( )

So x > 3 → (3, ∞), and x ≤ −1 → (−∞, −1]

What is the difference between < and ≤?

< means "strictly less than" — the boundary value is NOT part of the solution. On a number line, shown with an open circle ○.

≤ means "less than or equal to" — the boundary value IS included. On a number line, shown with a filled circle ●.

How do you solve −3x ≤ 9?

Divide both sides by −3 — and because you're dividing by a negative, flip the sign:
−3x ≤ 9 → x ≥ −3

Solution: x ≥ −3, or in interval notation: [−3, ∞)