Inequality Calculator Examples: Simple and Compound Problems

July 16, 2026

Four worked examples below, using the same step-by-step process the inequality calculator runs. For term definitions, see the inequality symbols glossary.

Example 1 — Simple Inequality, No Flip Needed

Solve 5x − 7 ≥ 13

Add 7 to both sides:   5x ≥ 20
Divide both sides by 5 (positive, no flip):  x ≥ 4
Interval notation: [4, ∞)

Example 2 — Simple Inequality, Sign Flip Required

Solve −4x + 9 > −3

Subtract 9 from both sides:   −4x > −12
Divide both sides by −4 — negative, so > flips to <:   x < 3
Interval notation: (−∞, 3)

Example 3 — Compound Inequality, Positive Coefficient

Solve 2 ≤ 3x − 1 < 11

Add 1 to all three parts:   3 ≤ 3x < 12
Divide all three parts by 3 (positive, no flip):   1 ≤ x < 4
Interval notation: [1, 4)

Notice the left bracket is square (≤ stays inclusive) while the right stays a parenthesis (< stays exclusive) — see the inequality symbols glossary for why the two bounds of a compound inequality can use different bracket types.

Example 4 — Real-World Word Problem

“A rental car costs $40 plus $0.25 per mile. Your budget is at most $100. How many miles can you drive?”

Let x = miles driven
Cost = 0.25x + 40
"At most $100" → 0.25x + 40 ≤ 100

Subtract 40 from both sides:   0.25x ≤ 60
Divide both sides by 0.25 (positive, no flip):   x ≤ 240

Since a negative number of miles doesn’t make sense in context, the practical solution is 0 ≤ x ≤ 240 — up to 240 miles. See solving word problems with linear inequalities for the full phrase-translation method used to set this problem up.

Try Your Own Problem

Enter your own coefficients, constants, and comparison operators into the inequality calculator — Simple mode for a single inequality, Compound mode for a three-part chained inequality — to see every step, including an explicit flip warning whenever one applies.