Inequality Calculator

Quick Answer

Last updated: 10 November 2026 ·How we calculate this: inverse operations and sign analysis

How This Inequality Calculator Works

Type an inequality such as 3x − 7 > 2x + 5 and press Calculate. The calculator moves every term to one side, then works out whether it is linear or quadratic. For a linear inequality it isolates x and flips the sign if it divides by a negative number. For a quadratic it finds the roots and checks where the expression is positive or negative. It gives the answer as an inequality and in interval notation, and checks one value.

How to Use the Inequality Calculator

1. Type the inequality using x and one of <, >, <= or >=.
2. Use ^ for powers and brackets where needed.
3. Press Calculate.
4. Read the solution and the interval notation.
5. Follow the steps to see each move, including any sign flip.

Rules for Solving Inequalities

• Add or subtract the same number on both sides: the sign stays the same.
• Multiply or divide by a positive number: the sign stays the same.
• Multiply or divide by a negative number: reverse the sign, so < becomes > and ≤ becomes ≥.
• Quadratics: find the roots of ax² + bx + c = 0. If a > 0, the expression is negative between the roots and positive outside them.

Key Terms

• Inequality: a statement that one expression is less than or greater than another.
• Solution set: all values of x that make the inequality true.
• Interval notation: a short way to write a range, such as [2, 5).
• Strict inequality: < or >, where the end point is not included.
• Compound inequality: two conditions joined, such as 2 ≤ x ≤ 3 or x < 1 or x > 4.

Worked Examples

Linear: solve −2x + 4 ≤ 10.

• Subtract 4: −2x ≤ 6
• Divide by −2 and flip the sign: x ≥ −3
• Interval notation: [−3, ∞)

Quadratic: solve x² − 5x + 6 > 0.

• Roots: x² − 5x + 6 = (x − 2)(x − 3) = 0, so x = 2 and x = 3.
• The parabola opens upward, so it is positive outside the roots.
• Answer: x < 2 or x > 3, which is (−∞, 2) ∪ (3, ∞).

Real-Life Examples

• Budgets: with $50 to spend and $8 tickets plus a $6 fee, 8x + 6 ≤ 50 gives x ≤ 5.5, so at most 5 tickets.
• Grades: to average at least 80 over four tests with 72, 85 and 78, (235 + x) ÷ 4 ≥ 80 gives x ≥ 85.
• Profit: if profit is −x² + 40x − 300, it is positive when 10 < x < 30.
• Homework: check each step of a linear or quadratic inequality.

Inequality vs Equation

• An equation such as 2x + 3 = 9 usually has one answer, x = 3.
• An inequality such as 2x + 3 < 9 has a range of answers, x < 3.
• The steps are the same, except that multiplying or dividing by a negative number flips the inequality sign.

Inequality Signs and Interval Notation

Showing the Answer on a Number Line

On a number line, draw an open circle at an end point that is not included (< or >) and a filled circle at one that is included (≤ or ≥). Then shade the side that works: to the right for greater than, to the left for less than. For an answer between two numbers, shade the part between the circles; for an “or” answer, shade both outer parts.

You can also see a quadratic inequality on a graph. Plot y = x² − 5x + 6 in the Graphing Calculator: the curve is above the x-axis where the expression is positive and below it where it is negative.

Common Mistakes to Avoid

• Forgetting to flip the sign. Dividing by a negative reverses the inequality.
• Wrong brackets. Use [ ] only when the end point is included (≤ or ≥). Infinity always gets ( ).
• Taking a square root of both sides. For x² > 9 the answer is x < −3 or x > 3, not just x > 3.
• Multiplying by x. x might be negative, so this can flip the sign without you knowing.
• Not checking. Put a value from your answer back into the original inequality.

Frequently Asked Questions

References & Sources