Inequality Calculator

Understanding Inequalities in UK Maths

An inequality is a mathematical statement that compares two expressions using one of the symbols < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike equations, which have a single solution, inequalities typically have a range of solutions represented as an interval on the number line.

Inequalities appear throughout UK mathematics — from GCSE Foundation and Higher tiers through to A-Level Pure Mathematics. Mastering inequalities requires understanding algebraic manipulation, number line representation, and the special rules that distinguish inequalities from equations.

Mode 1: Linear Inequalities (ax + b OP c)

A linear inequality involves x to the first power only. The method for solving is identical to solving a linear equation, with one crucial exception: if you multiply or divide both sides by a negative number, you must reverse (flip) the inequality sign.

Example: Solve 3x − 4 > 8

Example with sign flip: Solve −2x + 1 ≤ 7

Mode 2: Compound Inequalities (a OP ax + b OP c)

A compound inequality of the form a < ax + b < c represents two conditions simultaneously. To solve, apply the same operation to all three parts of the inequality at once.

Example: Solve 1 < 2x + 1 ≤ 13

The solution set is the interval (0, 6] — all values of x strictly greater than 0 and up to and including 6.

Mode 3: Quadratic Inequalities (ax² + bx + c OP 0)

Quadratic inequalities require finding the roots of the corresponding equation and then determining which intervals satisfy the original inequality. The shape of the parabola (opening up or down) determines the structure of the solution.

Example: Solve x² − 5x + 6 > 0

Example 2: Solve x² − 5x + 6 < 0

Number Line Representation: Key Conventions

UK GCSE and A-Level exams require you to represent inequalities on a number line correctly:

• Open circle (○): The value is NOT included. Used for strict inequalities < and >.
• Closed/filled circle (●): The value IS included. Used for ≤ and ≥.
• Arrow pointing right (→): x extends to +∞ (x > a or x ≥ a).
• Arrow pointing left (←): x extends to −∞ (x < a or x ≤ a).
• Solid line between two circles: For compound/bounded inequalities (a ≤ x ≤ b).

Integer Solutions of Inequalities

Some GCSE questions ask for the integer (whole number) solutions within an inequality. For example, if the solution is −2 < x ≤ 5, the integer solutions are −1, 0, 1, 2, 3, 4, 5. Note that −2 is not included (strict inequality) but 5 is included (≤). Always read the inequality signs carefully when listing integers.

Common Inequality Mistakes at GCSE and A-Level

• Forgetting to flip the sign: The most common error. Any multiplication or division by a negative number reverses the inequality direction.
• Incorrect number line circles: Confusing open and closed circles is a common mark-losing error in GCSE exams.
• Treating quadratic inequalities like equations: Writing x > 2 and x > 3 instead of x > 3 (only the outermost condition is valid for "greater than both").
• Using 'and' instead of 'or' for quadratic solutions: x < 2 OR x > 3 is not the same as x < 2 AND x > 3 (which is impossible).
• Squaring both sides: Squaring both sides of an inequality can introduce errors if both sides are not guaranteed to be non-negative. Always verify solutions.

Set Notation for Inequalities

UK A-Level students must be familiar with set notation for expressing solution sets:

• x > 3: Set notation: {x ∈ ℝ : x > 3} | Interval notation: (3, ∞)
• x ≤ −2: Set notation: {x ∈ ℝ : x ≤ −2} | Interval notation: (−∞, −2]
• 2 ≤ x < 7: Set notation: {x ∈ ℝ : 2 ≤ x < 7} | Interval notation: [2, 7)
• x < −1 or x > 4: Set notation: {x ∈ ℝ : x < −1} ∪ {x ∈ ℝ : x > 4}

Frequently Asked Questions: Inequalities