What you'll learn
Linear inequalities describe a range of values rather than a single answer. In this guide you will learn the inequality symbols, how to solve linear inequalities like equations, the special rule when multiplying or dividing by a negative, how to represent solutions on a number line, and how to list integer solutions. These skills are widely tested in GCSE algebra.
Key terms and definitions
Inequality — a statement that one quantity is greater or less than another, using <, >, ≤ or ≥.
Solution set — the range of values that satisfy the inequality.
Number line — a line used to represent the solution set.
Strict inequality — < or > (the endpoint is not included).
Inclusive inequality — ≤ or ≥ (the endpoint is included).
Core concepts
Inequality symbols
The symbols are: <** (less than), **> (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). For example, x > 3 means x is any value bigger than 3, while x ≤ 5 means x is 5 or less.
Solving like an equation
Solve a linear inequality using the same steps as an equation — add, subtract, multiply or divide both sides — keeping the inequality sign. For example, 2x + 1 < 9 → 2x < 8 → x < 4. The solution is a range, not a single value.
The negative rule
There is one crucial difference: if you multiply or divide both sides by a negative number, you must reverse the inequality sign. For example, −2x < 6 → divide by −2 and flip: x > −3. Forgetting to flip is the most common error.
Representing on a number line
Show the solution on a number line using:
- An open circle (○) for < or > (endpoint not included).
- A filled circle (●) for ≤ or ≥ (endpoint included).
- An arrow or line showing the direction of all included values.
For a "between" inequality (e.g. −2 ≤ x < 3), mark both endpoints with the correct circles and shade between them.
Integer solutions
Some questions ask for integer (whole-number) solutions. List the integers that lie in the range. For example, for −1 < x ≤ 3, the integer solutions are 0, 1, 2, 3 (note −1 is excluded because of the strict <, but 3 is included because of ≤).
Worked examples
Example 1: Solving
Solve 3x − 4 ≥ 11.
Add 4: 3x ≥ 15. Divide by 3: x ≥ 5.
Example 2: Negative rule
Solve 7 − 2x > 1.
Subtract 7: −2x > −6. Divide by −2 and flip: x < 3.
Example 3: Integer solutions
List the integers satisfying −2 ≤ x < 2.
The integers are −2, −1, 0, 1 (−2 included, 2 excluded).
Common mistakes and how to avoid them
Forgetting to flip the sign. Reverse the inequality when multiplying or dividing by a negative.
Wrong circle on the number line. Open circle for < or >; filled for ≤ or ≥.
Including the wrong endpoints in integer lists. Check whether each end is strict or inclusive.
Treating the answer as a single value. Inequalities give a range of solutions.
Mixing up < and >. Read which way the symbol points carefully.
Exam technique for Linear Inequalities
Solve like an equation, keeping the inequality.
Flip the sign whenever you multiply or divide by a negative.
Draw number lines with the correct open/filled circles and direction.
Check endpoints when listing integer solutions.
State the range clearly, not just a single number.
Quick revision summary
Linear inequalities use < (less than), > (greater than), ≤ (≤), ≥ (≥) and describe a range of values. Solve them like equations — add, subtract, multiply or divide both sides — keeping the inequality sign, except when you multiply or divide by a negative number, where you must reverse the sign (−2x < 6 → x > −3). Represent the solution on a number line: an open circle for <** or **> (endpoint excluded), a filled circle for ≤ or ≥ (endpoint included), with an arrow or line showing all included values. For integer solutions, list the whole numbers in the range, checking carefully whether each endpoint is included (≤/≥) or excluded (</>). The most common mistake by far is forgetting to flip the sign with negatives; others are wrong circles, wrong endpoints, and treating the answer as a single value. Solve like an equation, flip for negatives, draw accurate number lines, and check endpoints for integer lists.