Linear inequalities

Solve and interpret linear inequalities, including flipping the sign, integer solutions and real-world constraints.

Also known as: solving inequalities, inequality word problems, systems of inequalities.

SAT · Algebra GCSE Maths · Algebra IGCSE Maths · Algebra and graphs

Key ideas

Solve an inequality like an equation: do the same thing to both sides.
Multiplying or dividing both sides by a negative number reverses the inequality sign.
$<$ and $>$ exclude the boundary; $\le$ and $\ge$ include it.
In context, "at most" means ≤ and "at least" means ≥.

Solving inequalities

Treat $3x - 4 < 11$ like an equation: add $4$ to get $3x < 15$ , then divide by $3$ : $x < 5$ .

The one new rule: when you multiply or divide by a negative, reverse the sign. From $-2x \ge 6$ , dividing by $-2$ gives $x \le -3$ .

Integer solutions and double inequalities

For $-2 < x \le 3$ , the integers are $-1, 0, 1, 2, 3$ (not $-2$ , because the boundary is strict).

Solve a double inequality by doing the same thing to all three parts: $1 \le 2x + 3 < 9$ gives $-2 \le 2x < 6$ , so $-1 \le x < 3$ .

Inequalities in context

"A van can carry at most $900$ kg. Each box weighs $35$ kg and the driver weighs $80$ kg." gives $35b + 80 \le 900$ , so $b \le 23.4$ . The answer must be a whole number of boxes, so the maximum is $23$ . Always round down for "at most" questions about whole items.

Worked examples

Example 1 · Dividing by a negative

Solve $7 - 3x \ge 19$ .

Step 1.Subtract $7$ : $-3x \ge 12$ .
Step 2.Divide by $-3$ and reverse the sign: $x \le -4$ .

Answer: $x \le -4$

Example 2 · Maximum in context

A phone plan costs 12 per month plus 0.05 per text. Ali can spend at most 20 per month. What is the greatest number of texts Ali can send?

Step 1. $12 + 0.05t \le 20$ , so $0.05t \le 8$ .
Step 2. $t \le 160$ .

Answer: 160 texts

Common mistakes

Forgetting to reverse the sign after dividing by a negative number.
Including a boundary value that a strict inequality (< or >) excludes.
Rounding up in an "at most" problem, which breaks the constraint.

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Linear inequalitiesEasier

Which value of $x$ satisfies $3x - 4 < 11$ ?

Question 2Linear inequalitiesEasier

Solve $-2x \ge 6$ . What is the greatest possible value of $x$ ?