Algebra - Solving equations and inequalities: solve linear inequalities in one variable; represent the solution set on a number line (Foundation tier)

The idea

An inequality describes a range of values rather than a single one. Handle it the way you handle an equation, doing the same thing to both sides, with one extra rule: multiplying or dividing both sides by a negative number turns the sign round, so from −2x > 6 you get x < −3. For 3x + 4 ≤ 19, take 4 from both sides to reach 3x ≤ 15, then divide by 3 to reach x ≤ 5. On a number line, mark the boundary with an open circle when that value is not included and a filled circle when it is, then draw an arrow along the values that work. If integers are asked for, list them: the integers satisfying −2 < n ≤ 3 are −1, 0, 1, 2 and 3.

What earns the marks

• A correct first step that treats both sides alike, as in an equation.
• The answer written as an inequality rather than as an equation.
• Turning the sign round when both sides are divided by a negative number.
• A number line showing the right kind of circle at the boundary and an arrow in the right direction.
• A complete list of the integers in the range, with none missing and none extra.

Common mistakes and how to put them right

• Writing the answer as x = 5 when the solution is a whole range of values.

• Leaving the sign unchanged after dividing by a negative number, so −2x > 6 is given as x > −3.

• Filling in the circle for a strict inequality, or leaving it open when the boundary value is included.

• Including the boundary in a list of integers when the inequality is strict, so −2 is listed for −2 < n.

Exam tips

• Say your answer aloud, as in x is less than or equal to five, to check the sign is pointing the right way.
• To avoid dividing by a negative number, add the negative term to both sides instead and work from there.
• Label the numbers on your number line; an unlabelled line cannot show which value the circle sits on.