Algebra - Solving equations and inequalities: solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph (Higher tier)

The idea

This is Higher tier only, and it adds three things to the linear work. First, quadratic inequalities: factorise as though it were an equation, find the critical values, then decide which stretch of the curve you want. For x² − x − 6 < 0 the factors are (x − 3)(x + 2), the critical values are −2 and 3, and a sketch of this upward opening curve shows it below the horizontal axis between them, so −2 < x < 3. Second, set notation, which writes that same answer as {x : −2 < x < 3}. Third, inequalities in two letters, where each one gives a boundary line and one side of it: draw the line dashed when the boundary is not included and solid when it is, then shade or label the region where all of the inequalities hold at once.

What earns the marks

• The critical values of a quadratic inequality, found by factorising or by using the formula.
• Choosing the correct stretch of values, supported by a sketch of the curve or by testing a value.
• The answer in the notation asked for, whether that is an inequality, set notation or a number line.
• Each boundary line drawn correctly on a two letter question, dashed or solid to match its inequality.
• Identifying the region that satisfies every inequality at the same time.

Common mistakes and how to put them right

• Offering the critical values as the answer, so −2 and 3 are written down instead of the range between them.

• Taking the values outside the critical values when an upward opening quadratic is below zero, or the values between them when it is above zero.

• Writing a double inequality the wrong way round, such as 3 < x < −2, which no value can satisfy.

• Drawing a solid boundary line for a strict inequality on a region question.

Exam tips

• Sketch the curve, mark the critical values on it and shade the part you want; the sketch settles the direction of the signs for you.
• If you are unsure which stretch to take, test one value from each stretch in the original inequality.
• Set notation needs the letter as well as the condition, as in {x : x ≥ 4}, so do not write the condition on its own.