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 only)

The idea

At Higher tier, inequalities extend to linear inequalities in one or two variables and quadratic inequalities in one variable. A linear inequality in two variables, such as y < 2x + 1, is solved by rearranging to make y the subject and then representing the solution set as a region on a graph, using a dashed boundary line for strict inequalities and a solid line for ≤ or ≥. A quadratic inequality in one variable, such as x² − 5x + 6 < 0, is solved by finding the critical values where the quadratic equals zero, then testing intervals to determine where the inequality holds. The solution set can be shown on a number line, in set notation such as {x : 2 < x < 3}, and on a graph.

What earns the marks

• Solves a linear inequality in one or two variables by rearranging correctly and, where necessary, reversing the inequality sign when multiplying or dividing by a negative number.
• Finds the critical values of a quadratic inequality by solving the associated quadratic equation.
• Tests intervals between and beyond the critical values to determine where the quadratic inequality is satisfied.
• Represents the solution set on a number line with correct open or filled circles and direction.
• Uses set notation correctly, for example {x : 2 < x < 3} or {x : x ≤ −1} ∪ {x : x ≥ 4}.
• Draws the boundary line correctly on a graph, using a dashed line for strict inequalities and a solid line for inclusive inequalities, and shades the correct region.

Common mistakes and how to put them right

• Assuming a quadratic inequality is satisfied between the critical values without testing.

  Put it right: Test a value in each interval to determine the correct region.

• Using a solid boundary line for a strict inequality on a graph.

  Put it right: Use a dashed line for < and > and a solid line for ≤ and ≥.

• Writing set notation incorrectly, such as omitting the variable or using the wrong inequality signs.

  Put it right: Write the set clearly, for example {x : x > 2}.

• Forgetting to reverse the inequality sign when dividing by a negative number in a two-variable inequality.

  Put it right: Reverse the sign whenever the multiplier or divisor is negative.

Exam tips

• For quadratic inequalities, always find the critical values first and then test a value in each interval to confirm the solution set.
• When drawing a graph, label the boundary line clearly and use the correct line style for strict or inclusive inequalities.
• Write the solution set in more than one form where possible, such as on a number line and in set notation, to show full understanding.
• Check whether the inequality is strict or inclusive before deciding on open or filled circles and dashed or solid lines.