Graphs of inequalities - Higher
Click to explore refreshed revision resources for GCSE Maths: Graphs of inequalities, with step-by-step slideshows, quizzes, practice exam questions, and more!
An inequality can be represented graphically as a region on one side of a line.
For example, this graph shows the inequality \(x \textless -1\). This can be seen as there is a dashed line at \(x = \textless -1\), and the region where the \(x\) coordinates are less than −1 is shaded out.
Example
Show the region satisfied by the inequality \(-2 \textless x \leq 3\).
Identify the two regions shown by the inequalities. These are \(2 \textless x\) (or \(x \textgreater -2\)) and \(x \leq 3\).
\(x \textgreater -2\): draw a dotted line at \(x = -2\). \(x = -2\) is the graph made by coordinates points where \(x\) is equal to -2, for example (-2, 5), (-2, 4), (-2, 3), (-2, 2) and so on.
\(x \leq 3\): draw a solid line at \(x = 3\). \(x = 3\) is the graph made by coordinate points where \(x\) is equal to 3, for example (3, -4), (3, -3), (3,-2), (3, -1) and so on.
\(x\) is the values in between these two inequalities, so shade this region.
Question
Show the region satisfied by the inequalities \(-4 \leq y \textless 0\) and \(y \geq x\).
Identify the regions shown by the inequalities. These are \(y \geq -4\), \(y \textless 0\) and \(y \geq x\).
\(y \geq -4\): draw a solid line at \(y = -4\). This line is made by the coordinate points where \(y = -4\), for example (5, -4) or (-5, -4). \(y\) is greater than or equal to -4, so shade the points above the line.
\(y \textless 0\): draw a dashed line at \(y = 0\). This line goes through all the coordinate points where \(y = 0\) (it is the same as the \(x\)-axis). \(y\) is less than 0, so shade all the points below this line.
\(y \geq x\): draw a solid line at \(y = x\). To draw the line \(y = x\), plot coordinate points where the \(y\) coordinate is equal to the \(x\) coordinate, for example (4, 4), (2, 2), (-1, -1) and so on. \(y\) is greater than or equal to \(x\), so shade the region above this line.
The answer is the region where the three shaded regions all overlap.
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