Graphs of inequalities - Higher
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 \(x = -1\), and the region where the \(x\) coordinates are less than -1 is shaded.
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.
More guides on this topic
- Algebraic expressions - Edexcel
- Algebraic formulae - Edexcel
- Solving linear equations - Edexcel
- Solving simultaneous equations - Edexcel
- Solving quadratic equations - Edexcel
- Algebraic fractions - Edexcel
- Sequences - Edexcel
- Straight line graphs - Edexcel
- Other graphs - Edexcel
- Transformation of curves - Higher - Edexcel
- Using and interpreting graphs - Edexcel