Graphs of inequalities - Higher

An inequality can be represented graphically as a region on one side of a line.

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Inequalities that use < or > symbols are plotted with a dashed line to show that the line is not included in the region. Inequalities that use ≤ or ≥ symbols are plotted with a solid line to show that the line is included in the region.

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.

Graph showing the inequality 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.

Graph showing the inequalities of x = -2 and x = 3
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.

Graph showing, y is greater or equal to-4, y is less than 0 and y is greater than or equal to x

The answer is the region where the three shaded regions all overlap.

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