Inequalities and regions

Vertical regions

Graphs can be used to represent inequalities by regions (region is the area where all points obey a given rule).

For example, the region for x > 2 is represented by:

In the graph, the area where the graph is x > 2 is shown above. In this area, all of the points would be more than 2.

The line which intersects where x = 2 is drawn with a dotted line.

In the right region of the line x = 2, every point represents x > 2.

In the left region of the line x = 2, every point represents x < 2.

The rules for horizontal regions will follow the rules for the vertical regions. Therefore the graph which represents y ≤ 3 is:

In the graph, the area where the graph is y ≤ 3 is shown above. In this area, all of the points would be more than or equal to 3.

The line which intersects where y = 3 is drawn with a solid line.

In the top region of the line y = 3, every point represents y > 3 and y = 3

In the left region of the line y = 3, every point represents y < 3 and y = 3

When combining horizontal and vertical regions, the unwanted regions are shaded out. This enables the required area to be easily identified.

In the graph above, the regions represent –4 < x < –1 and –1 < y ≤ 3.

The regions which are unshaded represent –4 < x < –1 and –1 < y ≤ 3.