Inequality regions

In a nutshell

Shading graphs can help to identify regions on a curve or curves which satisfy given inequalities.

Shading regions

Shaded regions on a graph can be used to identify values which satisfy an inequality. Moreover, the curve being a solid or dotted line can be used to identify whether or not the curve of the function is included in the shaded region.

inequality	region	drawn with
$y\lt f(x)$	Below the graph.	Dotted line.
$y\gt f(x)$	Above the graph.	Dotted line.
$y\leq f(x)$	Below and including the graph.	Solid line.
$y\geq f(x)$	Above and including the graph.	Solid line.

Example 1

A function is given by $f(x)=0.5x^2$ . Sketch the function and shade in the area which satisfies the inequality $y \gt f(x)$ .

Sketch the function.

Note: As the curve is not included in the inequality, the curve is sketched with a dotted line.

Maths; Equations and inequalities; KS5 Year 12; Inequality regions

Example 2

Sketch and shade the region which satisfies the inequalities $y \le 2x$ and $y \gt x^2-9x+18$ .

For the quadratic, the $\gt$ symbol means that the curve isn't included in the shaded region, so sketch it with a dotted line.

For $y\leq 2x$ , the $\leq$ symbol means that the line is included in the shaded region, so sketch it with a solid line.

Shade the area above the quadratic and below the linear curve:

Maths; Equations and inequalities; KS5 Year 12; Inequality regions