# Inequalities on graphs

## In a nutshell

Inequalities with one or more variables can be represented on a graph by drawing a line and shading the area required. You can use your knowledge of straight line graphs to draw the line represented by the inequality, and then find which area to shade on the graph by testing co-ordinates. Quadratic inequalities are inequalities where the variable is squared, and can be solved by sketching the graph of the quadratic function and then finding which values satisfy the inequality.

## Inequalities on graphs

Inequalities can be represented on a grid. Draw the line and then pick which side to shade by testing a pair of co-ordinates. You may have to shade in the wanted or unwanted area. On the graph use a solid line for $\le$ or $\ge$ and use a dotted line for $\lt$ or $\gt$.

##### Example 1

*Shade the region on the graph that represents the inequality*

$x>3$

*First draw the line *$x=3$*, as a dotted line. *$x=3$* is a vertical line. Then pick a pair of co-ordinates, for example *$(1,2)$*. Check to see if the point satisfies the inequality. *$(1,2)$* has *$x$* coordinate *$1$* which is less than *$3$*. This is not in the required region, as the inequality states that *$x$* should be greater than *$3$*. Therefore, shade the region to the right of the line.*

##### Example 2

*Identify the region satisfied by these three inequalities, label the region R.*

$x \gt -2 \\y \ge -3 \\x + y \lt 4$

*First draw all the straight lines. *$x=-2$* is a vertical line, draw the line dotted. *$y=-3$* is a horizontal line, draw this line solid. *

$x+y=4$* is a diagonal line. The equation can be rearranged to *$y=-x+4$*, this has a gradient of *$-1$* and a y-intercept of *$+4$*.*

*To identify the region, shade the unwanted regions, so that the region that is left unshaded will be the area which satisfies all inequalities. The unwanted regions are to the left of *$x=-2$*, below *$y=-3$* and above *$x+y=4$*. *

## Quadratic inequalities

In order to solve a quadratic inequality, draw a graph of the quadratic function and a horizontal line and find which values satisfy the inequality.

##### Example 3

*Solve the inequality*

$x^2 \le 16$

*Sketch the graph of *$y=x^2$* which has the shape of a parabola and *$y=16$* which is a horizontal line. To find the solution to the inequality, find where the graph *$y=x^2$* is below the line *$y=16$* and read off the values of *$x$*. These values of *$x$* are the solution to the inequality.*

$\underline{-4 \le x \le 4}$