6.12 Regions defined by inequalities

Graphing linear inequalities

The graph of an inequality describes the set of all points that satisfy (solutions of) the inequality. For instance, the following graph shows a boundary line and a region. Together they describe the set of points $\left(x,y\right)$(x,y) that satisfy the inequality $y\ge2x+4$y≥2x+4. We call the region the required region, and use a key to indicate which side of the line satisfies the inequality.

To draw an inequality, we start by drawing the inequality as if it were an equation. If the inequality is strict ($>$> or $<$<) we draw a dashed line, and if the inequality is not strict ($\ge$≥ or $\le$≤) we draw a solid line. The points on a solid line indicate that they satisfy the inequality, while the points on a dashed line indicate that they don't satisfy the inequality.

Then, we shade the region, above or below the line, depending on the inequality sign.

 

Exploration

Let's look at this process using the above example, $y\ge2x+4$y≥2x+4.

The first step is to plot the inequality as if it was an equation. So we want to draw the line $y=2x+4$y=2x+4. The line passes through the $y$y-intercept at $y=4$y=4, and the $x$x-intercept at $x=-2$x=−2, so we can draw the line that passes through these two points. Because the inequality was not strict ($\ge$≥), we draw a solid line.

The next step is to identify which side of the line needs shading. We can decide which side of the line is the required region by checking whether a point either side of the line satisfies the inequality.

The origin, $\left(0,0\right)$(0,0), is a convenient choice. Let's also test a point on the other side of the line, say $\left(-5,5\right)$(−5,5). The two points, and the line are shown below.

To test $(0,0)$(0,0) , we substitute the set of coordinates into the given equation:

$y$y	$\ge$≥	$2x+4$2x+4
$0$0	$\ge$≥	$2\times0+4$2×0+4
$0$0	$\ge$≥	$4$4

Clearly the resulting inequality is not true, $0$0 is in fact less than $4$4. So the region below the line will not be shaded.

Testing the point $\left(-5,5\right)$(−5,5), we substitute the coordinates into the equation:

$y$y	$\ge$≥	$2x+4$2x+4
$5$5	$\ge$≥	$2\times\left(-5\right)+4$2×(−5)+4
$5$5	$\ge$≥	$-6$−6

The resulting inequality is true, that is, $5$5 is greater than or equal to $-6$−6. So the region above the line will be shaded. This gives the graph of the inequality, as we saw above.

It's true in general that if a point on one side of the line does not satisfy the inequality, then the region on the other side does. Which means in practice, we only ever need to draw the line of the equation and test one point to draw the graph of the inequality.

Alternatively, we can determine the region that requires shading by making $y$y the subject of the inequality. The inequality $y\ge2x+4$y≥2x+4 already has $y$y as the subject. Notice that any point $\left(x,y\right)$(x,y) above the line has a greater $y$y-value than the point on the line immediately below it. So points above the line will satisfy the inequality $y\ge2x+4$y≥2x+4.

This leads us to the following summary.

Practice questions

Question 1

Question 2

Question 3

Regions defined by reciprocal functions

We can also define regions using reciprocal functions. Consider the function $y=\frac{2}{x}$y=2x​ or $xy=2:$xy=2:

We can sketch the region defined by the inequality $xy\ge2$xy≥2 only if we also state which values of x and y we want to include, because of the two sections of the graph. For instance, below is the region defined by $xy\ge2$xy≥2 where $x\ge0:$x≥0:

And below is the region defined by $xy\le2$xy≤2 where $x\le0$x≤0: