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Middle Years

15.01 Linear inequalities

Lesson

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 $y\ge2x+4$y2x+4. We call the region the required region, and use a key to indicate which side of the line satisfies the inequality.

Required region

The graph of $y\ge2x+4$y2x+4.

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, $y\ge2x+4$y2x+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 graph of $y=2x+4$y=2x+4 passing through $\left(-2,0\right)$(2,0) and $\left(0,4\right)$(0,4).

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.

Required region

The graph of $y\ge2x+4$y2x+4.

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.

Summary
  1. Plot the line:
    • If the inequality is strict ($>$> or $<$<) use a dashed line.
    • If the inequality is not strict ($\ge$ or $\le$) use a solid line.
  2. Identify the required region:
    • Choose a test point on one side of the line, and determine whether it satisfies the inequality.
    • If the inequality statement is true, shade that side of the line.
    • If the inequality statement is false, shade the other side of the line.

Alternatively, we can determine the region that requires shading by making $y$y the subject of the inequality. The inequality $y\ge2x+4$y2x+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 $y\ge2x+4$y2x+4.

Points above the line will satisfy $y\ge2x+4$y2x+4.

This leads us to the following summary.

Summary

For inequalities of the form $y\ge mx+c$ymx+c or $y>mx+c$y>mx+c, the required region is above the line, and inequalities of the form $y\le mx+c$ymx+c or $yy<mx+c, the required region is below the line.

Practice questions

Question 1

Is $\left(3,2\right)$(3,2) a solution of $3x+2y$3x+2y $\ge$ $12$12?

  1. No

    A

    Yes

    B

Question 2

Write the inequality that describes the points in the shaded region.

Loading Graph...
A line is plotted on a Cartesian coordinate plane. The line is solid and is horizontal to the $x$x-axis where it intercepts $y$y-axis at $5$5. The region below the line is shaded.

Question 3

Consider the line $y=-2x+2$y=2x+2.

  1. Find the intercepts of the line.

    $x$x-intercept $\editable{}$
    $y$y-intercept $\editable{}$
  2. Which of the following points satisfies the inequality $y$y $\le$ $-2x+2$2x+2?

    $\left(2,3\right)$(2,3)

    A

    $\left(3,-6\right)$(3,6)

    B

    $\left(4,-2\right)$(4,2)

    C

    $\left(1,2\right)$(1,2)

    D
  3. Sketch a graph of $y$y$\le$$-2x+2$2x+2.

    Loading Graph...

  4. Do the points on the line satisfy the inequality $y$y $\le$ $-2x+2$2x+2?

    No

    A

    Yes

    B

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