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Recognizing the Relation Between the Solutions of an Inequality and Its Graph

Recognizing the Relation Between the Solutions of an Inequality and Its Graph

Now, we will look at how the solutions of an inequality relate to its graph.

The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity.

Let’s think about the number line in the figure above from the previous lesson again. The point \(x=3\) separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See the figure below.

The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity. An arrow above the number line extends from 3 and points to the left. It is labeled “numbers less than 3.” An arrow above the number line extends from 3 and points to the right. It is labeled “numbers greater than 3.”

The solution to \(x>3\) is the shaded part of the number line to the right of \(x=3\).

Similarly, the line \(y=x+4\) separates the plane into two regions. On one side of the line are points with \(yx+4\). We call the line \(y=x+4\) a boundary line.

Boundary Line

The line with equation \(Ax+By=C\) is the boundary line that separates the region where \(Ax+By>C\) from the region where \(Ax+By

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not \(a\) is included in the solution:

The figure shows two number lines. The number line on the left is labeled x is less than a. The number line shows a parenthesis at a and an arrow that points to the left. The number line on the right is labeled x is less than or equal to a. The number line shows a bracket at a and an arrow that points to the left.

Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to indicate whether or not it the line is included in the solution. This is summarized in the table below.

\(Ax+By \(Ax+By\le C\)
\(Ax+By>C\) \(Ax+By\ge C\)
Boundary line is not included in solution. Boundary line is included in solution.
Boundary line is dashed. Boundary line is solid.

Now, let’s take a look at what we found in the example from the previous lesson. We’ll start by graphing the line \(y=x+4\), and then we’ll plot the five points we tested. See the figure below.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as an arrow extending from the bottom left toward the upper right. The following points are plotted and labeled (negative 8, 12), (1, 6), (2, 6), (0, 0), and (negative 5, negative 15).

In the example from the previous lesson, we found that some of the points were solutions to the inequality \(y>x+4\) and some were not.

Which of the points we plotted are solutions to the inequality \(y>x+4\)? The points \(\left(1,6\right)\) and \(\left(-8,12\right)\) are solutions to the inequality \(y>x+4\). Notice that they are both on the same side of the boundary line \(y=x+4\).

The two points \(\left(0,0\right)\) and \(\left(-5,-15\right)\) are on the other side of the boundary line\(y=x+4\), and they are not solutions to the inequality \(y>x+4\). For those two points, \(y

What about the point \(\left(2,6\right)\)? Because \(6=2+4\), the point is a solution to the equation \(y=x+4\). So the point \(\left(2,6\right)\) is on the boundary line.

Let’s take another point on the left side of the boundary line and test whether or not it is a solution to the inequality \(y>x+4\). The point \(\left(0,10\right)\) clearly looks to be to the left of the boundary line, doesn’t it? Is it a solution to the inequality?

\(\begin{array}{cccc}y>x+4\hfill & & & \\ 10\stackrel{?}{>}0+4\hfill & & & \\ 10>4\hfill & & & \text{So},\phantom{\rule{0.2em}{0ex}}\left(0,10\right)\phantom{\rule{0.2em}{0ex}}\text{is a solution to}\phantom{\rule{0.2em}{0ex}}y>x+4.\hfill \end{array}\)

Any point you choose on the left side of the boundary line is a solution to the inequality \(y>x+4\). All points on the left are solutions.

Similarly, all points on the right side of the boundary line, the side with \(\left(0,0\right)\) and \(\left(-5,-15\right)\), are not solutions to \(y>x+4\). See the figure below.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as an arrow extending from the bottom left toward the upper right. The following points are plotted and labeled (negative 8, 12), (1, 6), (2, 6), (0, 0), and (negative 5, negative 15). To the upper left of the line is the inequality y is greater than x plus 4. To the right of the line is the inequality y is less than x plus 4.

The graph of the inequality \(y>x+4\) is shown in the figure below below. The line \(y=x+4\) divides the plane into two regions. The shaded side shows the solutions to the inequality \(y>x+4\).

The points on the boundary line, those where \(y=x+4\), are not solutions to the inequality \(y>x+4\), so the line itself is not part of the solution. We show that by making the line dashed, not solid.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as a dashed arrow extending from the bottom left toward the upper right. The coordinate plane to the upper left of the line is shaded.The graph of the inequality \(y>x+4\).

Example

The boundary line shown is \(y=2x-1\). Write the inequality shown by the graph.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2 x minus 1 is plotted as a solid arrow extending from the bottom left toward the upper right. The coordinate plane to the left of the line is shaded

Solution

The line \(y=2x-1\) is the boundary line. On one side of the line are the points with \(y>2x-1\) and on the other side of the line are the points with \(y<2x-1\).

Let’s test the point \(\left(0,0\right)\) and see which inequality describes its side of the boundary line.

At \(\left(0,0\right)\), which inequality is true:

\(\begin{array}{ccccc}\hfill y>2x-1\hfill & & \hfill \text{or}\hfill & & \hfill y<2x-1?\hfill \\ \hfill y>2x-1\hfill & & & & \hfill y<2x-1\hfill \\ \hfill 0\stackrel{?}{>}2·0-1\hfill & & & & \hfill 0\stackrel{?}{<}2·0-1\hfill \\ \hfill 0>-1\phantom{\rule{0.2em}{0ex}}\text{True}\hfill & & & & \hfill 0<-1\phantom{\rule{0.2em}{0ex}}\text{False}\hfill \end{array}\)

Since, \(y>2x-1\) is true, the side of the line with \(\left(0,0\right)\), is the solution. The shaded region shows the solution of the inequality \(y>2x-1\).

Since the boundary line is graphed with a solid line, the inequality includes the equal sign.

The graph shows the inequality \(y\ge 2x-1\).

We could use any point as a test point, provided it is not on the line. Why did we choose \(\left(0,0\right)\)? Because it’s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that \(y<2x-1\).

Example

The boundary line shown is \(2x+3y=6\). Write the inequality shown by the graph.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x plus 3 y equals 6 is plotted as a dashed arrow extending from the top left toward the bottom right. The coordinate plane to the bottom of the line is shaded.

Solution

The line \(2x+3y=6\) is the boundary line. On one side of the line are the points with \(2x+3y>6\) and on the other side of the line are the points with \(2x+3y<6\).

Let’s test the point \(\left(0,0\right)\) and see which inequality describes its side of the boundary line.

At \(\left(0,0\right)\), which inequality is true:

\(\begin{array}{ccccccccccc}\hfill 2x+3y& >\hfill & 6\hfill & & & \hfill \text{or}\hfill & & & \hfill 2x+3y& <\hfill & 6?\hfill \\ \hfill 2x+3y& >\hfill & 6\hfill & & & & & & \hfill 2x+3y& <\hfill & 6\hfill \\ \hfill 2\left(0\right)+3\left(0\right)& \stackrel{?}{>}\hfill & 6\hfill & & & & & & \hfill 2\left(0\right)+3\left(0\right)& \stackrel{?}{<}\hfill & 6\hfill \\ \hfill 0& >\hfill & 6\phantom{\rule{0.2em}{0ex}}\text{False}\hfill & & & & & & \hfill 0& <\hfill & 6\phantom{\rule{0.2em}{0ex}}\text{True}\hfill \end{array}\)

So the side with \(\left(0,0\right)\) is the side where \(2x+3y<6\).

(You may want to pick a point on the other side of the boundary line and check that \(2x+3y>6\).)

Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.

The graph shows the solution to the inequality \(2x+3y<6\).

This lesson is part of:

Graphs and Equations

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