Graphs of Linear Inequalities

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

Let's think about the number line in Figure \(4.30\) 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 Figure 4.31.

Figure 4.31

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 \(y < x+4\). On the other side of the line are the points with \(y > x+4\). We call the line \(y=x+4\) a boundary line.

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:

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 Table 4.48

Now, let's take a look at what we found in Example 4.69. We'll start by graphing the line \(y=x+4\), and then we'll plot the five points we tested. See Figure 4.32.

Figure 4.32

In Example 4.69 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 \((1,6)\) and \((-8,12)\) 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 \((0,0)\) and \((-5,-15)\) 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, \(y < x+4\).

What about the point \((2,6)\) ? Because \(6=2+4\), the point is a solution to the equation \(y=x+4\). So the point \((2,6)\) 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 \((0,10)\) clearly looks to be to the left of the boundary line, doesn't it? Is it a solution to the inequality?

\(\begin{array}{l} y > x+4 \\ 10 > 0+4 \\ 10 > 4 \qquad \qquad \qquad \text { So, }(0,10) \text { is a solution to } y > x+4. \end{array}\)

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 \((0,0)\) and \((-5,-15)\), are not solutions to \(y > x+4\). See Figure 4.33.

Figure 4.33

The graph of the inequality \(y > x+4\) is shown in Figure 4.34 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.

Figure 4.34 The graph of the inequality \(y > x +4\).