Exercise 22 Page 260

There are two major steps to writing an inequality when given its graph.

1. Write an equation for the boundary line.
2. Determine the inequality symbol and complete the inequality

In this exercise, we have been given the graph of a system of two linear inequalities. We will tackle them one at a time and bring them together in a system at the end.

The First Inequality

Let's take a look at the graph of the first inequality.

We can see that the boundary line is vertical and passes through $(2,0).$ Therefore, the equation of the line is $x=2.$ To finish forming the inequality, we need to determine the inequality symbol. This means substituting the equals sign with a blank space, since it is still unknown to us. To figure out what the symbol should be, let's substitute any point that lies within the solution set into the equation. We will substitute $(3,0)$ for this test, then make the inequality symbol fit the resulting statement. $3$ is greater than $2,$ so the symbol will be either $>$ or $\ge .$ Since the boundary line in the given graph is dashed, the inequality is strict, and we can form the first inequality in the system.

The Second Inequality

Writing the inequality for this region of the graph will involve the same steps as above.

Notice that the boundary line is vertical and passes through $(4,0).$ Therefore, the equation of the line is $x=4.$ Once more, we substitute the equals sign with a blank space. We will need another point that lies within the solution set to determine the sign of this inequality. We will substitute $(3,0)$ for this test, then make the inequality symbol fit the resulting statement. $3$ is less than $4,$ so the symbol will be either $<$ or $\le .$ Since the boundary line in the given graph is dashed, the inequality is strict, and we can form the second inequality in the system.

Writing the System

To complete the system of inequalities, we will bring both of our inequalities together in system notation.