Exercise 25 Page 403

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 a system consisting of two linear inequalities. We will tackle them one at a time and bring them together in a system at the end.

The Yellow Region

It only takes two points to create a unique equation for any line, so let's start by identifying two points on the boundary line.

Here we have identified two points, $(\text{-}1,0)$ and $(0,1),$ and indicated the horizontal and vertical changes between them. This gives us the $\text{rise}$ and $\text{run}$ of the graph, which will give us the slope $m.$ One of the points we selected, $(0,1),$ is also the $y\text{-}$intercept. With slope $m$ and the $y\text{-}$intercept at the point $(0,b),$ we can write an equation for the boundary line in slope-intercept form. To finish forming the inequality, we need to determine the inequality symbol. This means replacing the equals sign with a blank space, since the symbol 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 $(\text{-}1,1)$ for this test, then make the inequality symbol fit the resulting statement. Since $1$ is greater than 0, the symbol will be either $>$ or $\ge .$ The boundary line in the given graph is dashed, so the inequality is strict. We can now form the first inequality in the system.

The Blue Region

Since the boundary line of the second region of the graph is a horizontal line, writing the inequality for this region requires a different approach.

Notice that the $y\text{-}$coordinate of every point on the boundary line is equal to $2.$ This information is enough to write the corresponding equation. Once more, we replace the equals sign with a blank space. We will need a point that lies within the solution set to determine the sign of this inequality. We will substitute $(1,3)$ for this test, then make the inequality symbol fit the resulting statement. Since $3$ is greater than $2,$ the symbol will be either $>$ or $\ge .$ The boundary line in the given graph is solid, so the inequality is non-strict. We can now 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.