Exercise 25 Page 278

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 can see that there is no shaded region. This means the system has no solutions since the solution sets of the individual inequalities do not overlap. This can be true only if the solution set of the inequalities looks like the one below.

To write the system, we will tackle one inequality at a time and bring them together in a system at the end.

The First Inequality

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, $(0,\text{-}3)$ and $(1,\text{-}5),$ 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, $(0,\text{-}3),$ is also the $y\text{-}$intercept. With the 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 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 $(\text{-}2,0)$ for this test, then make the inequality symbol fit the resulting statement. $0$ is less than $1,$ 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 first inequality in the system.

The Second Inequality

Writing the inequality for this region of the graph will involve the same steps as above. We will start by identifying two points.

Again, we have denoted the $\text{rise}$ and $\text{run}$ of the graph, giving the slope $m.$ Since we chose the $y\text{-}$intercept at the point $(0,\text{-}1)$ as one of our points for this boundary line as well, we can write its equation. 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 $(1,0)$ for this test, then make the inequality symbol fit the resulting statement. $0$ is greater than $\text{-}3,$ 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 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.