Exercise 4 Page 267

To graph a linear inequality in two variables, you need to complete two things:

1. Graph the boundary line
2. Shade accordingly.

Graph the boundary line

Graphing a boundary line is just like graphing a linear equation. Let's look at an easy example, $y>x+1.$ Our boundary line is going to be the same function but with an equality symbol instead of an inequality symbol. If our inequality symbol was $\le$ or $\ge ,$ our boundary line will be solid because the values on the line should be included in the solution set. If our inequality symbol was $<$ or $>,$ our boundary line will be dashed because the values on the line should not be included in our solution set.

The above graph shows the boundary line for our inequality, dashed because it is $>.$

Shade accordingly

After you have your boundary line, you need to decide which side of the line to shade. There are two approaches you can take for making this decision. The first approach is just to think about the meaning of the inequality symbol. We have the inequality: which tells us that our values for $y$ should be greater than $x+1.$ This means that we should shade above the boundary line. Sometimes it is difficult to visualize which side to shade though. It is a much safer approach to choose a very simple point, such as $(0,0)$ and check if it satisfies the inequality. Because the point does not satisfy the inequality, we should shade on the side of the boundary line that does not contain the point $(0,0).$