91Ó°ÊÓ

We are given a linear inequality $y<\frac{2}{3}x-5$.

We are suppose to draw a graph that represents this inequality.

• If the inequality has $â‰\yen$or$≤$, the boundary line is solid.
• If the inequality has $<$or $>$, then the boundary line is dashed.

The given inequality is $y<\frac{2}{3}x-5$.

We will first replace the inequality with an equal sign and then graph the boundary line.

Here, the given inequality does not have an equal sign, therefore the boundary line must be dashed.

After replacing the inequality sign with an equal sign, we have the equation $y=\frac{2}{3}x-5$. Thus, the graph of this equation is,

Consider a point that is not on the boundary line.

Let us consider the point $(5,-5)$

In order to check if the inequality holds at this point, we will substitute $x=5,y=-5$in $y<\frac{2}{3}x-5$.

$$\begin{gathered} y<\frac{2}{3}x-5 \\ -5<\frac{2}{3}\left(5\right)-5 \\ -5<\frac{10}{3}-5 \\ -5<\frac{10-15}{3} \\ -5<\frac{-5}{3} \\ -5<-1.67 \end{gathered}$$

which is true.

Thus the point $(5,-5)$is a solution for the inequality $y<\frac{2}{3}x-5$.

In the earlier step, we have seen that the point $(5,-5)$is a solution for the inequality $y<\frac{2}{3}x-5$.

Therefore, we will shade the side of the boundary line that contains the point $(5,-5)$.

Therefore the graph of the inequality $y<\frac{2}{3}x-5$is

The graph of the linear inequality $y<\frac{2}{3}x-5$ is,