Write the inequality shown by the graph with the boundary line$y=\frac{1}{2}x-4$

We are given a graph as follows:

The equation of the boundary line is$y=\frac{1}{2}x-4$.

The boundary line is $y=\frac{1}{2}x-4$.

From this, we can note that on one side of the line are the points with $y>\frac{1}{2}x-4$and on the other side of the line are the points with$y<\frac{1}{2}x-4$.

Let us consider the point $(2,-4)$. We will check which inequality is true for this point.

Take $x=2,y=-4$

Let us consider the inequality $y>\frac{1}{2}x-4$

We have,

$$\begin{gathered} y>\frac{1}{2}x-4 \\ -4>\frac{1}{2}\left(2\right)-4 \\ -4>1-4 \\ -4>-3 \end{gathered}$$

which is not true.

Consider the inequality, $y<\frac{1}{2}x-4$

Substituting the values, we have

$$\begin{gathered} y<\frac{1}{2}x-4 \\ -4<\frac{1}{2}\left(2\right)-4 \\ -4<1-4 \\ -4<-3 \end{gathered}$$

which is true.

From the above two cases, we can note that inequality$y<\frac{1}{2}x-4$holds at the point$(2,-4)$.

We have already seen that the inequality $y<\frac{1}{2}x-4$is true at the point $(2,-4)$.

Therefore the side of the line which contains the point $(2,-4)$is the solution. Hence, the shaded region shows the solution to the inequality $y<\frac{1}{2}x-4$.

Since the boundary line is a solid line as well, we need to include the equal sign in the inequality.

Thus, the required inequality is$y\le \frac{1}{2}x-4$.

The graph shows the inequality$y\le \frac{1}{2}x-4$.