Question

Write the inequality shown by the shaded region in the graph with the boundary line $x-4y=8$.

Short answer

The inequality shown by the shaded region in the graph with the boundary line$x-4y=8$is$x-4y\le 8$.

Step-by-step solution

Step 1. Given information

The given graph is,

The equation of the boundary line is$x-4y=8$.

Step 2. Identify two inequalities from the boundary line  

The boundary line is $x-4y=8$.

On one side of the boundary line are the points with $x-4y>8$whereas on the other side of the boundary line are the points with$x-4y<8$.

Step 3. Checking which inequality is true for a given point  

Let us consider the point $(0,0)$. Here $x=0,y=0$.

We will check which inequality holds at this point.

Let us consider, $x-4y>8$

Substituting the values, we have

$$\begin{gathered} x-4y>8 \\ 0-4\left(0\right)>8 \\ 0>8 \end{gathered}$$

which is false.

Now, consider the inequality, $x-4y<8$

Substituting the values, we have

$$\begin{gathered} x-4y<8 \\ 0-4\left(0\right)<8 \\ 0<8 \end{gathered}$$

which is true.

From the above two cases, it can be noted that the inequality$x-4y<8$holds at the point$(0,0)$.

Step 4. The equation of the inequality that is represented by the graph  

We have seen that the inequality $x-4y<8$is true at the point $(0,0)$.

Therefore, the side with the point $(0,0)$is the side where $x-4y<8$. Thus the shaded region represents the solution to the inequality $x-4y<8$.

Along with this, it can be noted that the boundary line $x-4y=8$is a solid line. Therefore, we need to include the equal sign in the inequality as well.

Hence the inequality is$x-4y\le 8$

Step 5. Final answer

The inequality shown by the shaded region in the graph with the boundary line$x-4y=8$is$x-4y\le 8$.