Question

Write the inequality shown by the graph with the boundary line$y=-2x+3$

Short answer

The inequality shown by the given graph with the boundary line$y=-2x+3$is$y\ge -2x+3$.

Step-by-step solution

Step 1. Given information

The given graph is,

The equation of the boundary line is $y=-2x+3$.

We need to find the inequality that represents the above graph.

Step 2. Identify two inequalities from the boundary line

The boundary line is $y=-2x+3$.

This implies that, at one side of the line are the points with $y>-2x+3$and on the other side are the points with$y<-2x+3$.

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

Let us consider the point $(2,1)$. We will test this point and see which inequality is true at this point.

Here,$x=2,y=1$

Let us consider the inequality $y>-2x+3$

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

which is true.

Now, consider the inequality, $y<-2x+3$

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

which is false.

Thus we can observe that the inequality,$y>-2x+3$is true.

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

In the previous step, it was observed that the inequality $y>-2x+3$is true.

Therefore the side of the line which contains the point $(2,1)$is the solution. In the given graph, the shaded region represents the solution to the inequality $y>-2x+3$.

Also, it can be noted that the boundary line is not a dashed line, or rather it is a solid line. This implies that the points on the boundary line should also belong to the solution set. Therefore the inequality contains the equal sign.

Thus the inequality is$y\ge -2x+3$.

Step 5. Final Answer

The graph shows the inequality$y\ge -2x+3$