Question

Graph the linear inequality:$y\ge -2x$.

Short answer

The graph of the linear inequality $y\ge -2x$is,

Step-by-step solution

Step 1. Given Information

We are given a linear inequality $y\ge -2x$.

We have to draw a graph that represents this inequality.

Step 2. Concept

• If the linear inequality contains $<$or $>$, then the boundary line is dashed.
• If the linear inequality contains$\ge$or$\le$, then the boundary line is solid.

Step 3. Identify the boundary line and graph it

The given inequality is $y\ge -2x$.

We will first replace the inequality sign with an equal sign. Then the equation can be written as, $y=-2x$.

Since we have $\ge$sign in the inequality, the boundary line is solid.

Then the boundary line of the graph, $y=-2x$is

Step 4. Test a point that is not on the boundary line

Consider a point that is not on the boundary line.

Let us consider, $(1,1)$. Here, $x=1,y=1$.

Substituting these values in the inequality $y\ge -2x$, we have

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

which is true.

This implies that the point $(1,1)$is a solution for the inequality$y\ge -2x$.

Step 5. Shading the solution area on one side of the boundary line

From the previous step, we have seen that the point $(1,1)$is a solution to the inequality $y\ge -2x$.

Therefore, we will shade the side of the boundary line that contains the point $(1,1)$.

Therefore the graph of the inequality is,

The points in the shaded region and also the points on the boundary line represent the solution for the inequality$y\ge -2x$.

Step 6. Final Answer

The graph of the linear inequality $y\ge -2x$is,