Question

Write the inequality shown by the graph with the boundary line$2x-y=4$

Short answer

The graph shows the inequality$2x-y\le 4$

Step-by-step solution

Step 1. Given information

The boundary line shown in the graph is,

$2x-y=4$

Step 2. Calculation

The line $2x-y=4$is the boundary line.

On one side of the line are the points with $2x-y>4$, and on the other side of the line are the points with $2x-y<4$.

Let's test the point $(0,0)$and see which inequality describes its position relative to the boundary line.

At $(0,0)$which inequality is true: $2x-y>4$or $2x-y<4$

Checking $2x-y>4$, at $x=0,y=0$

$$\begin{gathered} 2x-y>4 \\ \Rightarrow 2.0-0>4 \\ \Rightarrow 0>4 \end{gathered}$$

which is false

Checking $2x-y<4$, at $x=0,y=0$

$$\begin{gathered} 2x-y<4 \\ \Rightarrow 2.0-0<4 \\ \Rightarrow 0<4 \end{gathered}$$

which is true

Since, $2x-y<4$is true, the side of the line with $(0,0)$is the solution. The shaded region shows the solution of the inequality $2x-y<4$.

Since, the boundary line is graphed with a solid line, the inequality includes the equal sign.

The graph shows the inequality$2x-y\le 4$.