Question

Write the inequality shown by the graph with the boundary line $3x-y=6$

Short answer

The graph shows the inequality$3x-y\le 6$

Step-by-step solution

Step 1. Given information

The boundary line shown in the graph is,

$3x-y=6$

Step 2. Calculation

The line $3x-y=6$is the boundary line.

On one side of the line are the points with $3x-y>6$, and on the other side of the line are the points with $3x-y<6$

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: $3x-y>6$or $3x-y<6$

Checking $3x-y>6$, at $x=0,y=0$

$$\begin{gathered} 3x-y>6 \\ \Rightarrow 3.0-0>6 \\ \Rightarrow 0>6 \end{gathered}$$

which is false

Now checking$3x-y<6$, at$x=0,y=0$

role="math" localid="1645557426639" $$\begin{gathered} 3x-y<6 \\ \Rightarrow 3.0-0<6 \\ \Rightarrow 0<6 \end{gathered}$$

which is true.

Since, $3x-y<6$is true, the side of the line with $(0,0)$is the solution. The shaded region shows the solution of the inequality $3x-y<6$.

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

The graph shows the inequality$3x-y\le 6$