Question

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

Short answer

The graph shows the inequality$x+y\le 3$

Step-by-step solution

Step 1. Given information

The boundary line shown in the graph is,

$x+y=3$

Step 2. Calculation

The line $x+y=3$is the boundary line.

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

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: $x+y>3$or $x+y<3$

Checking if $x+y>3$, at $x=0,y=0$

$$\begin{gathered} x+y>3 \\ \Rightarrow 0+0>3 \\ \Rightarrow 0>3 \end{gathered}$$

which is false

Now checking if $x+y<3$, at $x=0,y=0$

$$\begin{gathered} x+y<3 \\ \Rightarrow 0+0<3 \\ \Rightarrow 0<3 \end{gathered}$$

which is true

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

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

The graph shows the inequality$x+y\le 3$