4.7 Graphs of linear inequalities  (Page 3/10)

The line $2x+3y=6$ is the boundary line. On one side of the line are the points with $2x+3y>6$ and on the other side of the line are the points with $2x+3y<6$ .

Let’s test the point $\left(0,0\right)$ and see which inequality describes its side of the boundary line.

At $\left(0,0\right)$ , which inequality is true:

So the side with $\left(0,0\right)$ is the side where $2x+3y<6$ .

(You may want to pick a point on the other side of the boundary line and check that $2x+3y>6$ .)

Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.

The graph shows the solution to the inequality $2x+3y<6$ .

Solution

Solution

First we graph the boundary line $x-2y=5$ . The inequality is $<$ so we draw a dashed line.

Then we test a point. We’ll use $\left(0,0\right)$ again because it is easy to evaluate and it is not on the boundary line.

Is $\left(0,0\right)$ a solution of $x-2y<5$ ?

The point $\left(0,0\right)$ is a solution of $x-2y<5$ , so we shade in that side of the boundary line.

Solution

First we graph the boundary line $y=\mathrm{-4}x$ . It is in slope–intercept form, with $m=\mathrm{-4}\text{and}b=0$ . The inequality is $\le$ so we draw a solid line.

Now, we need a test point. We can see that the point $\left(1,0\right)$ is not on the boundary line.

Is $\left(1,0\right)$ a solution of $y\le \mathrm{-4}x$ ?

The point $\left(1,0\right)$ is not a solution to $y\le \mathrm{-4}x$ , so we shade in the opposite side of the boundary line. See [link] .

Solution

First we graph the boundary line $y=3$ . It is a horizontal line. The inequality is>so we draw a dashed line.

We test the point $\left(0,0\right)$ .

$\left(0,0\right)$ is not a solution to $y>3$ .

So we shade the side that does not include (0, 0).