Write Systems of Linear Inequalities from a Graph

When writing an inequality from a graph, there are a few things we need to do:

1. Determine the equation of the line. This gives us the general form of our inequality once we remove the equal sign.

2. Note whether the line is dotted or solid. Recall that dotted lines indicate $<$ or $>$ in an inequality and solid lines indicate $\le$ or $\ge$.

3. Use the graph's shading to determine which way your inequality sign will face.

This works for single inequalities as well as systems of inequalities. If you have more than one line on the graph, you'll need to use these steps to write more than one inequality.

To go over how to graph inequalities on a coordinate plane, check out this lesson.

Let's look at an example:

STEP 1: Determine the equation of the line

The slope (represented as $m$) appears to be $2$. Each time we rise by two units, our line runs to the right by $1$ unit. Since slope is $\frac{\text{rise}}{\text{run}}$, this is $\frac{2}{1}$, which is equal to $2$.

$$y=2x+1$$

STEP 2: Note whether the line is dotted or solid

The line in our graph is dotted. This means our inequality will include either $>$ or $<$.

STEP 3: Use shading to determine direction of inequality sign (i.e., $<$ or $>$)

Recall that the shaded area of the graph represents the solutions to the inequality. This is the area of the graph we are interested in.

We can see that the area "above" our line (even though it's tilted) is shaded. This means that the solutions are greater than the points on the line.

As a result, we can write our inequality as:

There's another way to verify the direction of your inequality sign. Pick any point in the shaded area of the graph. We'll choose $(-1, 4)$.

Plug these values into the equation of the line:

$$y=2x+1$$ $$4=2(-1) +1$$ $$4=-1$$

Decide which inequality sign ($<$ or $>$) makes the expression true when you replace the equals sign with it.

This tells us we need to write our inequality using the $>$ sign.

Writing a System of Linear Inequalities from a Graph

We can graph systems of inequalities to visualize the solutions. We can also look at a graph of a system and reverse engineer the inequalities.

Take the graph above for example. There are two lines so you know the system has two equations. Use whatever method you prefer to find the equations each line. I like to take two sets of points for each line and use $y-y_1=m(x-x_1)$ where $m=\dfrac{y_2-y_1}{x_2-x_1}$.

So the equation for the red (vertical) line is: $$x=1$$ And the equation for the blue (horizontal) line is: $$y=2$$

Once we have our equations, we look at the highlighted region and determine if the equations are $<$ or $>$.

The highlighted purple region has $x$ points that are larger than $1$ which is the equation of the line. So we can determine: $$x>1$$ But the red line isn't dashed. This means the equation is greater than or equal to instead of just greater than. So the equation is actually: $$x\geq1$$

For the second equation we notice that the $y$ points for the highlighted region are larger than $2$. So we can determine: $$y>2$$ Since the blue line is dashed, we know that its only greater than and not greater than or equal to.

Given the above graph we can determine the system of linear inequalities to be: $$x\geq1$$ $$y>2$$

(Video) Writing a System of Linear Inequalities from a Graph

by Rick Faulk

In this video by Rick Faulk, he goes over how to go from a graph to a system of linear inequalities.

Summary

In the graph below, we are working to figure out what the system of inequalities is that is represented by the graph.

We want to tackle this problem one inequality at a time.

We are first going to look at the dotted line. We can see that it's shaded below the line, so we can assume a less than sign: $$y<$$ We can see that the $y$-intercept is $+5$. So, we add that to our inequality: $$y< +5$$ We need a slope. We figure out that the slope is $\frac{-1}{2}$. $$y<\frac{-x}{2} +5$$

We will do something really similar for the second line, the one going diagonally from the bottom-left to the top-right. We know that there is shading above the line, so we will use a greater than or equal to sign. $$y \ge$$ We can see that the $y$-intercept is $-1$. Our next inequality becomes: $$y \ge -1$$ We can calculate the slope to be $1$. So, our second inequality is: $$y \ge x-1$$

Therefore, the system of linear inequalities that is represented by the graph are: $$y<\frac{-1}{2} +5$$ $$y \ge x-1$$

Try the next one on your own! Then follow along with the video to see how you did.