Print-friendly page

Graphing Linear Inequalities: y > mx + b, etc

Think about how you've done inequalites on the number line. For instance, they'd ask you to graph something like x > 2. How did you do it? You would draw your number line, find the "equals" part (in this case, x = 2), mark this point with the appropriate notation (an open dot or a parenthesis, indicating that the point x = 2 wasn't included in the solution), and then you'd shade everything to the right, because "greater than" meant "everything off to the right". The steps for linear inequalities are very much the same.

• Graph the solution to y < 2x + 3. 

Just as for number-line inequalities, the first step is to find the "equals" part.  In this case, the "equals" part is the line y = 2x + 3. There are a couple ways you can graph this: you can use a T-chart, or you can graph from the y-intercept and the slope. Either way, you get a line that looks like this:

Now we're at the point where your book gets complicated, with talk of "test points" and such. When you did those one-variable inequalities (like x < 3), did you bother with "test points", or did you just shade one side or the other? Ignore the "test point" stuff, and look at the original inequality:  y < 2x + 3.

You've already graphed the "or equal to" part (it's just the line); now you're ready to do the "y less than" part. In other words, this is where you need to shade one side of the line or the other. Now think about it: If you need y LESS THAN the line, do you want ABOVE the line, or BELOW? Naturally, you want below the line. So shade it in:

And that's all there is to it: the side you shaded is the "solution region" they want.

Note that this technique worked because we had y alone on one side of the inequality. Just as with plain old lines, you always want to "solve" the inequality for y on one side.

• Graph the solution to 2x – 3y < 6.  

First, solve for y:

2x – 3y < 6
–3y < –2x + 6
y > ( ²/₃ )x – 2

[Note the flipped inequality sign in the last line. Don't forget to flip the inequality if you multiply or divide through by a negative!]   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

Now you need to find the "equals" part, which is the line y = ( ²/₃ )x – 2. It looks like this:

But this is what is called a "strict" inequality. That is, it isn't an "or equals to" inequality; it's only "y greater than". When you had strict inequalities on the number line (such as x < 3), you'd denote this by using a parenthesis (instead of a square bracket) or an open [unfilled] dot (instead of a closed [filled] dot). In the case of these linear inequalities, the notation for a strict inequality is a dashed line. So the border of the solution region actually looks like this:

By using a dashed line, you still know where the border is, but you also know that it isn't included in the solution. Since this is a "y greater than" inequality, you want to shade above the line, so the solution looks like this:

If you need to graph a set of two or more linear inequalities at once, view the lesson on systems of linear inequalities.

Top  |  Return to Index

  Copyright © 2006-2008  Elizabeth Stapel   |   About   |   Terms of Use

 Feedback   |   Error?