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Lesson 132: Graphing Linear Inequalities
In this very important lesson we'll learn more about graphing linear inequalities. We start with graphing a linear equation, but then we have to take into account if we are dealing with >, ≥, <, or ≤
Before starting this lesson, make sure that you review all of the previous lessons on graphing linear equations. Make sure that you fully understand them, or you will have difficulty with this lesson.
Very
often we are asked to graph not a linear equation, but a linear
inequality. An example would be something like y ≤ 3x - 2.
Here is what we do. First, we'll ignore our inequality sign,
and change it to an equals sign. Then we'll graph the line
like we've learned how to do. But ultimately, when we pick a
value of x, we're going to not just be interested in the value of y
that equals the value of the expression on the right. We're
also looking for all values of y that are less than or equal to the
value of the expression.
This is how we graph this inequality. First we graph the line itself, like we learned how. Now we see that we are dealing with a "less than or equal to" inequality. Since points that are on the line are OK, we'll make the line solid. Points that are on the line are values that satisfy the "equals" part of the inequality. Any point that is below the line will also satisfy the inequality. These are points for which the value of y is less than the expression. What we do is shade in the area below the line. It is often convenient to use slanted lines the way that I did. We're just trying to show that any point that is below the line is OK.
Let's look at another. Graph y > (-1/2)x + 1. As before, ignore the inequality, and just graph the line like we've learned how. Now let's look at the inequality. In this case, we want values of y that are greater than the expression, but not equal to it. That means that values of y that are on the line, and therefore equal to the expression, are not allowed.
We
use a dashed or dotted line to show this. Since we have a
greater than sign, we want values of y that are above the line.
Those points will satisfy the "greater than" component of the
inequality.
Sometimes we have to graph inequalities based on the special type of lines that we saw in the last lesson. For example, to graph x < 5, we'll draw a vertical line at x = 5. We then have to make it dashed, since points that are equal are not OK. Then we'll shade in the area to the left of the line, since we want values of x that are less than 5.
If we wanted to graph y ≥ -3, we would draw a horizontal line at y = -3. Then we would make it solid (not dashed), since points that are on the line are OK, and satisfy the equals part of the inequality. Then we'd shade in the area above the line, since we want values of y that are greater than -3.
Make sure that you study this lesson very carefully. In the next lessons, we'll start to work with graphs that have more than one line plotted simultaneously.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.