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Graphing Linear Inequalities

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Today we're going to talk about graphing linear inequalities. A linear inequality will look similar to a linear equation, except for instead of having an equal sign, we use one of our inequality symbols. So we're going to go over some examples of graphing linear inequalities. And we'll talk about how that's different and similar to graphing a linear equation.

So for my first example I've got an inequality. And I'm going to show how you would graph this inequality. If I would replace this inequality symbol with an equals sign, I would have an equation that was written in slope intercept form. So I'm going to start by identifying the y-intercept and the slope.

I can see that the y-intercept is at positive 1. So I'm going to go to my y-axis and place a point at positive 1. Then I see that my slope is 3, which is the same as 3 over 1. So thinking about slope as rise over run, I would start at my y-intercept and go up 3 and over 1, and place another point.

So now that I have two points, I know that I can connect them. However, because we're graphing an inequality that is just less than, I'm going to use a dotted line to connect my points. Similarly, if we were graphing an inequality that was simply greater than, I would also use a dotted line to connect my points. And when you are graphing an inequality that is less than or equal to or greater than or equal to, you use a solid line to connect your points in the way that you connect your points with a solid line when you're graphing an equation.

So I'm going to go ahead and connect my two points with my dotted line. And I can see that my line is cutting my coordinate plane into two different regions. So one of these two regions is going to represent all of the coordinate pairs that satisfy this inequality. And the other region represents all of the coordinate pairs that do not satisfy the inequality.

And when you are determining which region is going to satisfy the inequality, if you have a less than symbol, then the region that satisfies the inequality is going to be below the line. And if you have a greater than inequality, the region that satisfies the inequality is going to be above the line.

So again, because I have less than. I'm going to go ahead and shade all of this region, which means that any coordinate pair, any pair of x,y values in this region, is going to satisfy this inequality. And any coordinate pair, any x,y pair in this region, is not going to satisfy this inequality.

So for my second example I've got an inequality y is greater than or equal to 1 over 3 x plus 1. If I were to replace this inequality symbol with an equal sign, I would have an equation that corresponds to this line. It has a y-intercept of 1, and a slope of 1 over 3.

I also know that the inequality symbols greater than or equal to or less than or equal to, correspond to a solid line. And the inequality symbols less than or greater than, would correspond instead to a dotted or a dashed line. I also know that if the inequality symbol is greater than or equal to or greater than, I would shade the region above the line. And if the inequality symbol is less than or equal to or less than, I would shade the region below the line.

However, I want to show you another way that you can determine which region to shade. And that's by picking a point in either region, and using the x and y values of that point to evaluate the inequality. So let's say that I pick this point, 0,3. So this point 0,3 is in the top region of my inequality. And again, I know that this point, if I were to substitute the values for x and y into my inequality, it should give me a true statement. Because we already know that I should be able to shade this top region of my coordinate plane.

So let's test that out. So with this point I know that 0 is my x value and 3 is my y value. So substituting those values into my inequality, I'm going to have that 3 should be greater than or equal to 1/3 times 0 plus 1. So simplifying this, I've got 1/3 times 0, which is just going to give me 0, plus 1 and then simplifying this, 0 plus 1 is just 1. And 3, we see that 3 is indeed greater than or equal to 1.

So because-- by evaluating this inequality, I get a true statement using the x and y values of this point. I know that since this point satisfies that inequality that I need to shade this region above my line.

So let's go over our key points from today. As usual, make sure you have them in your notes if you don't already, so you can refer to them later.

When graphing a linear inequality, the symbols less than and greater than indicate using a dashed line. And the symbols less than or equal to and greater than or equal to, indicate using a solid line.

When determining which region to shade on the graph, the symbols less than and less than or equal to indicate shading below the line. And the symbols greater than and greater than or equal to indicate shading above the line.

The x and y coordinates of a point that is in the shaded region will yield a true statement when substituted into the inequality. And the x and y coordinates of a point that is not in the shaded region will yield a false statement when substituted into the inequality.

So I hope that these key points and examples helped you understand a little bit more about graphing linear inequalities. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.