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Graphing 2 Variable Inequalities - Problem 1 12,623 views

Teacher/Instructor Alissa Fong
Alissa Fong

MA, Stanford University
Teaching in the San Francisco Bay Area

Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts

To graph a linear inequality, start by graphing it as you would a line in slope-intercept form y=mx+b. First plot the y intercept (the "b" value) on the y-axis. Then use the slope (the "m" value) to plot a second point. If the inequality is "greater than" or "less than" (but not equal to), then the line connecting the points should be a dotted line. This indicates that the possible solutions to the inequality do not have values on the line. If the inequality is "greater than or equal to" or "less than or equal to", then the line connecting the points should be a solid line. This indicates that the possible solutions include the values on the line. Lastly, shade the appropriate region. For values that are "greater than" or "greater than or equal to" the line, shade the region above the line. For values that are "less than" or "less than or equal to", shade the region below the line.

This is an inequality that has both x and y variables. That's how I know that I need to graph it on the XY plane. So the first thing I want to do is graph the line as if it were an equals sign using my y equals mx plus b techniques. I'm lucky because this equation is already solved for y.

So in order to graph it the first thing I want to do is put a dot at the y intercept which is -1. So on my graph I'm going to put a dot at -1 from there I'm going to count the slope which is 2. So from that intercept I'm going to go up 2, right 1, up 2, right 1 and then connect them. There we go.

Okay that's the first step is to graph the line as if it were an equal sign using y equals mx plus b techniques.

The next thing I need to do is decide if this line should be solid or dashed and the way you tell is by looking at this inequality symbol. These guys that are strict in equalities which is greater than or less than, get a dashed line, those guys get a solid line. So my case I know it going to be dashed.

I want to go back here to my graph and make sure this is a clear dashed line. If you use pen you're kind of in trouble this is why you don't use pen on Math homework my friends. So I'm going to go through and make a dashy line and the reason why it's dashed is because the points on this line are not solutions to the inequality. I need to show that the line doesn't contain solutions.

Next thing I want to do is decide which way to shade. Am I going to be shading this half of the plane, that's above my diagonal line, or I'm I going to be shading this lower half of the plane?

Well the way you tell is by picking any point you want to as long as it's not on the line you substitute in your x and y values and you see if you get a true inequality or not. So like for me I'm going to pick (0,0). I like zeros I think they are easy to work with. I'm going to test that point if x is 0 and y is 0 in this equation let's see if it's true. Is it true that 0 is greater than -1? Yeah that's a true statement. That means yes. Shade the side that contains 0, 0.

What this means is that every point in this plane is a solution to that inequality statement. Like if I choose the point like way the heck out there (-100,0) if I plug in x equals -100, y equals 0 into this equation, it's going to work and the way I know is because it's in the shaded region.

So one more thing to leave you with, there is a three step process for graphing two variable inequalities. First thing graph it like it was y equals mx plus b, second thing decide if it needs to be a solid or dashy line by looking at the inequality and then the third thing is to pick a point, plug it in like testing your x and y values and see if you get a true inequality statement or not.

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