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Absolute Value Inequality Graphs in Two Variables - Concept

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

Graphing inequalities with two variables can be tricky and is made even more tricky when we graph inequalities with two variables and absolute value. With absolute value graphing, if the inequality is similar to the equation of a line, (for example y > m|x| + b), then we get a V shape, and we shade above or below the V. This is very similar to graphing inequalities with two variables. The difference is that we are graphing inequalities with absolute values which makes the V-shape.

What we're going to be working on here is how to graph Absolute Value inequalities, Absolute Value inequalities. What I want to do is show you guys a shortcut using a parent function so that when you guys see this in your homework hopefully it will go a little faster. What I mean by parent function is that all graphs of this type look the same, let me show you what they look like.
In order to show you that I've made an xy chart I just chose these x values and I'm going to substitute in them, substitute them in to find the the y values. Like if I put -2 in there, Absolute Value y's it I get +2. By the way Absolute Value y's is not a real word I kind of made that up just between you and me but you can use it if you want to. -1 Absolute Value y becomes +1 go through and fill it out like that point by point, then when you go to graph it you'll see you get this really interesting shape you get a v. What I did was plot the points -2 2, -1 1 like that and then if you connect them shoop you'll see you get a v shape be careful it's not a u it's not a parabola some of you guys already know about that. This is a strict v and that's really interesting all Absolute Value graphs are going to have that shape.
Now, there's a couple of other things to keep in mind, you know about inequalities if it's one of these two signs it's going to be a solid line, but when it's one of these two signs is going to be a dashed line so I might have like a solid v like in our case or sometimes you're going to have dashy v. The last thing you guys know about inequalities is that there is some shading that happens and the way you pick the shading is you get to choose any point you want to any point that's not on your Absolute Value v you substitute in your x and y pairs and see if you get a true inequality or not you're going to shade parts that are true here's what I mean. Let's just say I picked the point 5 1 I just randomly chose that point and you can use any point you want to, my x number is 5 my y number is 1, I'm going to put that in here. Is it true that 1 is greater than abso- excuse me, greater than or equal to 5? No that's not true no. This point is a no that means don't shade here you want to shade the other part instead of shading outside, you told me "don't shade outside, shade inside." What that means is that every single point inside of my v if I were to plug in my x and y pairs to this equation inequality I would get a true statement. So when you guys are working on these problems there's a couple of things to keep in mind I just want to summarize for you.
The first thing is that all Absolute Value graphs have this v shape.
The next thing is that all inequalities are either going to be solid if they are in greater than or equal to or less than or equal to or it might be dashy.
And the last thing you have to remember to do is the shading. You pick a point that's not on your v, plug in the x and y values and find the area that has true statements find the yeses.

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