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Absolute Value Inequality Graphs in Two Variables - Problem 1
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When graphing an absolute value linear inequality, first make sure the inequality is in terms of y. Absolute value linear inequalities are v-shaped. To find points on the graph, set up a table of values. Use a wide range of values in order to get points that cover both sides of the "v". Use a dotted or solid line, depending on the inequality sign, to connect the points. After graphing using a table of values, shade in the appropriate region. If y is greater than the absolute value quantity, then shade above the graph. If y is less than the absolute value quantity, then shade below the graph.

In this problem, you’re asked to graph the absolute value inequality Y the absolute of X-2. So in order to do that, the first thing I want you to remember is that, absolute values are V shape. Unless you have an X2 or something which you won’t see in your algebra, you’re going to have a V shaped graph.

The other thing I’m going to do, is set up a table, where I’m going to choose some X numbers and substitu6te them in to find the Y numbers, that go with the x. Again, it’s a good idea to use some negative X numbers in addition to positive numbers, just so you can find the whole V shape.

So let’s see. If I -1 in there, Y would be equal, I’m just going to treat this as a equal sign for now, even though it’s inequality. Y would be the absolute value of -1, take away 2 which is the absolute value of -3, absolute value of -3 we know is 3. So I’m going to have a dot at -1 for my X number, 3 for my one number. There it is.

Then I’m going to use 0 so if I put in X equals 0, I’ll have Y equals to the absolute value of 0 take away 2which is the same thing as the absolute value of -2 which is 2. Here’s my next point at (0, 2). Go through like that, I’m going to give you a couple in my head, I’ll try to talk you through what I’m doing, if X is one, I’ll do 1-2 so that’s -1, absolute value Y becomes +1, so I have a dot at (1, 1).

Then if I do X equals 2, 2 take away 2 is 0, absolute value of 0 is just plain old 0. I’m going to keep going a little bit further. One of the things with graphing using a table, is that, if you don’t find your whole shape. By whole shape I mean, I’m looking for a V, and I haven’t yet found where that V point happens, that’s how I know I need to keep going in my table.

What happens if X is 3? Let’s see, X is 3 you’ll have 3 minus 2 that’s 1, absolute value of Y is just still one. That’s good, I found one of my graphs turned around, I have that V shape. I’m just going to use symmetry to show that this V would continue out in the right direction, even though I didn’t do these points. Since I know about the parent function, since I know these graphs are V's, I can go ahead and like extend it out in that way.

First step done. Next thing I have to do, is decide if they should be a solid or dashed line, based on that inequality sign. Well since it’s a strict inequality, meaning Y is only bigger than X-2, that means I have to go back and make this a dashed line.

So I’m going to make that a dashed line. That just means like mathematically, when we see that, that means we know it’s a less than or greater. We know it couldn’t be equal to the points on this V.

The last thing I need to do, is find out shading. I’m either going to shade everything outside of the V, or else everything inside of the V. And the way to pick, or the way to tell, is to choose any point you want to, that’s not on your graph. Plug in the X and Y values and see if you get a true or false indemnity.

I’m just going to pick the point (0, 0) because I think (0, 0) is kind of an easy point to work with, having 0s for X and Y. Let’s substitute in. Is it true that 0 is greater than the absolute value of 0-2? Is 0 bigger than 2? No it’s not. So that means don’t shade here. What I need to do to finish this problem is to shade inside my V.

What that tells us is that, any point I chose on the graph, that showed up in that shaded area, would make this inequality statement true. Like I could use the point like (2, 4). If I plugged in the values, 2 for X 4 for Y, that would make this inequality statement true.

Again guys, once you know the parent function’s shape, you know this is a V, make your table. Continue it until you see the V and then it’s just graphing inequality techniques from there.

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