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Absolute Value Inequality Graphs in Two Variables - Problem 3

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

When graphing an absolute value linear inequality, first make sure the inequality is in terms of y. Use inverse operations to isolate y. To find points on the graph, set up a table of values. Keep in mind that absolute value linear inequalities are v-shaped. 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. Pick any point on the coordinate plane and plug it into the inequality. If the points make the inequality true, then shade in that region of the graph. If it is not true, then shade the opposite region of the graph. In other words, 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.

When you’re wanting to graph absolute value inequalities, it’s really important that you keep in mind, it’s going to be a V shape, as long as you have just X and Y and nothing squared or anything. So what I’m going to do, and this problem is asking me to make a table, I’m going to choose X and Y numbers, hoping that the numbers that I choose, will show me the V shape. I want to see where this graph turns around.

So let’s see if I plug -2 in for my X number. -2 absolute value of Y becomes +2 plus 2 more is 4. -1 becomes +1 plus 2, 0 plus 2 equals 2, 1 plus 2 equals 3. Those are the points that are going to go on my graph. Let’s go ahead and get them on the grid.

(-2, 4), (-1, 3), (0, 2), (1, 3) and (2, 4). There it is, I’m lucky. I’m not lucky I’m happy I got my V shape. I can see where this graph turns around. Where it becomes a complete V.

The next thing I want to do, since this is an inequality, I need to decide if it’s going to be a solid line or a dashy line. Since that’s less than, it’s not less than or greater than, I know it’s going to be a dashed line.

So go back and use your eraser, and make sure that that’s clearly a dashy line, a dashy V. And that tells your teacher or whoever is communicating through this graph, that that’s going to be a less than sign with now less than are equal to.

Last thing is, choose any point you want to help you with the shading. I’m going to pick (0, 0) and plug in my X number 0 and my Y number 0, and see if I get a true or false statement, like yes or no. Like is it true that 0 is less than the absolute value of 0 plus 2? Is it true that 0 is less than 2? Yeah it is. That means this point right here, gave me a yes, like a true solution. That means shade here.

My solutions are going to be any point that I could choose that’s outside of my V. Any point that I could choose like way out here even, like (100, 0). If I were to plug in those X and Y coordinates, it would be a true solution for that inequality absolute value statement.

When you see these problems, the last thing I’m going to tell you is to keep in mind your absolute value V shape, making a table, so you can see your whole V. Decide if it’s a solid line or dashy line. And then last but not least, pick a point that helps you do the shading, to know if you shape inside the V or outside the V.

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