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Absolute Value Inequalities - Problem 1

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 solve an absolute value equation, start by isolating the absolute value. Just like you would isolate a variable using inverse operations, the same rules apply to isolate the absolute value -- eliminate everything else on the same side of the equal sign except for the quantity inside the absolute value signs using inverse operations. Next, split the inequality into two inequalities and remove the absolute value signs. At this point, remember that one equality should be compared to the positive quantity on the other side, and the second equation should be compared to the negative quantity. However, the first inequality will have the same sign while in the second inequality, you must flip to sign. Solve for the variable in both inequalities and graph on a number line (since there is only one variable, you know to graph on a number line and not a coordinate plane). Don't forget to check your solutions by choosing a value within the range of each inequality and substituting them back into the original inequality.

This type of problem is an absolute value inequality and I have to graph it. That's a lot of stuff to keep in mind and so I want to be really careful to work though step by step.

First thing I notice is that when I graph it its probably going to be on a number line. So I only have one letter I have x so when you graph it's going to be the open circle closed circle business just on the number line it's not going to be one of these graphs, these are when you have two variables. Okay so let's go through and solve for x.

First thing I want to do with absolute values is isolate the absolute value that means get it all by itself mathematically.

Right now it's being multiplied by three so I need to undo that by dividing by 3 so now it looks like this. X minus 2 in the absolute value is greater than 5.

Next thing I want to do with absolute values is find my 2 solutions, I'll have x minus 2 and it's going to be compared to 5. And I'll have x minus 2 going to be compared to -5. But the trick is working with this inequality sign. This first one I didn't do anything to the 5, it stayed as +5

In the second one the 5 was kind of multiplied by a -1 that's why I need to change the direction of the inequality sign. These two inequality pieces are going to help me find my solutions.

Next thing go through and solve for x. X is greater than 7 that's going to be part of my answer also I'm going to have values that are looks like less than -3.

X could be any number that's bigger than 7 or it can be any number that's less than negative 3. I could use -1800, I could use -4, I could use -4.5, I could use +10.9 but I could not use 6.

I'm going to show you what that looks like on the graph. Again I know it's going to be a number line because there's only the x letter, there's not the y variable to go with it.

I'm going to set up my number line marking my important numbers -3 and +7. These are going to be open circles because I have the strict inequality. And then I want to mark numbers that are bigger than 7 that's everything out here bigger than 7. And then these are numbers that are smaller than -3. So it could be -4, -5, -100, any number that's out on that left direction. Again I use the open circle because these are both strict inequalities, are not greater than an equal to. That's how I know these are open circle and it goes out. Like oars, personally how I remember it in oar problem, the things look like oars for rowing along, I don't know if that helps.

But so again the last thing you want to keep in mind when you see these problems is like absolute value tricks, inequality tricks including making that guy negative when you change the direction of the sign and the open circle, closed circle business.

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