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Solving "Greater Than" Absolute Value Inequalities - Problem 2

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Solving a more involved absolute value greater than problem. The first step for this is to always isolate your absolute value term. A common mistake is to want to make your two equations without isolating that. It’s always going to end wrong for you.

So what you first always want to do is to get this by itself. First step in that, adding the 9 over, the 3 absolute value stays the same, 15 plus 9 is 24. Next step in isolating that absolute value, is dividing that by 3. 2x plus 4 is greater than 8.

So once we have our absolute value by itself, is then that we can actually split these up and make our two equations. First equation is just going to stay the same. So this just goes to 2x plus 4 is greater than 8. The other equation 2x plus 4 stays the same. We need to flip our inequality sign, it becomes a less than and take the opposite of the number we are dealing with -8. Remember that we are dealing with a absolute value greater, think of as or which turns into a union. It's either going to be one or the other.

Now we just have a union of two linear inequalities, solve them out as you would before. So subtract 4, 2x is greater than 4, x is greater than 2. Taking the union of that, put the answer from this subtract 4 from both sides. 2x is less than -12, x is less than -6. We are dealing with the union of, it's a little bit spread out. Let’s rewrite over here.

We are dealing with the union of x greater than 2, and x is less than -6. We want a visual representation to see actually what’s going on. We can draw our number line -6. Open circle because it’s not equal to. We're dealing with 2 going up. And remember the union is just where one element is. So in this case, one element is less than -6, one element is greater than 2, we're just dealing with the union though. I tend to write things in interval notation. So that will be negative infinity to -6, soft bracket, union of 2 to infinity.

There you go. Always make sure you get your absolute value by itself before you do anything else.

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