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Solving "Greater Than" Absolute Value Inequalities - Problem 1

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 absolute value greater than a number problem. So, for here we have a 3x minus 1 absolute value that is greater than or equal to 14. First thing we have to do is make this into two equations. So the first one is just going to stay the same. 3x minus 1 greater than 14.

The second one is going to be, the absolute value stays the same, but then we have to flip the sign and flip the number. So by flip I mean just take the opposite. It was a greater than, so now it turns to less than or equals to, numbers 14 take the opposite-14.

And the trick with greater than is remembering that it's an odd. It’s a union statement, so we are actually taking the union between these two things. So going back to sort of what we did in solving the union of two equations. Solve it out as we would any other.

So add 1 to this one, 3x is greater than 15. X is greater than or equal to 5. Same idea over here, add 1 to both sides, -14 plus 1 is -13, divide by 3, x is less than or equal to -13 over 3.

We can plot these on a number line to see if there is any union, that we can simplify, union at all. So you are dealing with every number is greater than 5 or equal to it. So I filled in going up we have -13/3. X is less than that so I filled in going down.

And remember that union is just where one element is represented. So in this case we are just dealing with these two outside points. So, depending on how your teacher wants you written, you can either just write the union between these two pieces of information. Or if you want to do it in interval notation, negative infinity to -13/3, hard bracket because we are including that on point, union of 5 to infinity. Best putting up the absolute value, remembering that greater is a union. You are able to solve it out.

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