Abs Value

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
Solve for x

\(\displaystyle \left\vert x+1 \right\vert-\left\vert x \right\vert+2\left\vert x-1 \right\vert=2x-1\)
 
Aug 2007
3,171
860
USA
You must solve this four times.

1) x < -1
2) -1 < x < 0
3) 0 < x < 1
4) x > 1

Where did I get those values? Why are they important? What's the best way to go about it?
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
Not sure why it is broken up like that.
 
Aug 2007
3,171
860
USA
You didn't expect to get off THAT easy, did you? Look at those absolute values. Look hard. How do they behave?

For |x|, What is different about this symbol for x > 0 and x < 0?

For x > 0, |x| = ?????
For x < 0, |x| = ?????
 
Apr 2010
384
153
Canada
You didn't expect to get off THAT easy, did you? Look at those absolute values. Look hard. How do they behave?

For |x|, What is different about this symbol for x > 0 and x < 0?

For x > 0, |x| = ?????
For x < 0, |x| = ?????
For x > 0, |x| = x
For x < 0, |x| = -x

We basically have to compute your problem for all possible values of the absolutes...which sucks, this is a long problem. I'm far to lazy lol
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
I pretty much said screw it too. This is a practice question for the GRE Math subject test.
 
Aug 2007
3,171
860
USA
Come on, it's not that bad.

You can give it a good guess.

1) Let's look at the left side. For any 'x', only that middle term will have any influence as far as producing a negative number. However, the other two terms contribute three times the influence to the positive. In other words, it's safe to say that this thing is NEVER negative.

2) Still on the right side -- For x > 1, ALL THREE TERMS are positive so we can just chuck the Absolute Values, giving 2x-1

3) Looking at the right side: Hey! That has the same slope and the same y-intercept. There aremany solutions to this guy.

This kind of thinking can be done rather quickly. You are right - you don't have time for a formal examination in the middle of the GRE, but you may be able to spot something useful with a little creative thinking.

Good luck.
 
Last edited:
Nov 2009
517
130
Big Red, NY
Yeah, 3 was just the first number I threw in there.
 
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