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Inequality absolute value help

  • #1
## x - |x-|x|| > 2 ##

how would I go about solving something like this?

my initial thoughts was to consider if x >= 0

I get 2-x < 0 then x > 2 in that case

then consider if x < 0 which I get -|x+x| > 2-x then 2x > 2-x then x > 2/3 but I'm having troubles deciding which one is correct, and if there is another way to do it (I can't seem to sketch |x-|x||

edit: I know it's pretty obvious from looking at the equation that x > 2, but just wondering why I get the x >2/3 part from, and if I got given a question which is not obvious, then how would I know x > 2/3 would be wrong (for example), and which inequality would be the correct bit
 
Last edited:

Answers and Replies

  • #2
statdad
Homework Helper
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You do need to consider cases, and your first bit is correct since, if [itex] x \ge 0 [/itex], the absolute value term gives
[tex]
| x - |x|| = |x - x| = 0
[/tex]

and your problem reduces to [itex] x > 2 [/itex].

For the case [itex] x < 0 [/itex], remember [itex] |x| = -x [/itex], so
[tex]
|x - |x|= |x - (-x)| = |2x| = \text{ what?}
[/tex]

Simplify remembering [itex] x < 0 [/itex]: then take that simplification and plug into your original inequality statement and see what happens.
 
  • #3
You do need to consider cases, and your first bit is correct since, if [itex] x \ge 0 [/itex], the absolute value term gives
[tex]
| x - |x|| = |x - x| = 0
[/tex]

and your problem reduces to [itex] x > 2 [/itex].

For the case [itex] x < 0 [/itex], remember [itex] |x| = -x [/itex], so
[tex]
|x - |x|= |x - (-x)| = |2x| = \text{ what?}
[/tex]

Simplify remembering [itex] x < 0 [/itex]: then take that simplification and plug into your original inequality statement and see what happens.
## |x - |x|= |x - (-x)| = |2x| = -2x ## if x < 0

plugging this in I get x - (-2x) > 2 so x > 2/3
 
  • #4
statdad
Homework Helper
1,495
35
Ok, so for the inequality to be satisfied you need to have both [itex] x > \frac 2 3 [/itex] and [itex] x > 2 [/itex] true. Where on the number line do those two inequalities intersect (where do their graphs overlap)? The answer to that gives the solution set to the problem.
 
  • #5
x>2 obviously! So stupid

thanks, lol.
 
  • #6
statdad
Homework Helper
1,495
35
"So stupid"

Not at all. At some time we've all benefited from a little push to look at a problem with a different eye.
 

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