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**1. Homework Statement**

Determine all x in R such that the following hold:

1) (x+4/x-2) < x

2) |x+4/x-2| < x

3) |2x| > |5-2x|

**2. Homework Equations**

We have the triangle inequality, |a + b| [tex]\leq[/tex] |a| + |b|, which also implies the relation ||a| - |b|| [tex]\leq[/tex] |a-b|.

Also, relations such as: |x| < a implies -a < x < a, etc.

|ab| = |a|*|b|

**3. The Attempt at a Solution**

I feel very bad asking about these, I'm sure I should have learned this sort of thing earlier than analysis I (call it my weak spot). Anyway, here is what is bothering me:

1) (x+4/x-2) < x --> 0 < x - (x+4/x-2) --> 0 < [x(x-2) - (x+4)]/(x-2)

--> 0 < (x^2 - 3x + 4)/2. Now if we use the quadratic formula to find the roots of the numerator, we get (3 +/- [tex]\sqrt{9-16}[/tex] )/2, so the roots are complex. So am I to assume that there are no x in R such that this inequality is true? I can accept that, but then the next one is the same, except the absolute value. So for that one:

2) |x+4/x-2| < x --> -x < (x+4/x-2) < x --> -x(x-2) < x + 4 < x(x-2)

--> -x^2 + x -4 < 0 < x^2 - 3x - 4. Now the lhs doesn't factor, but the quadratic formula results in complex roots again, and the rhs roots, which would be x=4 and x=-1, pose problems: if you substitute x = 4 or x = -1 into the inequality you get 4 < 4 and -1 < -1 which is obviously never true! So - do I have to assume again that there are no real x in R such that this inequality holds? I'm getting worried that all my problems (there are more very similar to these) seem to have no real x possible! It makes me think I am doing something wrong...

3) |2x| > |5-2x| --> 2|x| > |5-2x| >= ||5| + |-2x|| --> I'm not sure what to do next! There is obviously some very simple answer that I should know. Maybe I am just having a slow night (lack of sleep does that!). I thought I'd be using the triangle inequality, but then, it's a difference and not a sum... so I'm just not sure what I'm supposed to do.

Any tips for those of us who are SLOW SLOW SLOW with this type of inequality? I feel like I can handle proofs pretty well, with sup/inf concepts, proving other similar sorts of things, and have always been fine at abstract and linear algebra. But concrete math problems with numbers, whoa! No good at that... :grumpy: