Rational Inequality with Absolute Value
posted by Randy on .
Here's a question I'm having trouble with:
x/x+2<2
Solve for x.
I'm trying to use the case method to solve this problem. Basically I just need confirmation that I did this correctly. I'm doing this through distance learning so I can't ask a teacher until Monday.
Case 1:
x is greater than or equal to zero, and x+2 is greater than zero.
x/(x+2)<2
x<2(x+2)
x<2x+4
x<4
x>4
Now here's where I get confused... since the assumptions for this case were x>=0 and x+2>0, x must be greater than or equal to zero. When I solved for x in this case, I got x>4. Reconciling the constraints leaves me back at x>=0. Did I do that correctly?
Case 2:
x is greater than or equal to zero, x+2 is less than xero (x is less than 2.) This isn't possible since there are no numbers that are both greater than or equal to zero and less than 2.
Case 3:
x is less than zero, and x+2 is greater than zero (x>2)
x/(x+2)<2
x<2(x+2)
x<2x+4
3x<4
x>4/3
When I reconcile the constraints, I end up with x is greater than 4/3 but less than zero. Again I am not sure if I did that correctly.
Case 4:
x is less than zero, and x+2 is less than zero (x<2.)
x/(x+2)<2
x<2(x+2)
x<2x4
x<4
Reconciling the constraints leaves me with x<4.
When I combine all of the solutions from all of the cases, I get a solution set of {xx<4 or x>4/3}
This seems to be correct when I enter test points, but I was wondering if someone could check my work and confirm that I've done it correctly.
Thanks
Randy

Your approach is impeccable.
There is only one little glitch for which you have to watch out.
In case one, you assumed x>0, and x+2>0, and you obtained x>4, which is inconsistent with your assumption of x>0.
So it should have been revised to x>0.
If you submitted your answer as
(∞,4)∪(4/3,∞)
you would have the correct answer, since [0,∞) from case 1 is absorbed in case 3.
If you had included x>4 (from case 1), the answer would have been (∞,∞) which is incorrect. 
Thanks for you help. I would have expressed my answer in the way you suggested, but I wasn't sure how to type the infinity symbol. I will watch out for the glitch you mentioned.
Thanks again,
Randy 
:)
The infinity symbol at this forum can be typed as follows:
& i n f i n ;
but without the spaces between the characters.