Inequalities with absolute values and fractions
I am facing a couple of issues with two inequalities, would appreciate help here.
1) |2x + 3| ≤ 3x − 5
the first case would be 2x + 3 ≤ 3x - 5, solves easily, but no approach to the second case makes sense to me:
-2x - 3 ≤ 3x - 5 ??
2x + 3 >= -3x + 5 ??
=> x >= 2/5 but that doesn't even come close to fitting the original inequality?
If I square them, I get
4x^2 + 12x + 9 <= 9x^2 - 30x + 25, solving that with the usual formula I end up with x1 = 2 and x2 = 8/5, which is also somewhat strange.
2) 3 / | x - 9 | > 2 / (x + 2)
The essential questions here are how many cases I have to distinguish between and if anyone could be so kind to formulate a general rule when the inequality sign changes direction and the signs on one (or both?) sides of the inequality change from + > - and vice versa.