How to solve this rational inequality?

Question

$$\Big|\frac{2x - 1}{x + 1}\Big| \geq \frac{5x}{2}$$

First I attempted the positive case. I tried moving everything to one side of the equation and then factoring, but I am left with an un-factorable equation in the numerator. What do you do once you reach this point?

Answer 1

HINT:

$$|F(x)| \geq a \Rightarrow F(x) \geq a \text{ or } F(x) \leq -a$$

Answer 2

Hint:

Square it, then you get: $$4(2x-1)^2\geq 25x^2(x+1)^2$$

so $$(4x-2-5x^2-5x)(4x-2+5x^2+5x)\geq0$$

or $$(5x^2+9x-2)(5x^2+x+2)\leq 0$$

Since discriminat for the second one is $-39$ second factor is always $>0$ so you have to solve: $$5x^2+9x-2\leq 0$$

Then you have to solve quadratic inequality...