# Thread: Inequality with x in the denominator (rational inequality)

1. ## Inequality with x in the denominator (rational inequality)

Hey all,

I am having problems trying to figure out the following inequality. 1/x<2. I keep getting inverse x and not sure how to proceed from there.

2. Are you trying to solve for x?

If so multiply both sides by x, then divide both sides by 2.

3. Originally Posted by pickslides
Are you trying to solve for x?

If so multiply both sides by x, then divide both sides by 2.
You can't do that Pickslides, because you don't know if $\displaystyle \displaystyle x$ is positive or negative. You will need to consider each case.

Case 1: $\displaystyle \displaystyle x < 0$...

Clearly $\displaystyle \displaystyle \frac{1}{x} < 0$, and therefore $\displaystyle \displaystyle < 2$.

So $\displaystyle \displaystyle x < 0$ satisfies the inequality.

Case 2: $\displaystyle \displaystyle x > 0$...

$\displaystyle \displaystyle \frac{1}{x} < 2$

$\displaystyle \displaystyle 1 < 2x$

$\displaystyle \displaystyle \frac{1}{2} < x$.

So the solution is $\displaystyle \displaystyle x \in (-\infty, 0) \cup \left(\frac{1}{2}, \infty\right)$.