# Inequality with x in the denominator (rational inequality)

• Jan 11th 2011, 07:22 PM
Oiler
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.
• Jan 11th 2011, 07:35 PM
pickslides
Are you trying to solve for x?

If so multiply both sides by x, then divide both sides by 2.
• Jan 11th 2011, 08:20 PM
Prove It
Quote:

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