Inequality with x in the denominator (rational inequality)

**Oiler** · January 11th 2011, 07:22 PM

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.

**pickslides** · January 11th 2011, 07:35 PM

Are you trying to solve for x?

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

**Prove It** · January 11th 2011, 08:20 PM

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 $\displaystyle x$ is positive or negative. You will need to consider each case.

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

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

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

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

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

$\displaystyle 1 < 2x$

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

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

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