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.

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- Jan 11th 2011, 07:22 PMOilerInequality 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 PMpickslides
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 PMProve It
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 $\displaystyle \displaystyle x \in (-\infty, 0) \cup \left(\frac{1}{2}, \infty\right)$.