How would I go about solving
(1/x) + 4 < 2/x?
Any help would be greatly appreciated!!
$\displaystyle \frac{1}{x} + 4 < \frac{2}{x} $
Subtract $\displaystyle \frac{2}{x} $ from both sides!
$\displaystyle \frac{1}{x} + 4 - \frac{2}{x} < 0$
Subtract 4 from both sides!
$\displaystyle \frac{1}{x} - \frac{2}{x} < -4$
Now you have to re-write the LHS as a single fraction, which is easy since they have the same denominator!
$\displaystyle \frac{1-2}{x} < -4$
$\displaystyle \frac{-1}{x} < -4$
Now if you multiply through by negative 1, you'll get a nicer expression, but remember that when you multiply through by negative 1, the direction of the inequality changes!
$\displaystyle \frac{1}{x} > 4$
Can you draw any conclusions from this?
I recommend these steps:
1. Rewrite $\displaystyle 4$ as $\displaystyle 4x/x$
2. Combine the two terms on the left hand side
3. Cancel out the $\displaystyle x$ terms in the denominator
4. Subtract 1 from both sides
5. Divide by 4 on both sides
Assuming that $\displaystyle x$ is nonzero, of course.