Solving inequalities

Random-Hero- · #1 $Old$ January 12th, 2009, 05:22 PM

How would I go about solving
(1/x) + 4 < 2/x?

Any help would be greatly appreciated!!

Mush · #2 $Old$ January 12th, 2009, 06:04 PM

Quote:

Originally Posted by Random-Hero- $View Post$

How would I go about solving
(1/x) + 4 < 2/x?

Any help would be greatly appreciated!!

$\frac{1}{x} + 4 < \frac{2}{x}$

Subtract $\frac{2}{x}$ from both sides!

$\frac{1}{x} + 4 - \frac{2}{x} < 0$

Subtract 4 from both sides!

$\frac{1}{x} - \frac{2}{x} < -4$

Now you have to re-write the LHS as a single fraction, which is easy since they have the same denominator!

$\frac{1-2}{x} < -4$

$\frac{-1}{x} < -4$

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!

$\frac{1}{x} > 4$

Can you draw any conclusions from this?

Last_Singularity · #3 $Old$ January 12th, 2009, 06:04 PM

Quote:

Originally Posted by Random-Hero- $View Post$

How would I go about solving
(1/x) + 4 < 2/x?

Any help would be greatly appreciated!!

I recommend these steps:

1. Rewrite $4$ as $4x/x$
2. Combine the two terms on the left hand side
3. Cancel out the $x$ terms in the denominator
4. Subtract 1 from both sides
5. Divide by 4 on both sides

Assuming that $x$ is nonzero, of course.