# Find the range of values of x

• May 7th 2011, 08:31 AM
Punch
Find the range of values of x
Find the range of values of $x$ for which $|\frac{2x+5}{x^2-4}|\geqslant \frac{1}{5}$
• May 7th 2011, 10:44 AM
topsquark
Quote:

Originally Posted by Punch
Find the range of values of $x$ for which |\frac{2x+5}{x^2-4}|≥\frac{1}{5}

Show us what you know. Hint: Can you find any horizontal asymptotes? Vertical asymptotes?

-Dan
• May 7th 2011, 10:45 AM
HallsofIvy
So you want to find x so that $\frac{2x+ 5}{x^2- 4}\ge \frac{1}{5}$ or $\frac{2x+ 5}{x^2- 4}\le -\frac{1}{5}$

If |x|> 2, then $x^2- 4> 0$ so those are the same as
$5(2x+ 5)= 10x+ 25> x^2- 4$ and
$5(2x+ 5)= 10x+ 25> 4- x^2$
which are, of course, the same as
$x^2- 10x- 29> 0$ and
$x^2+ 10x+ 21> 0$
What values of x> 2 or x< -2 satisfy those?

If |x|< 2 then $x^2- 4< 0$ so those are the same as
$5(2x+ 5)= 10x+ 25< x^2- 4$ and
$5(2x+ 5)= 10x+ 25< 4- x^2$ which are the same as
$x^2- 10x- 29< 0$ and
$x^2+ 10x+ 21< 0$
What values of x, -2< x< 2, satisfy those?
• May 7th 2011, 11:31 AM
mcp2
Multiply out and get it all on one side, you will have a quadratic.