# Solving Rational Inequalities

• November 7th 2012, 12:46 PM
misiaizeska
Solving Rational Inequalities
I'm having a hard time solving this question:
(2t-10)/t > t + 5

I multiplied by the squared lowest common denominator value and tried to solve for x, except that I cannot find any real solutions. I'm not sure what else to do?
• November 7th 2012, 01:01 PM
MarkFL
Re: Solving Rational Inequalities
$\frac{2t-10}{t}>t+5$

$2-\frac{10}{t}>t+5$

$\frac{10}{t}+t+3<0$

$\frac{t^2+3t+10}{t}<0$

The numerator has no real roots (the discriminant is negative), so we only have the critical number $t=0$. So, test the expression on the two intervals:

$(-\infty,0)$ and $(0,\infty)$. If the expression is negative on either interval, then that interval is the solution.