Solving Rational Inequalities

**misiaizeska** · November 7th 2012, 12:46 PM

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?

**MarkFL** · November 7th 2012, 01:01 PM

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

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

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

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

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.

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