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Thread: Solving Rational Inequalities

  1. #1
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    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?
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Solving Rational Inequalities

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

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

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

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

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

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