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Math Help - Quadratic Inequalities

  1. #1
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    Exclamation Quadratic Inequalities

    Find the range of values of x for which (1+x)(6-x) < = -8
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  2. #2
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    Quadratic Inequalities are ALWAYS easiest to solve by completing the square...

    \displaystyle (1 + x)(6 - x) \leq -8

    \displaystyle 6 - x + 6x - x^2 \leq -8

    \displaystyle x^2 - 5x - 6 \geq 8

    \displaystyle x^2 - 5x \geq 14

    \displaystyle x^2 - 5x + \left(-\frac{5}{2}\right)^2 \geq 14 + \left(-\frac{5}{2}\right)^2

    \displaystyle \left(x - \frac{5}{2}\right)^2 \geq \frac{56}{4} + \frac{25}{4}

    \displaystyle \left(x - \frac{5}{2}\right)^2 \geq \frac{81}{4}.


    You should be able to go from here. Also, note that " \displaystyle \sqrt{X^2} = |X|" and " \displaystyle |X| \geq a" is equivalent to " \displaystyle X \leq -a and \displaystyle X \geq a".
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