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Math Help - Problem with inequalities

  1. #1
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    Problem with inequalities

    Well, I'm not done yet. Am I on the right track?

    \sqrt{x^2-9x+20}\leq\sqrt{x-1}\leq\sqrt{x^2-13}

    x^2-9x+20\leq{x-1}
    x-1\leq{x^2-13}

    x^2-10x+21\leq{0}
    0\leq{x^2-x-12}

    (x-7)(x-3)\leq{0}
    0\leq{(x-4)(x+3)}
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  2. #2
    Senior Member mollymcf2009's Avatar
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    Hi Nathan02079!

    Are you supposed to be solving the inequality for x? Or just evaluating the function until you find what is \leq 0 \leq ?
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  3. #3
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    I am solving for x.
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  4. #4
    Senior Member mollymcf2009's Avatar
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    Looks good to me!
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  5. #5
    Senior Member pankaj's Avatar
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    Quote Originally Posted by nathan02079 View Post
    Well, I'm not done yet. Am I on the right track?

    \sqrt{x^2-9x+20}\leq\sqrt{x-1}\leq\sqrt{x^2-13}

    x^2-9x+20\leq{x-1}
    x-1\leq{x^2-13}

    x^2-10x+21\leq{0}
    0\leq{x^2-x-12}

    (x-7)(x-3)\leq{0}
    0\leq{(x-4)(x+3)}
    NOT yet
    First of all you need to find the values of x for which all the square roots are possible.i.e quantities under the square root signs must be non-negative.

    x^2-9x+20\geq 0,i.e.x\in(-\infty,4]\cup[5,\infty)

    x-1\geq 0,i.e x\in[1,\infty)

     <br />
x^2-13\geq 0,i.e x\in(-\infty,-\sqrt{13}]\cup[\sqrt{13},\infty)<br />

    On taking the intersection of the above conditions you will get

     <br />
x\in[1,4]\cup[5,\infty).<br />
    which can be said to be the domain of the inequality.

    Now the solutions you obtain by solving

    (x-7)(x-3)\leq{0} i.e x\in[3,7]

    0\leq{(x-4)(x+3)} i.e x\in(-\infty,-3]\cup[4,\infty)

    Therefore the values of x satisfying both the inequalities are x\in[4,7]

    which must satisfy x\in[1,4]\cup[5,\infty).

    Therefore the answer to the given question must be [5,7]
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  6. #6
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    i think answer is [4 7].
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  7. #7
    Senior Member pankaj's Avatar
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    Take x=4.5 and put it in x^2-9x+20 ,you will obtain -0.25 which is impossible since \sqrt{-0.25} is not possible in real numbers
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  8. #8
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    Therefore the answer to the given question must be {4}u[5,7]
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  9. #9
    Senior Member pankaj's Avatar
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    That's correct.
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