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Math Help - Another algebra exercise

  1. #1
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    Another algebra exercise

    \dfrac{x+5}{4} <= (\dfrac{x-7}{2})^{2}

    \dfrac{x+5}{4} <= \dfrac{x^{2}-14x+49}{4}

    -x^{2}+15x <= 44

    How do I continue?
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  2. #2
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    Quote Originally Posted by klik11 View Post
    \dfrac{x+5}{4} <= (\dfrac{x-7}{2})^{2}

    \dfrac{x+5}{4} <= \dfrac{x^{2}-14x+49}{4}

    -x^{2}+15x <= 44 no need to bring the terms to the left

    How do I continue?
    x^2-15x+44\ \ge\ 0

    (x-11)(x-4)\ \ge\ 0

    This is a U-shaped quadratic, which lies above the x-axis to the left of the smaller root
    and to the right of the larger root.
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  3. #3
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    Thanks!!!!!
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  4. #4
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    I'm sorry but I still can't solve the whole exercise.

    The whole exercise is this:
    "Solve the next systems:" (sorry for the bad translation)



    The solution is
    1 \ge\ x \ge\ 2

    What am I doing wrong?
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  5. #5
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    Quote Originally Posted by klik11 View Post
    I'm sorry but I still can't solve the whole exercise.

    The whole exercise is this:
    "Solve the next systems:" (sorry for the bad translation)



    The solution is
    1 \ge\ x \ge\ 2

    What am I doing wrong?
    Hi,

    the one on the right is correct, because the graph is below the x-axis between x=1 and x=2.

    1\ \le\ x\ \le\ 2

    For the equation on the left, -\infty\ < x\ \le4 and 11\ \le\ x\ < \infty

    The values you have written is the range of x on or below the x-axis for that graph.
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  6. #6
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    I see, thanks again!
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  7. #7
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    If you want to find the solutions without reference to the graphs, you can reason thus...

    (x-4)(x-11) is positive if x>11

    as that causes both factors to be positive

    or (x-4)(x-11) is positive if x<4

    as that causes both factors to be negative.

    If x is in between 4 and 11, one factor is negative and the other positive.


    For (x-1)(x-2) both factors are positive if x>2

    and both factors are negative if x<1

    Hence, the factors have opposite sign if x is between 1 and 2.
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