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Math Help - Inequalities help

  1. #1
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    Inequalities help

    I need help with these two inequalities Inequalities help-inequalities.png

    For c) I can get to x>or=5 but I know that it is true when x<0 too but I don't know how to show that algebraically

    For d) I can get to x>3/2 but I also know that it is true when x<1 but cant show it please help
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  2. #2
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    Re: Inequalities help

    Hello, calmo11!

    (c)\;\;e^{x^2} \:\ge\:e^{5x}

    We have: . x^2 \:\ge\:5x \quad\Rightarrow\quad x^2 - 5x \:\ge\:0

    . . . . . . . . x(x-5)\:\ge\:0

    It says: the product of two number is positive.
    This is true if both factors are positive or both factors are negative.
    We must consider these two cases.

    Both positive:
    . . x \,\ge\, 0 \:\text{ and }\:x-5\,\ge\,0 \quad\Rightarrow\quad x \,\ge\,5
    . . Both are true if x\,\ge\,5

    Both negative:
    . . x\,\le\,0\:\text{ and }\:x-5 \,\le\,0 \quad\Rightarrow\quad x\,\le\,5
    . . Both are true if x\,\le\,0

    Answer: . (x\,\ge\,5)\,\text{ or }\,(x\,\le\,0)




    (d)\;\;|4x-5| \:>\:1

    The absolute inequality gives us two statements:

    . . [1]\;4x-5 \,>\,1\;\text{ or }\;[2]\;4x-5\,<\,-1

    Solve them separately:

    . . [1]\;4x - 5 \,>\,1 \quad\Rightarrow\quad 4x\,>\.6 \quad\Rightarrow\quad x \,>\,\tfrac{3}{2}

    . . [2]\;4x-5 \,<\,-1 \quad\Rightarrow\quad 4x \,<\,4 \quad\Rightarrow\quad x \,<\,1

    Answer: . (x\,<\,1)\:\text{ or }\:(x\,>\,\tfrac{3}{2})
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  3. #3
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    Re: Inequalities help

    You are a genius! Why can't you teach me my uni course!
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