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Thread: Algebra

  1. #1
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    Algebra

    Given two real numbers $\displaystyle x$ and $\displaystyle y$ such that
    $\displaystyle (x+y)^2\leq2xy$
    find $\displaystyle x+y$.

    If $\displaystyle x+y=6$ and $\displaystyle xy=3$, find $\displaystyle \displaystyle\frac{1}{x}+\frac{1}{y}$.

    Given $\displaystyle |x-y|=2$ and $\displaystyle |y-z|=7$, find the minimum possible value of $\displaystyle |x-z|$.
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  2. #2
    o_O
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    1. $\displaystyle (x+y)^{2} \leq 2xy$
    $\displaystyle x^{2} + 2xy + y^{2} \leq 2xy$
    $\displaystyle x^{2} + y^{2} \leq 0$
    $\displaystyle x^{2} \leq -y^{2}$

    What values of x and y make this inequality even possible?

    2. $\displaystyle \frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{x + y}{xy}$

    3. $\displaystyle |x - y| = 2 \quad \Rightarrow \quad x - y = \pm 2 \quad \Rightarrow \quad x = y \pm 2$
    $\displaystyle |y - z| = 7 \quad \Rightarrow \quad y - z = \pm 7 \quad \Rightarrow \quad z = y \pm 7$

    Find x - z and see what the lowest value of |x - z| would be.
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  3. #3
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    Quote Originally Posted by o_O View Post
    1. $\displaystyle (x+y)^{2} \leq 2xy$
    $\displaystyle x^{2} + 2xy + y^{2} \leq 2xy$
    $\displaystyle x^{2} + y^{2} \leq 0$
    $\displaystyle x^{2} \leq -y^{2}$

    What values of x and y make this inequality even possible?
    Just skip your last line.
    $\displaystyle x^2+y^2 \leq 0$. Note $\displaystyle x^2+y^2\geq 0$ (since it is a sum of two squares). Thus, $\displaystyle x^2+y^2 = 0 \implies x=0,y=0$.
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