Chứng minh rằng x,y ∈ Q a)|x+y |$\leq$ |x|+|y| b) |x-y| $\geq$ |x|-|y|

Question

Chứng minh rằng x,y ∈ Q
a)|x+y |$\leq$ |x|+|y|
b) |x-y| $\geq$ |x|-|y|

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Valentina 1 năm 2021-07-17T16:52:28+00:00 1 Answers 6 views 0

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    2021-07-17T16:54:16+00:00

    Giải thích các bước giải:

    a)  $\forall x,y \in  Q :  $  
    $x\leq \left | x \right |,-x\leq \left | x \right |$
    $y\leq \left | y \right |,-y\leq \left | y \right |$
    $\Rightarrow x+y\leq \left | x \right |+\left | y \right |,-x-y\leq \left | x \right |+\left | y \right |$
    $\Rightarrow x+y\geq -\left ( \left | x \right |+\left | y \right | \right )$
    $\Rightarrow -\left ( \left | x \right |+\left | y \right | \right )\leq x+y\leq \left | x \right |+\left | y \right |$
    $\Rightarrow \left | x+y \right |\leq \left | x \right |+\left | y \right |$

    b) Theo câu a : 

    $\left | x-y \right |+\left | y \right |\geq \left | x-y+y \right |= \left | x \right |$
    $\Rightarrow \left | x-y \right |\geq \left | x \right |-\left | y \right |$

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