For any two real numbers x and y- such that xy- 0- the value of -x-y- equals

Given: Two real numbers x & y such that xy≥ 0. From the triangle inequality: |x+y|≤ |x|+|y| Case 1: x, y≥ 0 ⇒ |x|=x, |y|=y ⇒ |x+y|=x+y=|x|+|y| Case 2: x, ...

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For any two r...

Question

For any two real numbers $x$ and $y,$ such that $x y \geq 0$ , the value of $| x + y |$ equals

| x | + | y |

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- | x | + | y |

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- | x | - | y |

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Solution

The correct option is A $| x | + | y |$
Given: Two real numbers $x & y$ such that $x y \geq 0.$
From the triangle inequality:
$| x + y | \leq | x | + | y |$

Case 1: $x, y \geq 0$
$\Rightarrow | x | = x, | y | = y \Rightarrow | x + y | = x + y = | x | + | y |$

Case 2: $x, y \leq 0$
$\Rightarrow | x | = - x, | y | = - y \Rightarrow | x + y | = - (x + y) = - x - y = | x | + | y | \Rightarrow | x + y | = | x | + | y |$
Thus, we get: $| x + y | = | x | + | y | \forall x y \geq 0$

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For any two real numbers x and y, such that xy≥0, the value of |x+y| equals

For any two real numbers $x$ and $y,$ such that $x y \geq 0$ , the value of $| x + y |$ equals