Question

For any two real numbers $x\&y,$ the inequality that holds true is

• A. $|x+y|\le |x|-|y|$
• B. $|x+y|\le |x|+|y|$
• C. $|x+y|\ge |x|+|y|$

Answer 1

Answer: (B), (C)

$|x+y|\ge |x|+|y|$

Given two real numbers $x\&y.$
To find the relationship between $|x|,|y|\&|x+y|$
Let's consider the case where $x,y\ge 0$
$\Rightarrow |x|=x\&|y|=y$
Also, $|x+y|=x+y=|x|+|y|$
$\Rightarrow |x+y|=|x|+|y|\forall x,y\ge 0$

Let's consider the case where $x\ge 0\&y<0$
$\Rightarrow |x|=x\&|y|=-y$
$\Rightarrow |x|+|y|=x+(-y)$
Whose value is greater than $|x+y|$
$\Rightarrow |x+y|<|x|+|y|\forall x\ge 0,y<0$

Let's consider the case where $x,y\le 0$
$\Rightarrow |x|=-x\&|y|=-y$
$\Rightarrow |x|+|y|=-x+(-y)$
Also, $|x+y|=-(x+y)=|x|+|y|$
$\Rightarrow |x+y|=|x|+|y|\forall x,y\le 0$
Thus for any real number $x,y$
$|x+y|\le |x|+|y|$