Question

Verify that $\left|x+y\right|⩽\left|x\right|+\left|y\right|$ when $\left(i\right)$ $x=-3\&y=\frac{-7}{9},$ $\left(ii\right)$ $x=0.8\&y=\frac{-7}{9}$

Answer 1

$|x+y|\le |x|+|y|$ ..... $\left(1\right)$

$\left(i\right)$ $x=-3$ and $y=\frac{-7}{9}$

Now,

$|x+y|=|-3-\frac{7}{9}|=\frac{34}{9}$

$|x|=3$ and $|y|=\frac{7}{9}$

$|x|+|y|=3+\frac{7}{9}=\frac{34}{9}$

Therefore verified.

$\left(ii\right)$ $x=0.8=\frac{4}{5}$ and $y=\frac{-7}{9}$

Now,

$|x+y|=|\frac{4}{5}-\frac{7}{9}|=\frac{1}{45}$

$|x|=\frac{4}{5}$ and $|y|=\frac{7}{9}$

$|x|+|y|=\frac{4}{5}+\frac{7}{9}=\frac{71}{45}$

Therefore verified.