TRUE OR FALSE: For all real x, y and z we have that |x − y + z| ≤ |x| + |y| + |z|

Question

TRUE OR FALSE: For all real x, y and z we have that |x − y + z| ≤ |x| + |y| + |z|

Answer 1

According to the Triangle Inequality, l A + B + C l ≤ lAl + lBl + lCl

So, l x - y + z l = l x + (-y) + z l ≤ lxl + l-yl + [zl = lxl + lyl + lzl

Therefore, the given inequality is TRUE.

Answer 2

By contraction:

Assuming x - y + z > 0. and x, y, and z each > 0

x - y + z ≤ x + y + z

-y ≤ y

Not necessarily, therefore the statement is false.