Answer to Question #88937 in Algebra for RAKESH DEY

Question

|x1-x2|= |x1|-|x2| for all x1,x2€R.
Is the statement true or false? Justify your answer.

Answer 1

For all $a,b\in R$ we have the triangle inequality

$$|a|+|b|\ge |a+b|$$

Setting $a={{x}_{1}}-{{x}_{2}}$ and $b={{x}_{2}}$, we obtain

$$|{{x}_{1}}-{{x}_{2}}|+|{{x}_{2}}|\ge|{{x}_{1}}-{{x}_{2}}+{{x}_{2}}|$$

That is

$$|{{x}_{1}}-{{x}_{2}}|\ge |{{x}_{1}}|-|{{x}_{2}}|$$

It follows that there exist x₁ and x₂ for which inequality

$$|{{x}_{1}}-{{x}_{2}}|>|{{x}_{1}}|-|{{x}_{2}}|$$

holds. For example, if x₁=2 and x₂=-3 we have

$$|2-\left( -3 \right)|>|2|-|-3|$$

or

$$|5|>|2|-|3|$$

that is

$$5>-1$$

Hence, the statement

$$|{{x}_{1}}-{{x}_{2}}|\,=\,|{{x}_{1}}|\,-|{{x}_{2}}|$$

is false.