Is negative 20 equal to negative 20? May be not!

Question

Proof:

Using simple Algebra it can be proved that -20 $\neq$ -20. Is it Algebraically correct, can I treat this as a puzzle only?

Answer 1

The assertion that

$$\sqrt{\left(4 - \frac 9 2\right)^2} = \sqrt{\left(5 - \frac 9 2\right)^2} \implies 4 - \frac 9 2 = 5 - \frac 9 2$$ is false; in particular, this step assumes that

$$|x| = |y| \implies x = y$$ which is certainly false. Hence this is not algebraically correct.