For which $xx$ do we have $\sqrt{x^2}=-x\backslash sqrt\{x^2\}=-x$?

In general, we have $\sqrt{x^2}=\mid x\mid \backslash sqrt\{x^2\}=\backslash left|x\backslash right|$. So:

For which $xx$ is it true that $\mid x\mid =-x\backslash left|x\backslash right|=-x$?

Answer: For all values of $xx$ that are less than or equal to $00$.

This is true for all values $x\ge 1x\backslash geq1$ , since $\sqrt{{(x-1)}^2}=\mid x-1\mid \backslash sqrt\{\backslash left(x-1\backslash right)^2\}=\backslash left|x-1\backslash right|$ and the absolute value function is equal to the right-hand side exactly for $x-1\ge 0x-1\backslash geq0$ .