Chapter 6.5, Problem 4ST

To determine

The value of $\left(x\right)$ for which the expression $\sqrt{{\left(x-4\right)}^2}=4-x$ is true.

Answer to Problem 4ST

The expression is true for $x<0$ .

Explanation of Solution

Given information: The given expression is,

  $\sqrt{{\left(x-4\right)}^2}=4-x\mathrm{.....}\left(1\right)$

Calculation: Simplify left hand side of (1) gives,

  $\begin{array}{c}\sqrt{{\left(x-4\right)}^2}=\left|x-4\right|\\ =x-\mathrm{4......}\left(\text{For }x\ge 0\right)\\ =4-x\mathrm{......}\left(\text{For }x<0\right)\mathrm{......}\left(2\right)\end{array}$

From (2) the expression (1) holds true if $x<0$ . Therefore for $x<0$ the expression is true.