# How do you find the solution set for the equation |x-6|=7?

Oct 19, 2016

$x \in \left\{- 1 , 13\right\}$

#### Explanation:

Note that we define the absolute value function as

$| a | = \left\{\begin{matrix}a \mathmr{if} a \ge 0 \\ - a \mathmr{if} a < 0\end{matrix}\right.$

so we consider two cases:

Case 1) $x - 6 \ge 0$

$\implies | x - 6 | = x - 6$

$\implies x - 6 = 7$

$\implies x - 6 + 6 = 7 + 6$

$\implies x = 13$

Case 2) $x - 6 < 0$

$\implies | x - 6 | = - \left(x - 6\right)$

$\implies - \left(x - 6\right) = 7$

$\implies - x + 6 = 7$

$\implies - x + 6 - 6 = 7 - 6$

$\implies - x = 1$

$\implies x = - 1$

As these cases account for all possibilities, we have found all possible solutions.

$x \in \left\{- 1 , 13\right\}$