# Question #3e6fe

Sep 24, 2016

$x \in \left[2 , \infty\right)$

#### Explanation:

The absolute value function is defined as

$| a | = \left\{\begin{matrix}a \text{ if "a>=0 \\ -a" if } a < 0\end{matrix}\right.$

Thus, we consider two cases:

Case 1: $x - 2 \ge 0$

Then $| x - 2 | = x - 2$, so

$3 \left(x - 2\right) = 3 x - 6$

$\implies 3 x - 6 = 3 x - 6$

As this is a tautology, the equation is true for any $x$ satisfying $x - 2 \ge 0$... that is, for any $x \ge 2$

Case 2: $x - 2 < 0$

Then $| x - 2 | = - \left(x - 2\right) = 2 - x$, so

$3 \left(2 - x\right) = 3 x - 6$

$\implies 6 - 3 x = 3 x - 6$

$\implies 6 x = 12$

$\implies x = 2$

As we began this case with the restriction $x - 2 < 0$, that is, $x < 2$, there are no solutions within the interval $\left(- \infty , 2\right)$.

Taken together, we can see that the equation holds true if and only if $x \ge 2$, so we have the solution set $x \in \left[2 , \infty\right)$.