Precalc find the solution set

Question

Find the solution set $$\left|\frac{2}{3}x-6\right|=2$$

The solution is $$x=6,12$$

Looking for some help finding this solution, do I need to consider $x$ from two ends?

Answer 1

Whenever there is an absolute value equation you can consider it to be two equations. If $$\frac23 x-6\geq 0$$ then $$\frac23x-6 = 2$$ since $|y|=y$ if $y\geq 0$, and if $$\frac23x-6<0$$ then $$6-\frac23x=2$$ since $|y| = -y$ if $y<0$. You can see by solving each of these that you get the desired results.

Answer 2

Well, if $\displaystyle \left | \frac{2}{3}x-6 \right| = 2$, then $\displaystyle \frac{2}{3}x-6 =-2\quad (x=6)\;$ or $\;\displaystyle \frac{2}{3}x-6 =2 \quad(x=12)$

Answer 3

Hint: Consider the cases: $$\dfrac23x-6\geq 0$$ and you will get $$\frac{2}{3}x-6=2$$ in the other case you will have $$-\frac{2}{3}x+6=2$$

Answer 4

The equality is equivalent to $|x-9|=3$. And now read that out loud: "The distance from $x$ to $9$ equals $3$." There are obviously exactly two numbers satisfying that equation.