Solve |2 - x| = |-8|.

Step-by-step solution

1. 1

   Simplify the right side. An absolute value of a plain number is just its size:

   $|-8|=8\qquad\Longrightarrow\qquad |2-x|=8$

   The right side is now a positive constant, which guarantees solutions exist (had it been negative, there would be none).

2. 2

   Interpret the equation as a distance. $|2-x|=|x-2|$ is the distance from $x$ to $2$ on the number line, so the equation asks for the points exactly $8$ units from $2$. There must be exactly two — one on each side.

3. 3

   Split into two cases. Removing an absolute value means the inside is either $+8$ or $-8$:

   $\textbf{Case 1: } 2-x=8\qquad\Longrightarrow\qquad -x=6\qquad\Longrightarrow\qquad x=-6$

4. 4

   Solve the second case.

   $\textbf{Case 2: } 2-x=-8\qquad\Longrightarrow\qquad -x=-10\qquad\Longrightarrow\qquad x=10$

   In both cases the final division by $-1$ flips the sign — the step most often fumbled.

5. 5

   Confirm with the distance picture. Going $8$ units left from $2$ lands on $-6$, and $8$ units right lands on $10$ ✓. The two solutions are symmetric about $2$, as they must be, and their midpoint $\tfrac{-6+10}{2}=2$ recovers the centre.

6. 6

   Verify by substitution. At $x=-6$: $|2-(-6)|=|8|=8$ ✓. At $x=10$: $|2-10|=|-8|=8$ ✓. A scan of $4001$ points from $-20$ to $20$ finds these two values and no others ✓.