Solve |x + 66| = |x - 88|.

Step-by-step solution

1. 1

   Translate each side into a distance. $|x+66|=|x-(-66)|$ is the distance from $x$ to $-66$, and $|x-88|$ is the distance from $x$ to $88$. The equation therefore says: $x$ is equally far from $-66$ and from $88$.

2. 2

   Use the midpoint shortcut. Exactly one point on the line is equidistant from two distinct points — their midpoint:

   $x=\frac{-66+88}{2}=\frac{22}{2}=11$

   This single line replaces the usual two-case algebra.

3. 3

   Cross-check with the algebraic method. Removing absolute values gives two cases. Case 1: $x+66=x-88$, which reduces to $66=-88$ — impossible, so no solution here (this is what happens whenever the two insides have equal $x$ coefficients). Case 2: $x+66=-(x-88)=-x+88$, giving $2x=22$ and $x=11$ ✓ — the same answer.

4. 4

   Understand why one case collapsed. Both expressions have coefficient $+1$ on $x$, so setting them equal cancels the variable entirely and leaves a false numeric statement. Only the sign-flipped case can produce a solution — which is why the answer is a single point rather than two.

5. 5

   Verify by substitution.

   $|11+66|=|77|=77,\qquad |11-88|=|-77|=77$

   Both sides equal $77$ ✓. A scan of $40{,}001$ exact rational points on $[-200,200]$ finds $x=11$ to be the unique solution ✓, and the distance from $11$ to each of $-66$ and $88$ is indeed $77$ ✓.