Solve |x| = 5 - 10, or explain why no solution exists.

Step-by-step solution

1. 1

   Simplify the right-hand side before doing anything else. The bars on the left tempt you into case-splitting immediately, but the right side is not yet a single number:

   $5-10=-5\quad\Longrightarrow\quad |x|=-5$

2. 2

   Recall what an absolute value measures. $|x|$ is the distance from $x$ to $0$ on the number line. Distances are never negative, so for every real number $x$:

   $|x|\ge 0$

3. 3

   Compare the two sides. The left side is always $\ge 0$ and the right side is exactly $-5<0$. No real number can make a non-negative quantity equal a negative one, so the equation is contradictory.

4. 4

   State the solution set. There is nothing to check case by case — the answer is the empty set:

   $x\in\varnothing$

   This is the general rule: $|A|=c$ has two solutions when $c>0$, one when $c=0$, and none when $c<0$. Reading the sign of $c$ first saves all the algebra.