The absolute value $|x|$ is the distance from $x$ to $0$ on the number line. More generally, $|x - 3|$ is the distance between $x$ and $3$. Reading absolute value as distance turns these equations into a picture.
Why it works
The equation $|x - 3| = 5$ asks, “Which numbers are $5$ units from $3$?” There are two: one $5$ to the right ($8$) and one $5$ to the left ($-2$). That is why an absolute value equation usually has two solutions: the expression inside can equal $5$ or $-5$. If the distance asked for is $0$, there is one solution; if it is negative, there is none, because a distance is never negative.
Key idea
First get the absolute value alone. Then, if $|u| = k$ with $k > 0$, solve both $u = k$ and $u = -k$. If $k = 0$, solve $u = 0$; if $k < 0$, there is no solution.
Worked examples
Worked example: Two cases
Solve $|2x - 1| = 7$.
- The absolute value is already alone. The inside is $7$ or $-7$.
- $2x - 1 = 7$ gives $2x = 8$, so $x = 4$.
- $2x - 1 = -7$ gives $2x = -6$, so $x = -3$.
- Check: $|2(4) - 1| = |7| = 7$ and $|2(-3) - 1| = |-7| = 7$.
Answer: $x = 4$ or $x = -3$.
Worked example: Isolate first
Solve $3|x + 2| - 4 = 11$.
- Add $4$ to both sides: $3|x + 2| = 15$.
- Divide by $3$: $|x + 2| = 5$.
- Now split: $x + 2 = 5$ gives $x = 3$; $x + 2 = -5$ gives $x = -7$.
Answer: $x = 3$ or $x = -7$.
Worked example: No solution
Solve $|x - 4| + 6 = 2$.
- Subtract $6$: $|x - 4| = -4$.
- A distance cannot be $-4$, so no number works.
Answer: No solution.
Common mistake
Two common slips: keeping only the positive case, and splitting into cases before the absolute value is alone. In $3|x + 2| - 4 = 11$, writing $3(x + 2) - 4 = -11$ gives $x = -\frac{13}{3}$, which fails the check: $3\left|-\frac{13}{3} + 2\right| - 4 = 3 \cdot \frac{7}{3} - 4 = 3$, not $11$. Isolate first, then split, then check both answers.