Solve the equation |7x - 1| = 3.

Step-by-step solution

1. 1

   Check the right-hand side is non-negative. An absolute value can never be negative, so $|A|=c$ has solutions only when $c\ge 0$. Here $c=3>0$, which means there will be exactly two solutions (a right-hand side of $0$ would give one, and a negative one none at all).

2. 2

   Split into the two cases. $|A|=3$ means the quantity inside is either $3$ units above zero or $3$ units below:

   $7x-1=3\qquad\text{or}\qquad 7x-1=-3$

   Both must be solved; discarding the negative branch is the usual mistake.

3. 3

   Solve the positive branch.

   $7x-1=3\;\Longrightarrow\;7x=4\;\Longrightarrow\;x=\frac47$

4. 4

   Solve the negative branch.

   $7x-1=-3\;\Longrightarrow\;7x=-2\;\Longrightarrow\;x=-\frac27$

5. 5

   Substitute back and note the symmetry. For $x=\frac47$: $\left|7\cdot\frac47-1\right|=|4-1|=3$ ✓. For $x=-\frac27$: $\left|7\cdot\left(-\frac27\right)-1\right|=|-2-1|=3$ ✓. Their midpoint is $\frac12\left(\frac47-\frac27\right)=\frac17$, exactly where $7x-1=0$, and each root is $\frac37$ away from it — the expected mirror pattern.