A Coefficient Inside

In $\left|2x-1\right| = 7$, the expression inside the bars is $2x-1$. Split into two cases first. Each case is then a two-step equation: add 1 to both sides, then divide both sides by 2.

$$2x-1 = 7$$

$$2x = 8$$

$$x = 4$$

or

$$2x-1 = -7$$

$$2x = -6$$

$$x = -3$$

The two cases give $x = 4$ or $x = -3$. Check each answer in the original equation. For $x = -3$, the inside is $2\cdot \left(-3\right)-1 = -7$, and $\left|-7\right| = 7$, so the answer checks. Keep both answers.