Isolate, Then Split

As with equations, the absolute value must be alone before you split. In $2\left|x\right|+1 > 7$, subtract 1 from both sides, divide both sides by 2, and only then split.

$$2\left|x\right| > 6$$

$$\left|x\right| > 3$$

Now split into two cases: $x > 3$ or $x < -3$.

The same goes for a "less than". In $\left|x-3\right|-1 \le 1$, add 1 to both sides first to get $\left|x-3\right| \le 2$. Then write the chain $-2 \le x-3 \le 2$ and add 3 to all three parts, so $1 \le x \le 5$. The order is: isolate the absolute value, decide whether it is a "less than" (a chain) or a "greater than" (an "or"), split, then solve.