Isolate, Then Split

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

2|x| > 6

|x| > 3

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

The same goes for a "less than". In |x - 3| - 1 <= 1, add 1 to both sides first to get |x - 3| <= 2. Then write the chain -2 <= x - 3 <= 2 and add 3 to all three parts, so 1 <= x <= 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.