Isolate, Then Split

As with equations, the bars must be alone before the split. 2|x| + 1 > 7: subtract 1, divide by 2, and only then split.

2|x| > 6

|x| > 3

Now split: x > 3 or x < -3.

The same for a "less than": |x - 3| - 1 <= 1 needs the 1 added back first, |x - 3| <= 2, and then the chain -2 <= x - 3 <= 2, so 1 <= x <= 5. Isolate, decide whether it is a "less than" (chain) or a "greater than" (or), split, solve.