A Wide Absolute Value

Take $\left|3x-6\right| \le 9$. The sign is "less than or equal to", so the absolute value becomes a chain. Then do two steps on all three parts.

$$-9 \le 3x-6 \le 9$$

$$-3 \le 3x \le 15$$

$$-1 \le x \le 5$$

The condition says that $3x-6$ is within 9 of zero. For $x$ itself, this means within 3 of 2, so the solution is the segment from -1 to 5.

First get the absolute value bars alone on one side. Here they already are. Then look at the sign. A "less than" gives a chain, and a "greater than" gives two cases joined by "or". After the split, the steps are the same as for any chain or any pair of inequalities.