Brackets and a Flip

A longer inequality is solved with the same steps as a short one, done one after another. Take $3\left(x-2\right) \ge 5x-4$. Open the bracket, collect the $x$ terms on the left and the numbers on the right, then divide. The coefficient left in front of $x$ is negative, so the last division flips the sign.

$$3x-6 \ge 5x-4$$

$$-2x-6 \ge -4$$

$$-2x \ge 2$$

$$x \le -1$$

You can also collect the $x$ terms on the right, toward the $5x$. Then the coefficient is positive and there is no flip. You end with $-1 \ge x$, which reads as $x \le -1$. Both ways give the same answer. One way has a flip and the other has a swap of sides at the end. No way has both.

Before you start, count the parts of the inequality: a bracket, $x$ terms on both sides, numbers on both sides. Then you need four steps: open the bracket, collect the $x$ terms, collect the numbers, and divide. At each step, ask whether it flips the sign. Only multiplying or dividing by a negative number does.