Brackets and a Flip

A longer inequality is solved with the same steps as a short one, done one after another. Take 3(x - 2) >= 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 >= 5x - 4

-2x - 6 >= -4

-2x >= 2

x <= -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 >= x, which reads as x <= -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.