Linear Inequalities and the Sign Reversal

Solving a linear inequality works exactly like solving an equation: combine like terms, add or subtract the same thing from both sides, isolate x. None of that changes which number is bigger, so the inequality sign stays put through all of it. The only moment it changes is when you multiply or divide both sides by a negative number.

Here is why that one move is different. Picture the number line. Multiplying every point by -1 flips the whole line around zero: 5 becomes -5, and -2 becomes 2. Before, 5 was to the right of 2, meaning 5 > 2. After flipping, -5 sits to the left of -2, so -5 < -2. The order of the two values reversed because the whole line got mirrored. Dividing by a negative does the same mirroring, just scaled. So every time you divide or multiply an inequality by a negative, you must flip the sign to keep describing the same true relationship. Adding or subtracting a negative number, by contrast, just slides every point left or right by the same amount — it does not mirror anything, so the order of values never changes and the sign stays the same.

This rule sits right alongside solving linear equations and compound inequalities: the algebra steps are identical, but inequalities carry this one extra bookkeeping rule equations never need. It also sets up quadratic and rational inequalities later, where you track sign changes across multiple factors instead of just one negative coefficient.