Linear Equations & Inequalities

Linear equations and inequalities show up more than almost any other topic on the SAT Math section. The good news: they all boil down to the same handful of moves, done in the right order.

A linear equation like 3x + 5 = 20 is solved by "undoing" whatever was done to x, in reverse order of operations:

1. Move constants to one side (add or subtract the same value from both sides). 2. Divide both sides by the coefficient in front of the variable.

Inequalities (<, >, ≤, ≥) work exactly the same way, with one critical exception: if you multiply or divide both sides by a NEGATIVE number, you must flip the inequality sign. For example, -2x < 8 becomes x > -4 (not x < -4) once you divide by -2.

Compound and multi-step problems just repeat these two moves: simplify each side first (distribute, combine like terms), then isolate the variable.

Example 1: Solve 3x + 5 = 20. Subtract 5 from both sides: 3x = 15. Divide both sides by 3: x = 5.

Example 2: Solve -2x + 3 ≤ 11. Subtract 3 from both sides: -2x ≤ 8. Divide both sides by -2 and flip the inequality: x ≥ -4.

The single most common error is forgetting to flip the inequality sign when multiplying or dividing by a negative number. A close second: distributing a negative sign incorrectly, e.g. turning -2(x - 3) into -2x - 6 instead of the correct -2x + 6.