Linear Inequalities: Solving, Graphing, and the One Sign-Flip Rule
Inequalities solve like equations — except for one twist. Multiply or divide by a negative and the inequality sign flips. Master that single rule and you'll solve every CBE inequality on autopilot.
Same as equations — with one twist
Solving an inequality looks identical to solving an equation: distribute, combine, isolate, divide. But the moment you multiply or divide by a negative number, the inequality sign flips direction. Miss that one rule and you've miss-answered the whole question.
Multiply or divide by a negative → FLIP the inequality. > becomes <, ≤ becomes ≥. Adding or subtracting never flips the sign. Multiplying or dividing by a positive never flips it either. Only negative coefficients trigger the flip.
The four symbols
- < or >
- Strict inequality: the number itself is not included. Graph with an open circle.
- ≤ or ≥
- Includes equality: the number is included. Graph with a closed (filled) circle.
Worked example: standard solve
Sign-flip example
Graphing on a number line
Two-variable inequalities
Inequalities with both x and y graph as a shaded half-plane. The boundary line is dashed for < / >, solid for ≤ / ≥.
Pick a test point not on the line (the origin (0, 0) is easiest if it's not on the line). Substitute. If true, shade that side. If false, shade the other side.
3-second recap
- Solve like an equation — except flip the sign when you multiply/divide by a negative.
- Open circle → strict (<, >); closed circle → includes (≤, ≥).
- Two-variable: dashed for strict, solid for includes-equal; pick a test point to decide which side to shade.