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.
Read an inequality from a graph
Which inequality is represented by the shaded graph?
Open the question →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.