Solve inequalities like equations with one key difference: if you multiply or divide by a negative number, reverse the inequality symbol. In two variables, graph the boundary line and shade the side that satisfies the inequality.
- Reverse the inequality when multiplying or dividing by a negative.
- Use open or closed boundary points based on strict or inclusive symbols.
- Test a point to confirm the correct shaded region.
Keep track of symbol direction
Symbols < and > are strict, while <= and >= include boundary values. In one-variable graphs, strict inequalities use open circles and inclusive inequalities use closed circles.
Students often miss the sign flip after dividing by a negative. Write that step explicitly to reduce errors.
Graph in two variables
First graph the boundary equation, such as y = 2x - 1. Use a solid line for <= or >= and a dashed line for < or >. Then test a point not on the line, often (0, 0), to decide which side to shade.
- Boundary line: equation version of the inequality.
- Dashed line: boundary not included.
- Solid line: boundary included.
Solve and graph a one-variable inequality
A discount applies when 3x - 7 <= 11, where x is quantity purchased.
- Add 7 to both sides: 3x <= 18.
- Divide by 3: x <= 6.
- Graph with a closed circle at 6 and shading to the left.
- Check x = 6 and x = 7 in the original inequality.
Common mistakes
- Forgetting to reverse the symbol after dividing by -1.
- Using a solid boundary line for a strict inequality.
- Shading without testing a point.
- Treating one test point as proof for an algebraic manipulation error.
Try one
Solve -2x + 5 > 11.
Subtract 5: -2x > 6. Divide by -2 and reverse symbol: x < -3.
Sources
- OpenStax: College AlgebraOpen textbook with full chapter coverage, worked examples, and practice problems.
- Khan Academy: Algebra and Algebra 2Official lessons and practice sets for core algebra skills.