Math101 · Independent Ontario learning library
Math101Linear Inequalities
A linear inequality in one variable compares linear expressions and has a solution set that is an interval, ray, all real numbers, or no solution.
Linear inequalities model capacity, minimum requirements, error ranges, and constraints. They also introduce set operations and boundary reasoning used throughout algebra.
Intuition and core definition
A linear inequality in one variable compares linear expressions and has a solution set that is an interval, ray, all real numbers, or no solution. Solving uses the same balance operations as linear equations, except multiplying or dividing by a negative reverses the inequality because it reverses order.
Notation, language, and conditions
Forms include $ax+b<c$, $ax+b\ge c$, and compound conditions joined by “and” or “or.” An “and” solution is an intersection satisfying both inequalities; an “or” solution is a union satisfying at least one. Interval endpoints use brackets only when finite endpoints are included; infinity always uses parentheses.
Why this idea matters
Solving a linear inequality preserves an interval of solutions, and multiplying or dividing by a negative reverses order on the number line.
A dependable method
- Distribute and combine like terms on each side.
- Move variable terms to one side and constants to the other.
- When dividing by a negative coefficient, reverse the inequality.
- For compounds, solve each part and combine by intersection or union.
- Check with boundary values and test points, then graph the complete set.
