Algebraic Solutions
Linear inequalities in one variable can be solved using similar methods as equations, with one important difference: when multiplying or dividing by a negative number, the inequality sign reverses.
Example: Solve 3x - 5 < 1
Solution: 3x - 5 < 1 → 3x < 6 → x < 2
Graphical Solutions
Linear inequalities in two variables can be represented graphically. The solution is a half-plane bounded by the line representing the equality.
Steps:
- Graph the corresponding equation (solid line for ≤ or ≥, dashed for < or >)
- Test a point not on the line (usually (0,0))
- Shade the appropriate half-plane
Systems of Inequalities
The solution to a system of linear inequalities is the intersection of the solution sets of all inequalities in the system.
Key Points:
- The solution region is where all shaded areas overlap
- Corner points of the solution region are important in optimization problems
Real-World Applications
Linear inequalities are used in various fields to model constraints and limitations.
Practice Exercises
1. Solve the inequality and represent the solution on a number line: 2x + 5 ≤ 11
2. Graph the inequality: y > 2x - 1
3. Solve the system graphically: x + y ≤ 4, x ≥ 1, y ≥ 0