AP Precalculus: Systems of Inequalities & Linear Programming
1. Solve Systems of Linear Inequalities by Graphing
- Inequality form: \( y \le mx + b \), \( y \ge mx + b \), etc.
- Graph each boundary line (solid for ≤, ≥; dashed for <, >)
- Shade solution region for each inequality
- Final solution = overlapping region (intersection of all solution sets)
2. Linear and Absolute Value Inequalities
- Absolute value: \( |ax + by + c| \leq d \) or \( |ax + by + c| \geq d \)
- Split into two linear inequalities:
- \( ax + by + c \leq d \) and \( ax + by + c \geq -d \)
- Graph both solutions; intersection is the final region
3. Find Vertices of a Solution Set
- Vertices are intersection points of boundary lines
- To find: Solve pairs of equations (replace inequalities with equalities) to get points
- Test/intersect all pairs of boundaries within the feasible region
4. Linear Programming (Optimization)
- Model: Maximize or minimize \( Z = ax + by \)
- Subject to constraints (system of inequalities)
- Feasible region: intersection of inequalities (solution area)
- Optimal value: test \( Z \) at all vertices of solution region
- Formula: For vertex \( (x_0, y_0) \), compute \( Z(x_0, y_0) = a x_0 + b y_0 \)
- Best value is maximum or minimum \( Z \) among all vertices