📚 Topic Summary
Linear inequalities define regions on a graph, and the vertices are the corner points of these regions. Finding the vertices involves solving systems of equations formed by the intersecting lines that define the inequalities. These vertices are critical for optimization problems, where we want to find the maximum or minimum value of a function within the feasible region defined by the inequalities.
To find these vertices, graph the inequalities to visualize the feasible region. Then, identify the points where the lines intersect. Solve the system of equations for each intersection point to determine the coordinates of the vertices. These coordinates are crucial for solving linear programming problems.
🧠 Part A: Vocabulary
Match the following terms with their definitions:
| Term |
Definition |
| 1. Feasible Region |
A. The point where two or more lines intersect. |
| 2. Vertex |
B. A mathematical expression showing a relationship of greater than, less than, or equal to. |
| 3. Linear Inequality |
C. A method for finding the maximum or minimum value of a function subject to constraints. |
| 4. Intersection Point |
D. The area on a graph that satisfies all given inequalities. |
| 5. Linear Programming |
E. A straight line on a graph. |
✍️ Part B: Fill in the Blanks
The vertices of a feasible region are found by solving systems of equations formed by the intersecting _____. These vertices represent the _____ points of the region. The feasible region is the area that satisfies all the given _____. Finding these vertices is important for _____ problems, where we seek to maximize or minimize a function.
🤔 Part C: Critical Thinking
Explain how finding the vertices of a feasible region helps in solving optimization problems. Provide a real-world example where this process might be used.