Solve the system of inequalities by graphing. x ≤ – 2 y > 3 A. B. C. D. 5-Minute Check 1

Solve the system of inequalities by graphing. y ≤ 3x + 2 y > –x A. B. C. D. 5-Minute Check 2

Find the coordinates of the vertices of the figure formed by the system of inequalities. x ≥ 0 y ≤ 0 –3x + y = –6 A. (0, 0), (1, 0), (–3, 0) B. (0, 0), (2, 0), (0, –6) C. (0, 0), (–3, 0), (0, –6) D. (0, 0), (2, 0), (–3, 0) 5-Minute Check 3

Optimization with Linear Programming Chapter 3 Lesson 3 (Part 1) Optimization with Linear Programming

Mathematical Practices 4 Model with mathematics. Content Standards A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Mathematical Practices 4 Model with mathematics. 8 Look for and express regularity in repeated reasoning. CCSS

You solved systems of linear inequalities by graphing. Find the maximum and minimum values of a function over a region. Solve real-world optimization problems using linear programming. Then/Now

linear programming feasible region bounded unbounded optimize Vocabulary

Concept

Step 1 Graph the inequalities. Bounded Region Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function f(x, y) = 3x – 2y for this region. x ≤ 5 y ≤ 4 x + y ≥ 2 Step 1 Graph the inequalities. The polygon formed is a triangle with vertices at (–2, 4), (5, –3), and (5, 4). Example 1

Bounded Region Step 2 Use a table to find the maximum and minimum values of f(x, y). Substitute the coordinates of the vertices into the function. ← minimum ← maximum Answer: The vertices of the feasible region are (–2, 4), (5, –3), and (5, 4). The maximum value is 21 at (5, –3). The minimum value is –14 at (–2, 4). Example 1

A. maximum: f(4, 5) = 5 minimum: f(1, 5) = –11 Graph the following system of inequalities. What are the maximum and minimum values of the function f(x, y) = 4x – 3y for the feasible region of the graph? x ≤ 4 y ≤ 5 x + y ≥ 6 A. maximum: f(4, 5) = 5 minimum: f(1, 5) = –11 B. maximum: f(4, 2) = 10 minimum: f(1, 5) = –11 C. maximum: f(4, 2) = 10 minimum: f(4, 5) = 5 D. maximum: f(1, 5) = –11 minimum: f(4, 2) = 10 Example 1

Unbounded Region Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function f(x, y) = 2x + 3y for this region. –x + 2y ≤ 2 x – 2y ≤ 4 x + y ≥ –2 Graph the system of inequalities. There are only two points of intersection, (–2, 0) and (0, –2). Example 2

Unbounded Region The minimum value is –6 at (0, –2). Although f(–2, 0) is –4, it is not the maximum value since there are other points that produce greater values. For example, f(2,1) is 7 and f(3, 1) is 9. It appears that because the region is unbounded, f(x, y) has no maximum value. Answer: The vertices are at (–2, 0) and (0, –2). There is no maximum value. The minimum value is –6 at (0, –2). Example 2

A. maximum: no maximum minimum: f(6, 0) = 6 Graph the following system of inequalities. What are the maximum and minimum values of the function f(x, y) = x + 2y for the feasible region of the graph? x + 3y ≤ 6 –x – 3y ≤ 9 2y – x ≥ –6 A. maximum: no maximum minimum: f(6, 0) = 6 B. maximum: f(6, 0) = 6 minimum: f(0, –3) = –6 C. maximum: f(6, 0) = 6 minimum: no minimum D. maximum: no maximum minimum: f(0, –3) = –6 Example 2

Homework (Part 1): Pg 157: 2-12 even