Over Lesson 3–2

Over Lesson 3–2

Solve the system of inequalities by graphing.x ≤ – 2 y > 3

A. B.

C. D.

Over Lesson 3–2

Solve the system of inequalities by graphing.y ≤ 3x + 2y > –xA. B.

C. D.

Over Lesson 3–2

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)

Find the coordinates of the vertices of the figure formed by the system of inequalities.x ≥ 0y ≤ 0–3x + y = –6

Chapter 3 Lesson 3 (Part 1)

Optimization with Linear Programming

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.

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.

• linear programming

• feasible region

• bounded

• unbounded

• optimize

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 atriangle with vertices at (–2, 4), (5, –3), and (5, 4).

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.

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).

← minimum

← maximum

A. maximum: f(4, 5) = 5minimum: f(1, 5) = –11

B. maximum: f(4, 2) = 10minimum: f(1, 5) = –11

C. maximum: f(4, 2) = 10minimum: f(4, 5) = 5

D. maximum: f(1, 5) = –11minimum: f(4, 2) = 10

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

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).

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).

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 maximumminimum: f(6, 0) = 6

B. maximum: f(6, 0) = 6minimum: f(0, –3) = –6

C. maximum: f(6, 0) = 6minimum: no minimum

D. maximum: no maximumminimum: f(0, –3) = –6

Homework (Part 1):

Pg 157: 2-12 even