Lesson 5.7

1. Lesson 5.7 Systems of Inequalities

2. You have learned that a solution to a system of two linear equations, if there is exactly one solution is the coordinates of the point where the two lines intersect.

3. In this lesson you will learn about systems of inequalities and their solutions. • Many real world problems can be described by a system of inequalities. • When solving these problems, you’ll need to write inequalities, often called constraints, and graph them. • You’ll find a region, rather than a single point, that represents all solutions.

4. A Typical Envelope l = length of the envelop and w= width of the envelop • The US Postal Service imposes several constraints on the acceptable sizes for an envelope. • One constraint is that the ratio of length to width must be less than or equal to 2.5. • Another is that the ratio must be greater than or equal to 1.3. • Define variables and write inequalities for each constraint.

5. Solve each constraint for the variable representing length. Decide whether or not you have to reverse the directions on the inequality symbols. Write a system of inequalities to describe the Postal Service’s constraints on envelope sizes.

6. l • Decide on an appropriate scale for each axis and label the axes. • Decide if you should draw the boundary of the system with solid or dashed lines. • Graph each inequality on the same set of axes. • Shade each half-plane with a different color or pattern. • Where on the graph are the solutions to the system of inequalities? Discuss how to check that your answer is correct. 10 5 5 10 w

7. l • Decide if each envelope satisfies the constraints by locating the corresponding point on your graph. 10 5 5” x 8” 3” x 3” 2.5” x 7.5” 5.5” X 7.5” 5 10 w

8. l • Do the coordinates of the origin satisfy this system of inequalities? • Explain the real-world meaning of this point. • The postal service also has two other constraints: • Maximum length for 43c stamp is 11 ½ inches • Maximum width for 43c stamp is 6 1/8 inches • Illustrate these two additional constraints. 10 5 5 10 w

9. Example • Graph the system of inequalities • Graph the boundary lines and shade the half planes. • Indicate the solution area as the darkest region.

10. Example Lowest profit per box = $0.47 Lowest profit for x boxes=0.47x • A cereal company is including a change to win a $1000 scholarship in each box of cereal. In this promotional campaign, it will give away one scholarship each month, regardless of the number of boxes sold. • Because the cereal is priced differently at various locations, the profit from a single box is between $0.47 and $1.10. • Graph the expected profit, given the initial cost of the scholarship, for up to 5000 boxes sold in a month. • Show the solution region on a graph. If $1000 is given away then the lowest profit = 0.47x-1000, therefore y ≥ 0.47x-1000. Maximum profit per box = $1.10 Maximum profit for x boxes=1.10x If $1000 is given away then the maximum profit =1.10x-1000, therefore y ≤ 1.10x-1000.

11. Example 5000 • Is it possible to sell 3000 boxes and make a profit of $1000? 5000 y ≤ 1.10x-1000. y ≥ 0.47x-1000 The point (3000,1000) satisfies both inequalities: 1000 ≤ 1.10(3000)-1000 1000≤3300-1000 1000≤2300 1000≥ 0.47(3000)-1000 1000≥1410-1000 1000≥410

12. In this Section • You solved systems of inequalities by graphing. • You interpreted the mathematical solutions in terms of the problem context. • You wrote inequalities to represent constraints in application problems.