Systems of Inequalities

 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.

 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.

 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. l = length of the envelop and w= width of the envelop

 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.

 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. w l

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

 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. w l

 The grey region represents the solution set for the four inequality statements or system of inequalities. w l

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

 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. Lowest profit per box = $0.47 Lowest profit for x boxes=0.47x If $1000 is given away then the lowest profit = 0.47x-1000, therefore y ≥ 0.47x 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.

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

 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.