EDGE Math 8

Example 2

The Math Club officers plan to hold a miniconcert as part of their fund-raising project. Student tickets will be sold at ₱25 each and nonstudent tickets at ₱40 each. They should sell at least 120 tickets and make no less than ₱4000 from the total sales.

a.	Write a system of inequalities for the given situation, then graph the solution set of the system.
b.	Give one ordered pair that satisfies the system of inequalities. Show that this pair satisfies the conditions in the problem.

Solution

a.	Let x = number of student tickets and y = number of nonstudent tickets.

The number of student tickets and nonstudent tickets to be sold should be at least 120; hence,

x + y \geq 120.

The total sales from the student and nonstudent tickets should be no less than ₱4000. Thus,

25 x + 40 y \geq 4000.

Moreover, the number of student and nonstudent tickets cannot be negative, so

x \geq 0

and

y \geq 0.

The system of inequalities is as follows.

{\begin{array}{l} x + y \geq 120 \\ 25 x + 40 y \geq 4000 \\ x \geq 0 \\ y \geq 0 \end{array}

Next, graph the inequalities on the same coordinate plane.
Graph of

x + y \geq 120 :

region on or above the line

x + y = 120

Graph of

25 x + 40 y \geq 4000 :

region on or above the line

25 x + 40 y = 4000

Graph of

x \geq 0

and

y \geq 0 :

first quadrant region together with the nonnegative sides of the x- and y-axes

The graph of the solution set of the system of inequalities is the intersection of the respective graph of each inequality of the system. Refer to the figure below.

b.	The point $(130, 40)$ is a point inside the shaded region; hence, this point is a solution to the system. To check, the number of tickets sold is $130 + 40 = 170,$ which is greater than 120. The receipts for the tickets totaled

(130 \times P 25) + (40 \times P 40)

= P 3250 + P 1600

= ₱ 4850,

which is greater than ₱4000.

Word Problems on Systems of Linear Inequalities in Two Variables