1. A-REI Represent and solve equations and inequalities graphically
12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

2. Write and graph a system of linear inequalities to describe the problem.
An arena contains 1200 seats. For an upcoming concert, some seats will be priced $12.00 while other seats are $10.00.
500 $10 tickets have already been sold, and the total sales must be at least $7200 to make a profit. What are some possible combinations of tickets that can be sold to make a profit?

3. 7. An arena contains 1200 seats
7. An arena contains 1200 seats. For an upcoming concert, some seats will be priced $12.00 while other seats are $ $10 tickets have already been sold, and the total sales must be at least $7200 to make a profit. What are some possible combinations of tickets that can be sold to make a profit?
1. Ask students to assign variables and identify the unknowns Ask: “why does the first equation make sense when translating the first sentence”– Explain that since the arena can hold 1200 any combination of $12 seats and $10 that adds up to < or or = 1200 seats, meaning that not all seats must be filled.

4. Only the quadrant I Will be used. Since it is not possible to
Sell negative amount of
Tickets, what two other
Inequalities should be
Considered?
What do these additional inequalities
Tell you about the graph of this system?
The link will take you to an online graphing calculator. Ask students what values should be the Max and Min for x and y– since x and y are number of tickets the min should be zero for both since negative amount of tickets cannot be sold. The Max for both x and y is 1200 since there are only 1200 seats in the arena. Since there is such a large range then the scale can be counting by 100.
Only the quadrant I
Will be used.

5. At most 800 $10 tickets. If 400 $12 tickets where sold, how many
$10 tickets can be sold?
At most 800
$10 tickets.
Discuss with students why the answer to the first question is “at most” (< or +=)– not all seats need to be filled, therefore you can sell less than 800, however since there are only 1200 seats the most $10 tickets that can be sold is 800.

6. Would (700.5, 400) be a solution to this equation?
Not for the problem stated, because
You cannot purchase half a ticket.
Discuss with students that although (700.5, 400) is a solution to the inequality written, it is not a solution to the problem stated .

7. Can 900 $12 tickets be sold? No, because at least 500
tickets must be sold at
$10, and that would make the total number of tickets greater than 1200.
Many students may answer that the intersection means that if 700 $12 tickets are sold then only 500 $10 tickets can be sold. Explain that although this is true a better interpretation would be that since you must sell at least 500 $10 tickets then the maximum amount of $12 tickets that can be sold is 700.

8. Of $12 tickets that can be sold.
What is the intersection
Of the two lines, and what
Does it represent in this
Situation?
(700, 500) It represents
The maximum number
Of $12 tickets that can be sold.
Many students may answer that the intersection means that if 700 $12 tickets are sold then only 500 $10 tickets can be sold. Explain that although this is true a better interpretation would be that since you must sell at least 500 $10 tickets then the maximum amount of $12 tickets that can be sold is 700.

9. What is the maximum number Of $10 tickets that can be sold?
What would be the total sales?
The Max number of $10 tickets is 1200 (meaning zero
$12 tickets were sold). The total sales would be $12,000
Discuss the fact that although at least 500 $10 tickets need to be sold, there is no restriction for the $12 tickets which means that it is OK if none are sold.

10. What is the y-intercept of the
red line? What does it represent
in this situation?
The y-intercept is If you sold zero $12 tickets then you must sell 720 $10 tickets to make a profit.

11. What is the intersection of the Red and Blue lines? What does
It represent in this situation?
(183⅓, 500) This ordered
Pair is represents the minimum amount of tickets that must be sold in order to make a profit. However because you cannot sell ⅓ of a ticket an adjustment must be made.

12. 184 $12 tickets and 500 $10 tickets. Total Sales: $7208.
What is the minimum amount of
each ticket that must be sold in
order to make a profit? What
Is the amount of total sales?
184 $12 tickets and 500 $10 tickets.
Total Sales: $7208.
Some students will follow conventional rounding rules and round 183⅓ to 183– explain that this would not meet the profit requirement as it would yield a profit of $ Also mention that the point (183⅓, 500) falls below the shaded area that contains possible solutions.

13. 700 $12 tickets and 500 $10 tickets. Total Sales: $13,400
What combination of ticket sales
Would maximize the profit?
What would be the total sales?
700 $12 tickets and
500 $10 tickets.
Total Sales: $13,400
Discuss with students that the max profit would result from selling the max amount of $12 tickets as possible.

14. Within a finite area, and all combinations must be whole numbers.
Does this situation have infinitely
Many ticket combinations?
No, the solutions are
Within a finite area, and all combinations must be whole numbers.
Discuss the fact that this system of inequalities itself has infinitely many solutions, however because it represents this real world situation where the number of tickets must be whole numbers, the solutions become finite.