My question is linear inequalities

Question

tickets to a play cost $10 at the door and $8 in advance. The theatre club wants to raise at least $800 from the sale of the tickets from the play. Write and graph an inequality for the number of tickets the theatre club needs to sell. If the club sells 40 tickets in advance, how many does it need to sell at the door to reach its goal. use x to represent the number of tickets sold at the door. Use y to represent the number of tickets sold in advance

Answer 1

Hi Oceana hope this helps!

Let's use x to represent the number of tickets sold at the door and y to represent the number of tickets sold in advance.

The revenue from tickets sold at the door is 10x (since each ticket costs $10), and the revenue from tickets sold in advance is 8y (since each ticket costs $8).

The total revenue, R, is the sum of these two:

R = 10x + 8y

The theater club wants to raise at least $800, so the inequality is:

10x + 8y ≥ 800

This is the inequality that represents the goal of raising at least $800 from ticket sales.

Now, if the club sells 40 tickets in advance y = 40, you can substitute this into the inequality to find out how many tickets ( x )need to be sold at the door to reach the goal:

10x + 8(40) ≥ 800

Simplify the expression and solve for x:

10x + 320 ≥ 800

Subtract 320 from both sides:

10x ≥ 480

Divide by 10:

x ≥ 48

So, if the club sells 40 tickets in advance, they need to sell at least 48 tickets at the door to reach their goal.

Answer 2

There is a hint contained within the question that x represents the number of tickets sold at the door and y represents the number of tickets sold in advance. We want to write an inequality which is expressed in dollars since that is the value that we ultimately care about reaching.

Remember # of tickets * their dollar value will give us the overall value. Because we have 2 different types of tickets with 2 different dollar values then we can write the inequality:

10x + 8y ≥ 800

We can solve for y to get it into the format:

y ≥ 100 - (5/4)x

I'll leave this function for you to graph by yourself but all of the solutions to this inequality exist above and on this graphed line. You can express this as a shaded region and a solid line.

If we want to know how many tickets we need to sell at the door if we've already presold 40 tickets then simply plug 40 in for y and solve for x.

10x + 8(40) ≥ 800

10x ≥ 480

x ≥ 48

So we need to sell 48 or more tickets in order to meet our goal of $800.