Chapter 6.5, Problem 23P

Tickets to a play cost $5 at the door and $4 in advance. The theater club wants to raise at least $400 from the play. Write and graph an inequality for the number of tickets the theater club needs to sell. If the club sells 40 tickets in advance, how many do they need to sell at the door to reach their goal?

To determine

To find: The inequality and the graph satisfying the given condition.

Answer to Problem 23P

The required inequality is $5x+4y\ge 400$ .

If the club sells 40 tickets in advance then, at least 48 tickets needed to be sold at the door to reach the goal.

Explanation of Solution

Given information:

The cost of the ticket at the door to a play $=\$5$

The cost of the ticket in advance to a play $=\$4$

The theatre club wants to raise at least $\$400$ from the play.

Calculation:

Let $x$ tickets sold at the door and $y$ in advance.

Money raised from tickets sold at the door $=\$5x$

Money raised from tickets sold in advance $=\$4y$

A.T.Q

  $5x+4y\ge 400$

The graph of the obtained inequality is

If the club sells 40 tickets in advance, that is, $y=40$

Then,

  $\begin{array}{l}5x+4\left(40\right)\ge 400\\ 5x+160\ge 400\\ 5x+160-160\ge 400-160\\ 5x\ge 240\\ \frac{5x}{5}\ge \frac{240}{5}\\ x\ge 48\end{array}$

At least 48 tickets needed to be sold at the door to reach the goal.