You know that $x + y = 800$. How can you use this equation to write $y$ in terms of $x$ and then substitute that into your revenue inequality so everything is in one variable?

After substituting $y = 800 - x$ into the revenue inequality, combine like terms carefully (pay attention to the coefficients on $x$), solve for $x$, and then think about what the solution means when $x$ must be a whole number representing seats.

Even if the inequality gives you only a lower bound for $x$, what is the largest number of orchestra seats you could possibly sell, given there are 800 seats total?

In Desmos, type R(x) = 40x + 25(800 - x) to define the total revenue as a function of the number of orchestra seats $x$.

On the next line, type y = R(x) and on another line type y = 25000. Look at where the line $y = R(x)$ is above or on the horizontal line $y = 25000$.

Zoom or adjust the view so you can clearly see the intersection point of $y = R(x)$ and $y = 25000$. Note the x-coordinate of this intersection; revenue values for $x$ greater than or equal to this x-coordinate will meet the manager’s goal. Remember that $x$ must also be at most 800 and must be a whole number.

Each orchestra seat brings in $40 and each balcony seat brings in $25.

So the total revenue $R$ is

$R = 40x + 25y$.

The manager wants at least $25,000, so we write the inequality

We are told every seat will be sold and there are 800 seats, so

Solve for $y$:

Substitute this into the revenue inequality:

Now simplify the left side and solve step by step:

Compute the fraction:

So $x$ must be greater than or equal to about $333.3$.},{