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Question

# A publishing company publishes a total of no more than 100 books every year. At least 20 of these are nonfiction, but the company always publishes at least as much fiction as nonfiction. Find a system of inequalities that describes the possible numbers of fiction and nonfiction books that the company can produce each year consistent with these policies. Graph the solution set.

Solution

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Let $x$ be the number of fiction books published and $y$ be the number of nonfiction books published. The number of books published can't be negative so $x\ge0$ and $y\ge0$. Since a total of no more than 100 books is published, then $x+y\le100$. Since at least 20 nonfiction books are published, then $y\ge20$. Since the company published at least as many fiction books as nonfiction books, then $x\ge y$.

The system that models this problem is then

\left\{\begin{aligned}x+y&\le100\\y&\ge20\\y&\le x\\x&\ge0\\y&\ge0\end{aligned}\right.

We must then graph the equations $x+y=100\to y=-x+100$, $y=20$, $y=x$, $x=0$ and $y=0$. Since all the inequality symbols have equal signs, all the equations are graphed as solid lines.

Since $x\ge0$ and $y\ge0$, then we know the solution set is in the first quadrant. Since $y\ge20$, then we also know the solution set is above the line $y=20$. Testing (50,30) in the first and third inequalities gives:

\begin{align*} x+y&\le100&y&\le x\\ 50+30&\le100&30&\le50\\ 80&\le100 \end{align*}

Since both $80\le100$ and $30\le50$ are true statements, then the solution set is the region above $y=20$ and below both $y=x$ and $x+y=100$. The solution set of the system is then the shaded region shown below.

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