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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.


Answered 4 months ago
Answered 4 months ago
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Let xx be the number of fiction books published and yy be the number of nonfiction books published. The number of books published can't be negative so x0x\ge0 and y0y\ge0. Since a total of no more than 100 books is published, then x+y100x+y\le100. Since at least 20 nonfiction books are published, then y20y\ge20. Since the company published at least as many fiction books as nonfiction books, then xyx\ge y.

The system that models this problem is then

{x+y100y20yxx0y0\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=100y=x+100x+y=100\to y=-x+100, y=20y=20, y=xy=x, x=0x=0 and y=0y=0. Since all the inequality symbols have equal signs, all the equations are graphed as solid lines.

Since x0x\ge0 and y0y\ge0, then we know the solution set is in the first quadrant. Since y20y\ge20, then we also know the solution set is above the line y=20y=20. Testing (50,30) in the first and third inequalities gives:

x+y100yx50+30100305080100\begin{align*} x+y&\le100&y&\le x\\ 50+30&\le100&30&\le50\\ 80&\le100 \end{align*}

Since both 8010080\le100 and 305030\le50 are true statements, then the solution set is the region above y=20y=20 and below both y=xy=x and x+y=100x+y=100. The solution set of the system is then the shaded region shown below.

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