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A publishing company publishes a total of no more than 100 books every year. At least 20 of these are non-fiction, but the company always publishes at least as much fiction as non-fiction. Find a system of inequalities that describes the possible numbers of fiction and non-fiction books that the company can produce each year consistent with these policies. Graph the solution set.
Solution:
The given word problem is about getting the number of fiction and non-fiction books by sketching the graph of the system of inequalities.
Let x = be the number of fiction books
y = be the number of non-fiction books
From the word statement "A publishing company publishes a total of no more than 100 books every year.", then the working equation will be
Since the sign of inequality is less than or equal to, then all points along the line are included in the solution. If x = 0 and y = 0, then the given equation reduces to 0 ≤ 100 which is correct and that point is included in the solution.
From the word statement "At least 20 of these are non-fiction,..", then the working equation will be
Since the sign of inequality is greater than or equal to, then all points along the line are included in the solution. However, all points above the given line are included in the solution.
From the word statement "...but the company always publishes at least as much fiction as non-fiction.", then the working equation will be
Since the sign of inequality is greater than or equal to, then all points along the line are included in the solution. If x = 20 and y = 10, then the given equation reduces to 20 ≥ 10 which is correct and that point is included in the solution. Therefore, the graph of a set of inequalities is
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| Photo by Math Principles in Everyday Life |
The number of fiction and non-fiction books which are the coordinates of the vertices are (20, 20), (50, 50), and (80, 20). The three vertices are located at the intersection of the three shaded regions bounded by three lines.