Jafar is an author who has one year to write two books. The first book, a paperback romance novel, will probably be a strong seller. The other book is a serious philosophical novel and is likely to be a poor seller. Jafar wants to invest no more than 4 months of his time writing the serious novel and at least twice as much time working on the romance novel than the serious novel.
Let [tex]$x$[/tex] be the time to spend writing the romance novel, and let [tex]$y$[/tex] be the time to spend writing the serious novel.
Which system of inequalities describes Jafar's writing strategy?
[tex]\begin{array}{l}
x+y \leq 12 \\
y \leq 4 \\
x\ \textgreater \ 7 v
\end{array}[/tex]
Answer (1)
The total time spent on both novels must be less than or equal to 12 months: x + y ≤ 12 .
The time spent on the serious novel must be no more than 4 months: y ≤ 4 .
The time spent on the romance novel must be at least twice the time spent on the serious novel: x ≥ 2 y .
The system of inequalities describing Jafar's writing strategy is: ⎩ ⎨ ⎧ x + y ≤ 12 y ≤ 4 x ≥ 2 y .
Explanation
Understanding the Constraints Let's break down Jafar's writing constraints into mathematical inequalities to represent his writing strategy. We'll consider each constraint one by one and translate it into an inequality.
Total Time Constraint Jafar has one year (12 months) to write two books. The time he spends on the romance novel ( x ) plus the time he spends on the serious novel ( y ) must be no more than 12 months. This gives us the inequality: x + y ≤ 12
Serious Novel Time Constraint Jafar wants to invest no more than 4 months writing the serious novel. This means the time y he spends on the serious novel must be less than or equal to 4 months. This gives us the inequality: y ≤ 4
Romance Novel Time Constraint Jafar wants to spend at least twice as much time working on the romance novel than the serious novel. This means the time x he spends on the romance novel must be greater than or equal to twice the time y he spends on the serious novel. This gives us the inequality: x ≥ 2 y
System of Inequalities Combining these inequalities, we get the following system of inequalities that describes Jafar's writing strategy: ⎩ ⎨ ⎧ x + y ≤ 12 y ≤ 4 x ≥ 2 y
Final Answer Therefore, the system of inequalities that describes Jafar's writing strategy is: ⎩ ⎨ ⎧ x + y ≤ 12 y ≤ 4 x ≥ 2 y
Examples
Understanding systems of inequalities is crucial in resource allocation. For instance, a company might use them to optimize production, considering constraints like budget, labor, and material availability. Similarly, a farmer could use inequalities to determine the optimal mix of crops to plant, given limitations on land, water, and fertilizer. These mathematical tools help in making informed decisions to maximize efficiency and profits.
