Anna is no more than 3 years older than 2 times Jamie's age. Jamie is at least 14 and Anna is at most 35. Which system of linear inequalities can be used to find the possible ages of Anna, $a$, and Jamie, $j$?
A. $a \geq 3+2 j ; j \geq 14, a \leq 35$
B. $a \leq 3+2 j ; j \geq 14, a \leq 35$
C. $a \geq 3+2 j ; j \leq 14, a \leq 35$
D. $a \leq 3+2 j ; j \leq 14, a \leq 35$
Answer (2)
Represent Anna's age as a and Jamie's age as j .
Translate 'Anna is no more than 3 years older than 2 times Jamie's age' into a ≤ 3 + 2 j .
Translate 'Jamie is at least 14' into j g e q 14 .
Translate 'Anna is at most 35' into a ≤ 35 .
The system of inequalities is: a ≤ 3 + 2 j ; j g e q 14 , a ≤ 35 .
Explanation
Problem Analysis We are given a scenario involving Anna's and Jamie's ages, and we need to translate the given information into a system of linear inequalities. Let a represent Anna's age and j represent Jamie's age.
Translating the First Statement The first statement says, 'Anna is no more than 3 years older than 2 times Jamie's age.' This can be written as: a ≤ 3 + 2 j .
Translating the Second Statement The second statement says, 'Jamie is at least 14.' This can be written as: j g e q 14 .
Translating the Third Statement The third statement says, 'Anna is at most 35.' This can be written as: a ≤ 35 .
Final Answer Combining these inequalities, we get the system: a ≤ 3 + 2 j , j g e q 14 , and a ≤ 35 . Therefore, the correct answer is a ≤ 3 + 2 j ; j g e q 14 , a ≤ 35 .
Examples
Systems of inequalities are used in various real-world scenarios, such as resource allocation, production planning, and diet optimization. For instance, a company might use a system of inequalities to determine the optimal production levels of different products, given constraints on resources like labor, materials, and budget. Similarly, a nutritionist might use a system of inequalities to design a diet plan that meets certain nutritional requirements while staying within a specific calorie range. Understanding how to formulate and solve systems of inequalities is essential for making informed decisions in these and other practical applications.
The correct system of inequalities to determine the ages of Anna and Jamie is a ≤ 3 + 2 j ; j ≥ 14 , a ≤ 35 , which corresponds to option B. This captures all the provided conditions accurately regarding their ages.
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