System Of Linear Inequalities For Anna And Jamie's Ages

Determining age relationships often involves understanding how different ages relate to each other. In this scenario, we explore the relationship between Anna's age (a) and Jamie's age (j) using a system of linear inequalities. The problem states three key facts:

1. Anna is no more than 3 years older than 2 times Jamie's age.
2. Jamie is at least 14 years old.
3. Anna is at most 35 years old.

Our goal is to translate these statements into mathematical inequalities and then discuss how these inequalities can be used to find the possible ages of Anna and Jamie. This involves understanding the nuances of inequality symbols and how they represent real-world constraints.

Translating the Statements into Inequalities

Let's break down each statement and convert it into a mathematical inequality. This step is crucial for setting up the system of inequalities that accurately represents the given conditions.

Anna is no more than 3 years older than 2 times Jamie's age

This statement implies that Anna's age (a) is less than or equal to 3 plus 2 times Jamie's age (j). Mathematically, this can be written as:

$$a \leq 3 + 2j$$

Here, the phrase "no more than" is key. It indicates an upper limit, meaning Anna's age cannot exceed this value. The inequality symbol "$\leq$ " (less than or equal to) captures this condition perfectly. This inequality forms the cornerstone of our system, linking Anna's age directly to Jamie's age.

Jamie is at least 14 years old

The phrase "at least" means that Jamie's age (j) is greater than or equal to 14. This gives us our second inequality:

$$j \geq 14$$

This inequality sets a lower bound for Jamie's age, indicating that he cannot be younger than 14. This is a simple yet crucial constraint, as it limits the possible values of j and, consequently, influences the possible values of a.

Anna is at most 35 years old

Similarly, "at most" indicates an upper limit. Anna's age (a) is less than or equal to 35. This can be written as:

$$a \leq 35$$

This inequality places a cap on Anna's age, ensuring that she cannot be older than 35. This constraint is independent of Jamie's age but plays a vital role in defining the feasible region for the solution.

The System of Linear Inequalities

Combining the three inequalities, we get the following system of linear inequalities:

$$\begin{cases} a \leq 3 + 2j \\ j \geq 14 \\ a \leq 35 \end{cases}$$

This system of inequalities mathematically represents the given conditions. Each inequality contributes to defining the boundaries within which the possible ages of Anna and Jamie must lie. This system is the foundation for solving the problem and understanding the possible age combinations.

Understanding the Inequalities and Their Implications

Each inequality in the system provides specific information and constraints on the ages of Anna and Jamie. Understanding these implications is key to interpreting the solution set.

${a \leq 3 + 2j}$

This inequality is the most complex as it relates Anna's age to Jamie's age. It tells us that Anna's age is limited by twice Jamie's age plus 3 years. This means as Jamie gets older, the upper limit for Anna's age also increases, but at a rate of twice Jamie's age. This relationship is crucial for finding feasible age combinations.

${j \geq 14}$

This inequality is straightforward: Jamie's age must be 14 or older. This sets a clear minimum age for Jamie and restricts the possible solutions to those where Jamie is at least 14 years old. This is a simple lower bound but essential for realistic age scenarios.

${a \leq 35}$

This inequality sets an upper limit on Anna's age. She cannot be older than 35. This constraint is independent of Jamie's age but provides an overall maximum age for Anna, preventing solutions where Anna is unrealistically old. This upper bound helps define the solution space more precisely.

Graphical Representation and Solution Set

To visualize the solution set, we can graph these inequalities on a coordinate plane, where the x-axis represents Jamie's age (j) and the y-axis represents Anna's age (a). Each inequality will define a region on the plane, and the solution set will be the intersection of these regions. This graphical approach provides a visual understanding of the possible age combinations.

Graphing ${a \leq 3 + 2j}$

This inequality represents the region below the line ${a = 3 + 2j}$. To graph this line, we can find two points that satisfy the equation. For example, when ${j = 14}$, ${a = 3 + 2(14) = 31}$, and when ${j = 16}$, ${a = 3 + 2(16) = 35}$. Plotting these points and drawing a line through them, we shade the region below the line to represent the inequality.

Graphing ${j \geq 14}$

This inequality represents the region to the right of the vertical line ${j = 14}$. This is a straightforward vertical boundary, indicating that all feasible solutions must have Jamie's age greater than or equal to 14. This line is essential for defining the lower bound of Jamie's age.

Graphing ${a \leq 35}$

This inequality represents the region below the horizontal line ${a = 35}$. This sets a maximum limit for Anna's age, and the region below this line includes all possible ages for Anna that meet this condition. This horizontal boundary caps Anna's age at 35.

The Feasible Region

The feasible region, or the solution set, is the area where all three shaded regions overlap. This region represents all possible combinations of Anna's and Jamie's ages that satisfy all three conditions. Any point within this region is a valid solution to the system of inequalities. The shape and size of this region provide valuable insights into the age ranges that are possible.

Finding Possible Ages

To find specific possible ages, we can look for points within the feasible region. These points represent pairs of ages (a, j) that satisfy all the inequalities. For example, if Jamie is 15, then ${j = 15}$. We can substitute this into the first inequality to find the maximum possible age for Anna:

$$a \leq 3 + 2(15) = 33$$

So, if Jamie is 15, Anna can be at most 33. Since Anna must also be at most 35 (from the third inequality), this condition is met. Thus, (33, 15) is a possible solution.

Similarly, we can check other points within the feasible region to find other possible ages. For instance, if Jamie is 16, then:

$$a \leq 3 + 2(16) = 35$$

In this case, Anna can be at most 35, which is also the maximum age allowed by the third inequality. The point (35, 16) is another possible solution.

Importance of the System of Inequalities

The system of inequalities provides a concise and accurate way to represent the given information about Anna's and Jamie's ages. It allows us to:

• Mathematically define the relationships between their ages.
• Identify all possible age combinations that satisfy the given conditions.
• Visualize the solution set through graphical representation.
• Solve real-world problems involving constraints and limitations.

This approach is not only useful for solving mathematical problems but also for understanding and modeling real-world scenarios where multiple constraints exist. By translating verbal statements into mathematical inequalities, we can analyze and interpret complex situations more effectively.

Conclusion

The system of linear inequalities accurately represents the given conditions for Anna's and Jamie's ages. By understanding and interpreting these inequalities, we can determine the possible age ranges for both individuals. The system allows us to define mathematical boundaries and find solutions that satisfy all constraints. This method highlights the power of mathematical modeling in solving practical problems and understanding relationships between variables. The solution to this problem not only answers the specific question but also illustrates a broader approach to problem-solving using inequalities and graphical representation.

What system of linear inequalities accurately captures the age relationship between Anna and Jamie, given that Anna is no more than 3 years older than twice Jamie's age, Jamie is at least 14, and Anna is at most 35?