Representing the Relationships between Ages

Oct 13, 2025 by windailys.com 38 views

Introduction

In this article, we will delve into a mathematical problem involving the ages of three individuals: Carl, Caitlyn, and Daryl. The problem requires us to find the correct answer based on the given conditions. We will use algebraic equations to represent the relationships between their ages and solve for the unknown variable, Caitlyn's age, represented by $x$ .

Problem Statement

Carl is two years older than Caitlyn, and Daryl is five years older than Carl. The product of Carl and Daryl's ages is at least 160. If $x$ represents Caitlyn's age, we need to find the possible values of $x$ that satisfy the given conditions.

Representing the Relationships between Ages

Let's represent the ages of Carl, Caitlyn, and Daryl as $c$ , $x$ , and $d$ , respectively. Based on the given information, we can write the following equations:

$c = x + 2$ (Carl is two years older than Caitlyn)
$d = c + 5$ (Daryl is five years older than Carl)
$cd \geq 160$ (The product of Carl and Daryl's ages is at least 160)

Substituting the Expressions

We can substitute the expression for $c$ into the equation for $d$ to get:

$d = (x + 2) + 5$ $d = x + 7$

Now, we can substitute the expressions for $c$ and $d$ into the inequality $cd \geq 160$ :

$(x + 2)(x + 7) \geq 160$

Expanding and Simplifying the Inequality

Expanding the left-hand side of the inequality, we get:

$x^2 + 9x + 14 \geq 160$

Subtracting 160 from both sides, we get:

$x^2 + 9x - 146 \geq 0$

Factoring the Quadratic Expression

We can factor the quadratic expression on the left-hand side of the inequality:

$(x + 17)(x - 8) \geq 0$

Finding the Critical Points

The critical points occur when the expression $(x + 17)(x - 8)$ is equal to zero. Setting each factor equal to zero, we get:

$x + 17 = 0$ or $x - 8 = 0$

Solving for $x$ , we get:

$x = -17$ or $x = 8$

Determining the Solution Set

Since the inequality is greater than or equal to zero, we need to find the values of $x$ that make the expression $(x + 17)(x - 8)$ greater than or equal to zero. This occurs when both factors are positive or both factors are negative.

Case 1: Both Factors are Positive

When both factors are positive, we have:

$x + 17 > 0$ and $x - 8 > 0$

Solving for $x$ , we get:

$x > -17$ and $x > 8$

Since both conditions must be true, we can combine them to get:

$x > 8$

Case 2: Both Factors are Negative

When both factors are negative, we have:

$x + 17 < 0$ and $x - 8 < 0$

Solving for $x$ , we get:

$x < -17$ and $x < 8$

Since both conditions must be true, we can combine them to get:

$x < -17$

Conclusion

Based on the inequality $(x + 17)(x - 8) \geq 0$ , we have found that the solution set is $x \leq -17$ or $x \geq 8$ . However, since Caitlyn's age cannot be negative, we can discard the solution $x \leq -17$ . Therefore, the possible values of $x$ are $x \geq 8$ .

Final Answer

Q&A: Understanding the Age Problem

Q: What is the problem about? A: The problem involves three individuals: Carl, Caitlyn, and Daryl. They are comparing their ages, and we need to find the correct answer based on the given conditions.

Q: What are the conditions? A: The conditions are:

Carl is two years older than Caitlyn.
Daryl is five years older than Carl.
The product of Carl and Daryl's ages is at least 160.

Q: How do we represent the relationships between their ages? A: We can represent the ages of Carl, Caitlyn, and Daryl as $c$ , $x$ , and $d$ , respectively. Based on the given information, we can write the following equations:

$c = x + 2$ (Carl is two years older than Caitlyn)
$d = c + 5$ (Daryl is five years older than Carl)
$cd \geq 160$ (The product of Carl and Daryl's ages is at least 160)

Q: How do we solve the inequality? A: We can substitute the expressions for $c$ and $d$ into the inequality $cd \geq 160$ and then expand and simplify the inequality. We can also factor the quadratic expression on the left-hand side of the inequality.

Q: What are the critical points? A: The critical points occur when the expression $(x + 17)(x - 8)$ is equal to zero. Setting each factor equal to zero, we get:

$x + 17 = 0$ or $x - 8 = 0$ Solving for $x$ , we get:
$x = -17$ or $x = 8$

Q: What is the solution set? A: Based on the inequality $(x + 17)(x - 8) \geq 0$ , we have found that the solution set is $x \leq -17$ or $x \geq 8$ . However, since Caitlyn's age cannot be negative, we can discard the solution $x \leq -17$ . Therefore, the possible values of $x$ are $x \geq 8$ .

Q: What is the final answer? A: The final answer is $\boxed{8}$ , but we must consider all possible values of $x$ that satisfy the given conditions. Therefore, the correct answer is $x \geq 8$ .

Common Mistakes to Avoid

Not considering all possible values of $x$ that satisfy the given conditions.
Not discarding the solution $x \leq -17$ since Caitlyn's age cannot be negative.
Not factoring the quadratic expression on the left-hand side of the inequality.

Tips and Tricks

Use algebraic equations to represent the relationships between the ages of Carl, Caitlyn, and Daryl.
Substitute the expressions for $c$ and $d$ into the inequality $cd \geq 160$ .
Expand and simplify the inequality.
Factor the quadratic expression on the left-hand side of the inequality.
Find the critical points by setting each factor equal to zero.
Determine the solution set based on the inequality.

Conclusion

In this article, we have solved the age problem using algebraic equations and inequalities. We have represented the relationships between the ages of Carl, Caitlyn, and Daryl, substituted the expressions for $c$ and $d$ into the inequality $cd \geq 160$ , expanded and simplified the inequality, factored the quadratic expression on the left-hand side of the inequality, found the critical points, and determined the solution set. We have also provided tips and tricks to help you avoid common mistakes and solve the problem efficiently.