https://file.aiquickdraw.com/mathbot/file/1728751274886mq9fs2id.jpg

The given problem states that \( S = \{ X, 1, 7 \} \), where the mean of \( S \) is also the median of \( S \). We are tasked with finding the product of all possible values of \( X \).

### Step-by-step solution:

1. **Mean of Set \( S \):**
   The mean (average) of a set is the sum of its elements divided by the number of elements. For the set \( S = \{ X, 1, 7 \} \), the mean is:
   \[
   \text{Mean of } S = \frac{X + 1 + 7}{3} = \frac{X + 8}{3}
   \]

2. **Median of Set \( S \):**
   To find the median, we need to arrange the elements in increasing order:
   - If \( X \leq 1 \), then the order is \( \{ X, 1, 7 \} \), and the median is 1.
   - If \( 1 \leq X \leq 7 \), then the order is \( \{ 1, X, 7 \} \), and the median is \( X \).
   - If \( X \geq 7 \), then the order is \( \{ 1, 7, X \} \), and the median is 7.

3. **Condition: The Mean Equals the Median**
   We need to find values of \( X \) such that the mean is equal to the median in each case.

   - **Case 1: \( X \leq 1 \)**  
     In this case, the median is 1. Setting the mean equal to the median:
     \[
     \frac{X + 8}{3} = 1
     \]
     Solving for \( X \):
     \[
     X + 8 = 3 \quad \Rightarrow \quad X = -5
     \]

   - **Case 2: \( 1 \leq X \leq 7 \)**  
     In this case, the median is \( X \). Setting the mean equal to the median:
     \[
     \frac{X + 8}{3} = X
     \]
     Solving for \( X \):
     \[
     X + 8 = 3X \quad \Rightarrow \quad 8 = 2X \quad \Rightarrow \quad X = 4
     \]

   - **Case 3: \( X \geq 7 \)**  
     In this case, the median is 7. Setting the mean equal to the median:
     \[
     \frac{X + 8}{3} = 7
     \]
     Solving for \( X \):
     \[
     X + 8 = 21 \quad \Rightarrow \quad X = 13
     \]

4. **Possible Values of \( X \):**
   The possible values of \( X \) that satisfy the condition are \( X = -5 \), \( X = 4 \), and \( X = 13 \).

5. **Product of All Possible Values of \( X \):**
   The product of all possible values of \( X \) is:
   \[
   (-5) \times 4 \times 13 = -260
   \]

### Final Answer:
The product of all possible values of \( X \) is \( \boxed{-260} \).

---

Would you like further details on any part of this solution, or have additional questions? Here are some related questions for practice:

1. How do you calculate the mean of a set of numbers?
2. What is the difference between the mean and median of a dataset?
3. How would the solution change if \( S \) had four elements instead of three?
4. What are the general steps to find the median of a set?
5. How does changing the value of \( X \) influence the median?

**Tip:** When solving problems that involve both mean and median, carefully check how the median changes depending on the value of each element!

Let S = {X, 1, 7} where the mean of S is also the median of S. Find the product of all possible X.

Solve a problem where the mean and median of the set S = {X, 1, 7} are equal. Learn how to find all possible values of X and calculate the product of these values. The solution includes algebraic steps and explanations suitable for Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation