The list `{2, -1, x, 5, 3}` has a range of 13. What are the possible values for x ?

Answer

To solve the problem, we need to determine the possible values for \( x \) in the list \{2, -1, x, 5, 3\} such that the range of the list is 13. The range is defined as the difference between the largest and smallest values in the list.

### Step-by-Step Solution:

1. **Understand the concept of range**:
The range of a set of numbers is calculated as:
\[ \text{Range} = \text{Largest value} - \text{Smallest value} \]
We know that the range of the list is 13.

2. **Identify the current smallest and largest values**:
In the list \{2, -1, x, 5, 3\}, the smallest value without \( x \) is -1 and the largest value is 5.

3. **Consider the cases for \( x \)**:
Since the range must be 13, \( x \) must either be the largest value or the smallest value in the list. We will consider both cases.

**Case 1**: \( x \) is the largest value.
- If \( x \) is the largest, then the smallest value remains -1.
- The range can be expressed as:
\[ x - (-1) = 13 \]
Simplifying this gives:
\[ x + 1 = 13 \implies x = 12 \]

**Case 2**: \( x \) is the smallest value.
- If \( x \) is the smallest, then the largest value remains 5.
- The range can be expressed as:
\[ 5 - x = 13 \]
Simplifying this gives:
\[ -x = 13 - 5 \implies -x = 8 \implies x = -8 \]

4. **Conclusion**:
The possible values for \( x \) are:
\[ x = 12 \quad \text{or} \quad x = -8 \]

### Final Answer:
The possible values for \( x \) are \( 12 \) or \( -8 \).

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