To find the possible values for \( b \), we first need to understand how the range is calculated. The range of a set of numbers is defined as the difference between the maximum and minimum values.
Given the numbers: \( 14, 14, 10, 9, b \), we need to determine values for \( b \) such that the range is equal to 6.
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Identifying the Maximum and Minimum: In the data set, the current maximum is \( 14 \) (since \( 14 \) appears twice) and the current minimum is \( 9 \).
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Expressing the Range: The formula for the range is: \[ \text{Range} = \text{Max} - \text{Min} \] We want this range to equal \( 6 \): \[ 14 - 9 = 5 \]
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Considering the Value of \( b \):
- If \( b < 9 \): The minimum would be \( b \) and the maximum would still be \( 14 \), giving a range of \( 14 - b \).
- If \( b > 14 \): The maximum would be \( b \) and the minimum would still be \( 9 \), giving a range of \( b - 9 \).
- If \( 9 \leq b \leq 14 \): The minimum remains \( 9 \), and the maximum is \( 14 \), so the range remains \( 14 - 9 = 5\).
To make the range equal to \( 6 \), we consider the following two scenarios:
- If \( b < 9 \): \[ 14 - b = 6 \implies b = 8 \]
- If \( b > 14 \): \[ b - 9 = 6 \implies b = 15 \]
Thus, \( b \) can either be \( 8 \) or \( 15 \).
Conclusion: The values that \( b \) could take in order for the range to equal \( 6 \) are \( b = 8 \) or \( b = 15 \).
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