To determine the correct inequality showing the range of Darien's earnings, we need to consider the following:
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Earnings Calculation: The function \( f(w) = 15w \) represents Darien's earnings based on the number of times he walks dogs, where \( w \) is the number of times he walks dogs.
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Real World Constraints:
- The number of times Darien walks dogs, \( w \), must be a non-negative integer (i.e., he cannot walk dogs a negative number of times).
- The total earnings should also not exceed a certain threshold.
Assuming that the maximum earnings he can make is \( 1,050 \), we derive the maximum number of dog walks from this:
\[ 15w \leq 1,050 \implies w \leq 70 \]
This means he can walk the dog a maximum of 70 times.
Given these parameters, the range of his earnings is:
- \( f(w) \) can be zero if he does not walk any dogs (i.e., if \( w = 0 \)).
- \( f(w) \) can be up to \( 1,050 \) when he walks the dog \( 70 \) times.
Based on this analysis, the appropriate inequality to describe his range of earnings is:
\[ 0 \leq f(w) \leq 1,050 \]
Thus, the correct answer is:
C. \( 0 \leq f(w) \leq 1,050 \).