To determine the appropriate inequality showing the range of Dorian's earnings based on his dog walking equation \( f(w) = 15w \), we need to identify the constraints.
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Understanding the parameters:
- \( f(w) \) represents the total earnings from walking dogs.
- \( w \) represents the number of dogs walked, which cannot be negative (i.e., \( w \geq 0 \)).
- We need to assume a reasonable upper constraint on walking dogs. If we consider that Dorian walks dogs to earn up to a certain limit, let’s say he can potentially earn a maximum of $1,050.
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Finding how many dogs he needs to walk to reach $1,050: \[ 15w \leq 1,050 \] \[ w \leq \frac{1,050}{15} = 70 \]
So, he can walk up to 70 dogs to earn $1,050.
- Determining the inequality for earnings: Based on the realistic scenarios, the total earnings \( f(w) \) must be non-negative (since he cannot earn less than $0) and cannot exceed $1,050.
Thus, the correct inequality representing the range of his earnings, given these constraints, would be:
c. \( 0 \leq f(w) \leq 1,050 \).