To determine the inequality that shows the range of Dorian's earnings from walking dogs, we need to consider the potential constraints in the real world.
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Earnings Model: The function given is \( f(w) = 15w \), where \( w \) is the number of dogs Dorian walks. His earnings increase by $15 for each dog he walks.
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Constraints to Consider:
- \( w \) (the number of dogs walked) cannot be negative. Therefore, \( w \geq 0 \).
- There may also be an upper limit to the number of dogs he can walk, depending on time or other hypothetical constraints. For illustrative purposes, let's assume there isn’t a specific upper limit placed here, but in reality, he might only be able to walk a certain number of dogs in a day.
Given these considerations, if \( w \) is restricted to non-negative values, the inequality representing the number of dogs Dorian can walk is:
\[ w \geq 0 \]
And since his earnings will be calculated from this number of dogs, you can express his earnings as being non-negative:
\[ f(w) = 15w \geq 0 \]
Thus the appropriate inequality showing the range of his earnings, given \( w \geq 0 \), is:
\[ f(w) \geq 0 \]
In conclusion, if we consider only non-negative scenarios without a maximum number of dogs, we can summarize his earnings range as \( f(w) \geq 0 \), where \( w \) (number of dogs walked) must be a zero or positive integer. If there exists a maximum limit (say \( w \) cannot exceed a certain number), you would also include that in your range definition.
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