Question

Dorian earns $15 every time he walks a dog. He models the total amount of money he makes walking dogs with the equation f (w) = 15w. Which inequality shows the range of his earnings given the real-world constraints?

Answer 1

To find the appropriate inequality that represents the range of Dorian's earnings while walking dogs, we first need to consider the nature of the parameters involved in the function \( f(w) = 15w \), where \( w \) represents the number of dogs walked.

Since Dorian can only walk a non-negative number of dogs (he cannot walk a negative number of dogs), this gives us the constraint that \( w \geq 0 \).

Thus, we can represent his earnings as:

\[
f(w) = 15w
\]

For earnings, since he earns money for each dog walked and can walk any non-negative number of dogs, the range of his earnings can be described as:

\[
f(w) \geq 0
\]

This leads us to the inequality:

\[
15w \geq 0
\]

Moreover, as \( w \) can represent a whole number of dogs (0, 1, 2, 3, ...), Dorian's earnings will also have a lower limit of $0 (when he walks 0 dogs) and has no upper limit in theory, provided he is able to walk a sufficiently high number of dogs.

Thus, the final range of his earnings can be expressed with the inequality:

\[
f(w) \geq 0
\]

or more specifically related to \( w \):

\[
w \geq 0
\]

In summary, the inequality that shows the range of his earnings is:

\[
f(w) \geq 0 \quad \text{for } w \geq 0
\]