Mike's school is a 30-minute walk or a 15-minute bus ride away from his house. The bus runs once every 30 minutes, and the number of minutes, \(w\), that Mike waits for the bus varies between 0 and 30.
Which of the following inequalities gives the values of \(w\) for which is would be faster for Mike to walk to school?
We need to compare the time it would take to take the bus to the time it would take to walk to school. It would be better for Mike to walk to school when the time it takes to walk to school is less than the time it takes to ride the bus.
$$ \text{Bus wait and travel time} \gt \text{walking time}$$
$$ \underbrace{\text{$w$}}_\text{wait time} + \underbrace{15}_\text{travel time} \gt \underbrace{30}_\text{walk time} $$
$$ \boxed{w+15 \gt 30}$$
We can inspect the scenario and select an arbitrary value that must be true. For example, we expect the inequality to satisfy the condition \(w=25\)
(wait time of 25 minutes which means walking is faster).
The last option satisfies this condition.
$$ \boxed{w+15 \gt 30} $$
$$ 25+15 \gt 30 $$
$$ 40 \gt 30 \checkmark $$
Though the first option satisfies this condition, let's consider the other extreme. When the wait time is low, such as if \(w=1\), we would expect the inequality to not be true.
$$ w-15 \lt 30 $$
$$ 1-15 \lt 30 $$
$$ -14 \lt 30 \checkmark $$
This inequality would therefore be a poor representation, since it concludes that we should walk even if the wait time for the bus is low!