"At least 6" and "less than 8" describe the speed. How do you write these two phrases as inequality symbols for the speed variable?

Once you have an inequality for speed, multiply each part by 1.5 to change it into an inequality for distance. Be careful about whether the inequality symbols flip when you multiply.

Ask yourself: should the smaller endpoint distance be included or excluded? What about the larger endpoint? Match that pattern (≤ vs <) to the answer choices.

In Desmos, type 1.5*6 on one line and 1.5*8 on another line. These outputs give the minimum and maximum distances corresponding to the speeds 6 mph and 8 mph (before you think about whether each endpoint is included).

Use the problem wording: "at least 6" and "less than 8" to decide whether each of the two distances you found should be included ($\le$) or excluded ($<$) in the inequality for $d$, then pick the choice that matches that pattern and those numbers.

Use the basic distance formula:

• $d =$ distance (miles)
• $r =$ speed (miles per hour)
• $t =$ time (hours)

They are related by

Here, $t = 1.5$ hours.

The problem says the cyclist rides at a speed that is at least 6 miles per hour but less than 8 miles per hour.

• "At least 6" means $r \ge 6$.
• "Less than 8" means $r < 8$.

Together, this is a compound inequality:

Use $d = r \cdot 1.5$ and multiply every part of the inequality by $1.5$.

Since $1.5$ is positive, the inequality directions stay the same:

Because $d = 1.5r$, this becomes an inequality in terms of $d$.

Calculate the new bounds:

• $1.5 \cdot 6 = 9$
• $1.5 \cdot 8 = 12$

So the distances $d$ must satisfy

This matches answer choice D.