Question

1. on monday, you run on a treadmill for $\frac{1}{2}$ hour at $x$ miles per hour. on tuesday, you walk the same distance on the treadmill, at 2 miles per hour slower, and it takes you $\frac{3}{4}$ hour.

Explanation:

To solve for \( x \), we start by recalling the formula for distance: \( \text{Distance} = \text{Speed} \times \text{Time} \).

Step 1: Calculate the distance on Monday

On Monday, the speed is \( x \) miles per hour and the time is \( \frac{1}{2} \) hour. So the distance \( d \) on Monday is:
\[
d = x \times \frac{1}{2} = \frac{x}{2}
\]

Step 2: Calculate the distance on Tuesday

On Tuesday, the speed is \( (x - 2) \) miles per hour (since it's 2 miles per hour slower than Monday) and the time is \( \frac{3}{4} \) hour. So the distance \( d \) on Tuesday is:
\[
d = (x - 2) \times \frac{3}{4} = \frac{3(x - 2)}{4}
\]

Step 3: Set the distances equal (since they are the same)

Since the distance on Monday and Tuesday is the same, we can set the two expressions for distance equal to each other:
\[
\frac{x}{2} = \frac{3(x - 2)}{4}
\]

Step 4: Solve for \( x \)

Multiply both sides of the equation by 4 to eliminate the denominators:
\[
4 \times \frac{x}{2} = 4 \times \frac{3(x - 2)}{4}
\]
Simplify both sides:
\[
2x = 3(x - 2)
\]
Expand the right side:
\[
2x = 3x - 6
\]
Subtract \( 2x \) from both sides:
\[
0 = x - 6
\]
Add 6 to both sides:
\[
x = 6
\]

So the speed on Monday is 6 miles per hour.

Answer:

\( x = 6 \)