We are asked to find the amount of time $t$ (in minutes) that Manuel spends running such that Routine #1 burns fewer calories than Routine #2.

Step 1: Set up the inequality for each routine.

For Routine #1, Manuel burns: $\text{Calories burned} = 20 + 8.5t$ where $t$ is the time in minutes spent running.

For Routine #2, Manuel burns: $\text{Calories burned} = 43 + 2.75t$ where $t$ is again the time in minutes spent running.

Step 2: Set up the inequality.

We want Routine #1 to burn fewer calories than Routine #2: $20 + 8.5t < 43 + 2.75t$

Step 3: Solve the inequality.

First, subtract $2.75t$ from both sides: $20 + 8.5t - 2.75t < 43$ Simplify the terms involving $t$: $20 + 5.75t < 43$

Next, subtract 20 from both sides: $5.75t < 23$

Finally, divide both sides by 5.75: $t < \frac{23}{5.75}$

Simplify: $t < 4$

Conclusion:

Manuel must run for less than 4 minutes ($t < 4$) in order for Routine #1 to burn fewer calories than Routine #2.

Would you like further details on any step? Here are five questions that build on this solution:

1. What happens when $t = 4$ minutes? Will the calorie burn be the same?
2. How would the inequality change if the running rate in Routine #1 increased to 9 calories per minute?
3. What is the total number of calories burned in each routine if $t = 3$?
4. How many minutes does Manuel need to run in Routine #2 to burn more calories than Routine #1?
5. How would the answer change if the walking calories burned were different for both routines?

Tip: When solving inequalities, remember that dividing or multiplying by a negative number reverses the inequality sign.