Manuel is choosing between two exercise routines.
In Routine #1, he burns 20 calories walking. He then runs at a rate that burns 8.5 calories per minute.
In Routine #2, he burns 43 calories walking. He then runs at a rate that burns 2.75 calories per minute.
For what amounts of time spent running will Routine #1 burn fewer calories than Routine #2?
Use t for the number of minutes spent running, and solve your inequality for t.

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.

Manuel is choosing between two exercise routines. In Routine #1, he burns 20 calories walking. He then runs at a rate that burns 8.5 calories per minute. In Routine #2, he burns 43 calories walking. He then runs at a rate that burns 2.75 calories per minute. For what amounts of time spent running will Routine #1 burn fewer calories than Routine #2? Use t for the number of minutes spent running, and solve your inequality for t.

Learn how to solve an inequality that compares the calories burned between two exercise routines with different walking and running rates. This problem involves solving a linear inequality, making it perfect for students in Grades 7-9. Detailed steps guide you through the solution, including calorie rates and solving for the time t.

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