Solving a decimal word problem using a linear inequality with the variabl... Chris is choosing between two exercise routines. In Routine \#1, he burns 24 calories walking. He then runs at a rate that burns 12.5 calories per minute. In Routine \#2, he burns 42 calories walking. He then runs at a rate that burns 8.9 calories per minute. For what amounts of time spent running will Routine \#1 burn at most as many calories as Routine \#2? Use \( t \) for the number of minutes spent running, and solve your inequality for \( t \). In Un

To determine for how many minutes \( t \) Chris should spend running in Routine \#1 such that it burns **at most** as many calories as Routine \#2, we'll set up and solve a linear inequality. ### Step 1: Define the Calorie Equations 1. **Routine \#1:** - **Walking:** Burns 24 calories. - **Running:** Burns 12.5 calories per minute. **Total Calories for Routine \#1:** \[ 24 + 12.5t \] 2. **Routine \#2:** - **Walking:** Burns 42 calories. - **Running:** Burns 8.9 calories per minute. **Total Calories for Routine \#2:** \[ 42 + 8.9t \] ### Step 2: Set Up the Inequality We want Routine \#1 to burn **at most** as many calories as Routine \#2: \[ 24 + 12.5t \leq 42 + 8.9t \] ### Step 3: Solve the Inequality for \( t \) 1. **Subtract \( 8.9t \) from both sides** to get all \( t \)-terms on one side: \[ 24 + 12.5t - 8.9t \leq 42 \] \[ 24 + 3.6t \leq 42 \] 2. **Subtract 24 from both sides** to isolate the term with \( t \): \[ 3.6t \leq 42 - 24 \] \[ 3.6t \leq 18 \] 3. **Divide both sides by 3.6** to solve for \( t \): \[ t \leq \frac{18}{3.6} \] \[ t \leq 5 \] ### Conclusion Chris should spend **at most 5 minutes** running in Routine \#1 for it to burn **no more calories** than Routine \#2. \[ \boxed{t \leq 5} \]