Rafael is choosing between two exercise routines. In Routine \( \# 1 \), he does only running, burning 8.5 calories per minute. In Routine \( \# 2 \), he burns 22 calories walking. He then runs at a rate that burns 6.3 calories per minut For what amounts of time spent running will Routine \#1 burn more calories than Routine \#2? Use \( t \) for the number of minutes spent running, and solve your inequality for \( t \).

To determine for what amounts of time spent running Routine \( \# 1 \) burns more calories than Routine \( \# 2 \), we can set up the inequalities based on the information given. In Routine \( \# 1 \): - The total calories burned from running for \( t \) minutes is \( 8.5t \). In Routine \( \# 2 \): - The calories burned walking is 22 calories. - The calories burned from running \( t \) minutes is \( 6.3t \). - Therefore, the total calories burned in Routine \( \# 2 \) is \( 22 + 6.3t \). Now, we need to set up the inequality: \[ 8.5t > 22 + 6.3t \] To solve this inequality, we first subtract \( 6.3t \) from both sides: \[ 8.5t - 6.3t > 22 \] \[ 2.2t > 22 \] Now, divide both sides by 2.2: \[ t > \frac{22}{2.2} \] \[ t > 10 \] Thus, for Routine \( \# 1 \) to burn more calories than Routine \( \# 2 \), Rafael needs to run for more than \( 10 \) minutes.