1. **Problem statement:** Jack and Jon have a hot dog eating contest. After the first 2 minutes, Jack has eaten 8 hot dogs and Jon has eaten 7 hot dogs. From that point, Jack eats 3 hot dogs per minute and Jon eats 4 hot dogs per minute. We want to find after how many minutes $m$ Jack will have eaten more hot dogs than Jon. 2. **Set up the inequality:** Let $m$ be the number of minutes after the first 2 minutes. Jack's total hot dogs: $8 + 3m$ Jon's total hot dogs: $7 + 4m$ We want Jack to have eaten more than Jon, so: $$8 + 3m > 7 + 4m$$ 3. **Solve the inequality:** $$8 + 3m > 7 + 4m$$ Subtract $3m$ from both sides: $$8 + \cancel{3m} > 7 + 4m - \cancel{3m}$$ $$8 > 7 + m$$ Subtract 7 from both sides: $$8 - 7 > 7 + m - 7$$ $$1 > m$$ 4. **Interpretation:** Jack will have eaten more hot dogs than Jon after fewer than 1 minute from the 2-minute mark. Since $m$ represents minutes after the first 2 minutes, Jack will have more hot dogs immediately after the 2 minutes, but before 3 minutes total. **Final answer:** $$m < 1$$ Jack will have eaten more hot dogs than Jon after less than 1 minute from the 2-minute mark.