1. **State the problem:** We need to find the lowest integer value of $r$ such that the inequality $7r + 5 > 37 - 3r$ holds true. 2. **Write down the inequality:** $$7r + 5 > 37 - 3r$$ 3. **Add $3r$ to both sides to get all $r$ terms on one side:** $$7r + 3r + 5 > 37$$ $$10r + 5 > 37$$ 4. **Subtract 5 from both sides:** $$10r > 37 - 5$$ $$10r > 32$$ 5. **Divide both sides by 10 (positive number, so inequality direction stays the same):** $$r > \frac{32}{10}$$ $$r > 3.2$$ 6. **Interpretation:** The values of $r$ must be greater than 3.2. 7. **Find the lowest integer value:** Since $r$ must be greater than 3.2, the smallest integer satisfying this is $4$. **Final answer:** $$\boxed{4}$$