1. **State the problem:** Simplify the expression $0.5(b+7)-0.3b+4$. 2. **Apply the distributive property:** Multiply $0.5$ by each term inside the parentheses. $$0.5(b+7) = 0.5b + 0.5 \times 7 = 0.5b + 3.5$$ 3. **Rewrite the expression:** $$0.5b + 3.5 - 0.3b + 4$$ 4. **Combine like terms:** Combine the $b$ terms and the constant terms separately. $$0.5b - 0.3b = (\cancel{0.5} - \cancel{0.3})b = 0.2b$$ $$3.5 + 4 = 7.5$$ 5. **Final simplified expression:** $$0.2b + 7.5$$ This is the simplified form of the original expression.