1. **Problem:** Solve the equation $5 \cdot (x + 1) = 10 - 3 \cdot x$ for $x$. 2. **Formula and rules:** Use the distributive property $a(b+c) = ab + ac$ to expand expressions. 3. **Step 1:** Expand the left side: $$5 \cdot (x + 1) = 5x + 5$$ So the equation becomes: $$5x + 5 = 10 - 3x$$ 4. **Step 2:** Add $3x$ to both sides to get all $x$ terms on one side: $$5x + 3x + 5 = 10 - \cancel{3x} + 3x$$ $$8x + 5 = 10$$ 5. **Step 3:** Subtract 5 from both sides: $$8x + 5 - 5 = 10 - 5$$ $$8x = 5$$ 6. **Step 4:** Divide both sides by 8 to solve for $x$: $$\frac{8x}{\cancel{8}} = \frac{5}{8}$$ $$x = \frac{5}{8}$$ **Final answer:** $x = \frac{5}{8}$