1. **State the problem:** Solve the equation $4(x-3)-3(x-5)=3(x+1)$ for $x$. 2. **Apply the distributive property:** Multiply each term inside the parentheses by the factor outside. $$4(x-3) = 4x - 12$$ $$-3(x-5) = -3x + 15$$ $$3(x+1) = 3x + 3$$ 3. **Rewrite the equation with distributed terms:** $$4x - 12 - 3x + 15 = 3x + 3$$ 4. **Combine like terms on the left side:** $$ (4x - 3x) + (-12 + 15) = 3x + 3$$ $$x + 3 = 3x + 3$$ 5. **Isolate variable terms on one side:** Subtract $3x$ from both sides. $$x + 3 - 3x = 3x + 3 - 3x$$ $$\cancel{x} + 3 - \cancel{3x} = \cancel{3x} + 3 - \cancel{3x}$$ $$-2x + 3 = 3$$ 6. **Isolate the variable term:** Subtract 3 from both sides. $$-2x + 3 - 3 = 3 - 3$$ $$-2x + \cancel{3} - \cancel{3} = \cancel{3} - \cancel{3}$$ $$-2x = 0$$ 7. **Solve for $x$:** Divide both sides by $-2$. $$\frac{-2x}{-2} = \frac{0}{-2}$$ $$\cancel{-2}x / \cancel{-2} = 0$$ $$x = 0$$ **Final answer:** $x = 0$