1. **State the problem:** Solve the equation $$\frac{x-6}{3} + x + \frac{2}{4} = 32$$ for $x$. 2. **Identify the formula and rules:** To solve for $x$, we need to combine like terms and isolate $x$. Remember to handle fractions carefully by finding a common denominator or multiplying through to clear denominators. 3. **Clear the fractions:** Multiply every term by 12 (the least common multiple of 3 and 4) to eliminate denominators: $$12 \times \frac{x-6}{3} + 12 \times x + 12 \times \frac{2}{4} = 12 \times 32$$ 4. **Simplify each term:** $$4(x-6) + 12x + 3 \times 2 = 384$$ 5. **Distribute and simplify:** $$4x - 24 + 12x + 6 = 384$$ 6. **Combine like terms:** $$4x + 12x - 24 + 6 = 384$$ $$16x - 18 = 384$$ 7. **Isolate $x$:** Add 18 to both sides: $$16x - 18 + 18 = 384 + 18$$ $$16x = 402$$ 8. **Divide both sides by 16:** $$x = \frac{402}{16}$$ 9. **Simplify the fraction:** $$x = \frac{\cancel{402}^{\div 2}}{\cancel{16}^{\div 2}} = \frac{201}{8}$$ **Final answer:** $$x = \frac{201}{8}$$