Question: Solve for $x$ and write your answer in simplest form.
$-\frac{1}{2} x + 3 = -5 - \frac{1}{2} (9x + 1)$
1. **State the problem:** Solve the equation $$-\frac{1}{2} x + 3 = -5 - \frac{1}{2} (9x + 1)$$ for $x$ and simplify the answer.
2. **Distribute and simplify:** Apply the distributive property to the right side:
$$-\frac{1}{2} x + 3 = -5 - \frac{1}{2} \times 9x - \frac{1}{2} \times 1$$
$$-\frac{1}{2} x + 3 = -5 - \frac{9}{2} x - \frac{1}{2}$$
3. **Combine constants on the right side:**
$$-5 - \frac{1}{2} = -\frac{10}{2} - \frac{1}{2} = -\frac{11}{2}$$
So the equation becomes:
$$-\frac{1}{2} x + 3 = -\frac{11}{2} - \frac{9}{2} x$$
4. **Bring all $x$ terms to one side and constants to the other:**
Add $\frac{9}{2} x$ to both sides:
$$-\frac{1}{2} x + \frac{9}{2} x + 3 = -\frac{11}{2}$$
Combine like terms:
$$\left(-\frac{1}{2} + \frac{9}{2}\right) x + 3 = -\frac{11}{2}$$
$$\frac{8}{2} x + 3 = -\frac{11}{2}$$
$$4x + 3 = -\frac{11}{2}$$
5. **Isolate $x$:**
Subtract 3 from both sides:
$$4x = -\frac{11}{2} - 3$$
Convert 3 to fraction:
$$4x = -\frac{11}{2} - \frac{6}{2} = -\frac{17}{2}$$
6. **Divide both sides by 4:**
$$x = \frac{-\frac{17}{2}}{4} = -\frac{17}{2} \times \frac{1}{4} = -\frac{17}{8}$$
7. **Final answer:**
$$\boxed{-\frac{17}{8}}$$