Question: The equation $\frac{1}{2}x - 4 = 8$ is multiplied by $-1$. Which of the following are always true? Select all that apply.
The terms on the left side of the equation change signs.
The coefficient of the variable becomes an integer.
The left side of the equation will become $\frac{1}{2}x + 4$.
The right side of the equation will not change.
The right side of the equation will become negative.
1. **State the problem:** We start with the equation $$\frac{1}{2}x - 4 = 8$$ and multiply both sides by $$-1$$. We want to determine which statements about the resulting equation are always true.
2. **Multiply the entire equation by $$-1$$:**
$$-1 \times \left(\frac{1}{2}x - 4\right) = -1 \times 8$$
3. **Distribute the $$-1$$ on the left side:**
$$-1 \times \frac{1}{2}x + (-1) \times (-4) = -8$$
which simplifies to
$$-\frac{1}{2}x + 4 = -8$$
4. **Analyze each statement:**
- "The terms on the left side of the equation change signs."
Yes, $$\frac{1}{2}x$$ changed to $$-\frac{1}{2}x$$ and $$-4$$ changed to $$+4$$.
- "The coefficient of the variable becomes an integer."
No, the coefficient changed from $$\frac{1}{2}$$ to $$-\frac{1}{2}$$, which is still a fraction, not an integer.
- "The left side of the equation will become $$\frac{1}{2}x + 4$$."
No, the $$x$$ term changed sign to $$-\frac{1}{2}x$$, not $$\frac{1}{2}x$$.
- "The right side of the equation will not change."
No, the right side changed from $$8$$ to $$-8$$.
- "The right side of the equation will become negative."
Yes, it changed from $$8$$ to $$-8$$.
**Final answers:**
- The terms on the left side of the equation change signs. **True**
- The coefficient of the variable becomes an integer. **False**
- The left side of the equation will become $$\frac{1}{2}x + 4$$. **False**
- The right side of the equation will not change. **False**
- The right side of the equation will become negative. **True**