Question: Solve for $v$.
$\frac{10}{v + 1} = \frac{-10}{v + 7}$
There may be 1 or 2 solutions.
$v = $ or $v = $
1. **State the problem:** Solve for $v$ in the equation $$\frac{10}{v + 1} = \frac{-10}{v + 7}$$.
2. **Use the cross-multiplication formula:** When two fractions are equal, their cross products are equal:
$$10 \times (v + 7) = -10 \times (v + 1)$$
3. **Expand both sides:**
$$10v + 70 = -10v - 10$$
4. **Bring all terms involving $v$ to one side and constants to the other:**
$$10v + 10v = -10 - 70$$
$$20v = -80$$
5. **Divide both sides by 20 to isolate $v$:**
$$v = \frac{\cancel{20}v}{\cancel{20}} = \frac{-80}{20}$$
6. **Simplify the fraction:**
$$v = -4$$
7. **Check for restrictions:** The denominators $v+1$ and $v+7$ cannot be zero.
- $v + 1 \neq 0 \Rightarrow v \neq -1$
- $v + 7 \neq 0 \Rightarrow v \neq -7$
Since $v = -4$ does not violate these, it is a valid solution.
**Final answer:**
$$v = -4$$
There is only one solution for $v$.