Question: User: 14/x^2-4 -12/x+2 = 6/2-x
1. **State the problem:** Solve the equation $$\frac{14}{x^2-4} - \frac{12}{x+2} = \frac{6}{2-x}$$.
2. **Rewrite and analyze:** Note that $$x^2-4 = (x-2)(x+2)$$ and $$2-x = -(x-2)$$.
3. **Rewrite the equation using these factorizations:**
$$\frac{14}{(x-2)(x+2)} - \frac{12}{x+2} = \frac{6}{-(x-2)} = -\frac{6}{x-2}$$.
4. **Find a common denominator:** The common denominator is $$(x-2)(x+2)$$.
5. **Rewrite each term with the common denominator:**
$$\frac{14}{(x-2)(x+2)} - \frac{12(x-2)}{(x+2)(x-2)} = -\frac{6(x+2)}{(x-2)(x+2)}$$.
6. **Combine all terms over the common denominator:**
$$\frac{14 - 12(x-2) + 6(x+2)}{(x-2)(x+2)} = 0$$.
7. **Simplify the numerator:**
$$14 - 12x + 24 + 6x + 12 = 14 + 24 + 12 - 12x + 6x = 50 - 6x$$.
8. **Set numerator equal to zero:**
$$50 - 6x = 0$$.
9. **Solve for $x$:**
$$6x = 50$$
$$x = \frac{50}{6} = \frac{25}{3}$$.
10. **Check for restrictions:**
Denominators cannot be zero, so $x \neq 2$ and $x \neq -2$.
11. **Final answer:**
$$x = \frac{25}{3}$$ is the solution.