To solve the equation [tex]\(\frac{x-2}{3} = \frac{2}{x+3}\)[/tex], we can follow these steps:
1. Eliminate the denominators by cross-multiplying:
[tex]\[
(x - 2)(x + 3) = 3 \cdot 2
\][/tex]
2. Expand the left-hand side:
[tex]\[
x^2 + 3x - 2x - 6 = 6
\][/tex]
Simplify:
[tex]\[
x^2 + x - 6 = 6
\][/tex]
3. Set the equation to zero by moving all terms to one side:
[tex]\[
x^2 + x - 6 - 6 = 0
\][/tex]
Which simplifies to:
[tex]\[
x^2 + x - 12 = 0
\][/tex]
4. Factor the quadratic equation [tex]\(x^2 + x - 12 = 0\)[/tex]. We look for two numbers that multiply to [tex]\(-12\)[/tex] and add up to [tex]\(1\)[/tex]:
[tex]\[
(x + 4)(x - 3) = 0
\][/tex]
5. Solve for [tex]\(x\)[/tex] by setting each factor to zero:
[tex]\[
x + 4 = 0 \quad \text{or} \quad x - 3 = 0
\][/tex]
So,
[tex]\[
x = -4 \quad \text{or} \quad x = 3
\][/tex]
6. Check the possible values given in the options to see which match our solutions. The options are:
- [tex]\(-7\)[/tex]
- [tex]\(-6\)[/tex]
- [tex]\(-5\)[/tex]
- [tex]\(-4\)[/tex]
- [tex]\(-3\)[/tex]
7. Among the possible values, the value [tex]\(-4\)[/tex] is listed and matches one of our solutions.
Therefore, the possible value of [tex]\(x\)[/tex] is [tex]\(\boxed{-4}\)[/tex].
