1. **State the problem:** Find the value of $x$ such that $$\frac{x - 2}{x - 3} + \frac{x + 3}{x + 2} = 2.$$\n\n2. **Formula and rules:** To solve equations involving rational expressions, first find a common denominator to combine terms, then solve the resulting equation. Remember to check for values that make denominators zero, as these are excluded from the solution.\n\n3. **Find the common denominator:** The denominators are $x - 3$ and $x + 2$. The common denominator is $(x - 3)(x + 2)$.\n\n4. **Rewrite each fraction with the common denominator:**\n$$\frac{(x - 2)(x + 2)}{(x - 3)(x + 2)} + \frac{(x + 3)(x - 3)}{(x + 2)(x - 3)} = 2.$$\n\n5. **Combine the fractions:**\n$$\frac{(x - 2)(x + 2) + (x + 3)(x - 3)}{(x - 3)(x + 2)} = 2.$$\n\n6. **Expand the numerators:**\n$$(x - 2)(x + 2) = x^2 - 4,$$\n$$(x + 3)(x - 3) = x^2 - 9.$$\n\n7. **Sum the numerators:**\n$$x^2 - 4 + x^2 - 9 = 2x^2 - 13.$$\n\n8. **Rewrite the equation:**\n$$\frac{2x^2 - 13}{(x - 3)(x + 2)} = 2.$$\n\n9. **Multiply both sides by the denominator to clear fractions:**\n$$2x^2 - 13 = 2(x - 3)(x + 2).$$\n\n10. **Expand the right side:**\n$$(x - 3)(x + 2) = x^2 - 3x + 2x - 6 = x^2 - x - 6.$$\n\n11. **Substitute back:**\n$$2x^2 - 13 = 2(x^2 - x - 6) = 2x^2 - 2x - 12.$$\n\n12. **Bring all terms to one side:**\n$$2x^2 - 13 - 2x^2 + 2x + 12 = 0,$$\nwhich simplifies to\n$$2x - 1 = 0.$$\n\n13. **Solve for $x$:**\n$$2x = 1 \implies x = \frac{1}{2}.$$\n\n14. **Check for excluded values:**\nDenominators are zero if $x = 3$ or $x = -2$. Since $x = \frac{1}{2}$ is neither, it is valid.\n\n**Final answer:**\n$$\boxed{\frac{1}{2}}.$$