(x-3)/(x-2) + (x-2)/(x-3) = 2+1/2

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To solve this equation, we need to find a common denominator for the two fractions on the left side of the equation.

First, let's rewrite the equation:

(x-3)/(x-2) + (x-2)/(x-3) = 2 + 1/2

Now, let's find a common denominator. The common denominator for (x-2) and (x-3) is (x-2)(x-3).

So, rewrite the equation with the common denominator:

[(x-3)^2 + (x-2)^2] / [(x-2)(x-3)] = 2 + 1/2

Now expand the numerator:

[(x^2 - 6x + 9) + (x^2 - 4x + 4)] / [(x-2)(x-3)] = 2 + 1/2

Simplify the expression:

[2x^2 - 10x + 13] / [(x-2)(x-3)] = 5/2

Now, cross multiply to get rid of the denominator:

2x^2 - 10x + 13 = 5(x-2)(x-3)

Expand the right side:

2x^2 - 10x + 13 = 5(x^2 - 5x + 6)

2x^2 - 10x + 13 = 5x^2 - 25x + 30

Rearrange the equation to set it equal to zero:

3x^2 - 15x + 17 = 0

Unfortunately, this is a quadratic equation that does not factor easily. We can use the quadratic formula to solve for x:

x = [15 ± sqrt((-15)^2 - 4(3)(17))] / 6

x = [15 ± sqrt(225 - 204)] / 6

x = [15 ± sqrt(21)] / 6

Therefore, the solutions for x are:

x = (15 + sqrt(21)) / 6 and x = (15 - sqrt(21)) / 6