To solve this equation, we first need to find common denominators for all the fractions:

(2x - 7)/(x^2 - 9x + 14) - 1/(x^2 - 3x + 2) = 1/(x - 1)

The denominators are already factored, so the common denominator is (x - 7)(x - 2)(x - 1).

Next, rewrite the fractions with the common denominator:

[(2x - 7)(x - 2)]/[(x - 7)(x - 2)(x - 1)] - [(x - 7)(x - 1)]/[(x - 7)(x - 2)(x - 1)] = 1/(x - 1)

Multiply each term by the common denominator:

(2x - 7)(x - 2) - (x - 7)(x - 1) = (x - 7)(x - 2)

Expand and simplify each side:

2x^2 - 11x + 14 - x^2 + 8x - 7 = x^2 - 2x - 7

x^2 - 3x + 7 = x^2 - 2x - 7

Combine like terms:

-3x + 7 = -2x - 7

Add 2x to both sides:

-x + 7 = -7

Subtract 7 from both sides:

-x = -14

Multiply both sides by -1 to solve for x:

x = 14

Therefore, the solution to the equation is x = 14.