What is the difference between adding rational expressions and solving a rational equation?

When adding or subtracting rational expressions, you determine the LCD and then multiply both the numerator and denominator of each rational term by the appropriate factors(s) to get all fractions to have the same denominator. Then you can combine the numerators.

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

LCD = (x + 1)(x - 2)

1(x - 2)/[(x + 1)(x - 2)] + 1(x + 1)/[(x + 1)(x - 2)]

(x - 2 + x + 1)/[(x + 1)(x - 2)]

(2x-1)/[(x + 1)(x - 2)]

When solving an equation with rational expressions as terms, you determine the LCD and then multiply both sides of the equation by the LCD to eliminate all the denominators.

1/(x + 1) + 1/(x -2) = 5

LCD = (x + 1)(x - 2)

1[(x + 1)(x - 2)]/(x + 1) + 1[(x + 1)(x - 2)/(x - 2) = 5(x + 1)(x - 2)

(x - 2) + (x + 1) = 5(x^2 - x - 2)

2x - 1 = 5x^2 - 5x - 10

5x^2 - 7x - 9 = 0

Candidate for the quadratic formula