Adding and Subtracting Rational Expressions

Adding rational expressions means rewriting each over the least common denominator, combining the numerators, and simplifying the result.

Add 1/x + 2/(x + 1). The LCD is x(x + 1). Rewrite each fraction: (x + 1)/(x(x + 1)) plus 2x/(x(x + 1)). Combine the numerators: x + 1 + 2x = 3x + 1, so the sum is (3x + 1)/(x(x + 1)). Spot-check at x = 1: the original is 1/1 + 2/2 = 2, and the result gives (3 + 1)/(1 · 2) = 2. Keep the LCD in factored form throughout — factored denominators show what cancels, and the final simplification is a factor hunt. Subtraction runs identically, with the minus sign distributed across the second numerator whole.