Express \( \frac{1}{x+1} - \frac{1}{x-2} \) as a single algebraic fraction

Answer Details

To express the given expression as a single algebraic fraction, we first need to find a common denominator for all the terms. The common denominator is (x+1)(2-x). Then, we can simplify each term by multiplying the numerator and denominator by the missing factor in the denominator to obtain: 1(x+1)(2-x) / (x+1) + 1(2-x) / (2-x)(x+1) - 1x(x+1) / (2-x)(x+1) Simplifying further by combining like terms, we get: [(x+1)(2-x) + 1(1-x)(x+1) - x(x+1)] / [(2-x)(x+1)] Simplifying the numerator by distributing, we get: [-x^2 + 3x - 1] / [(2-x)(x+1)] Therefore, the expression 1x+1 / (x+1) + 1 / (2-x) - 1x?2 / (x+1)(2-x) can be simplified to -[-x^2 + 3x - 1] / [(2-x)(x+1)]. So, the answer is (a) -3(x+1)(2-x) / [-x^2 + 3x - 1].