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Adding 2x−2+5x+3\frac{2}{x-2} + \frac{5}{x+3}

To add the rational expressions 2x−2\frac{2}{x-2} and 5x+3\frac{5}{x+3}, first find the least common denominator (LCD). Since the denominators (x−2)(x-2) and (x+3)(x+3) share no common factors, the LCD is their product, (x−2)(x+3)(x-2)(x+3). Next, rewrite each expression as an equivalent fraction with this LCD by multiplying the numerator and denominator by the missing factor: 2(x+3)(x−2)(x+3)+5(x−2)(x+3)(x−2)\frac{2(x+3)}{(x-2)(x+3)} + \frac{5(x-2)}{(x+3)(x-2)}. Distribute the numerators to get 2x+6(x−2)(x+3)+5x−10(x−2)(x+3)\frac{2x+6}{(x-2)(x+3)} + \frac{5x-10}{(x-2)(x+3)}. Combine the numerators over the shared denominator: 2x+6+5x−10(x−2)(x+3)\frac{2x+6+5x-10}{(x-2)(x+3)}. Finally, simplify by combining like terms to obtain the final sum: 7x−4(x−2)(x+3)\frac{7x-4}{(x-2)(x+3)}.

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Updated 2026-04-30

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References

Tags

OpenStax

Intermediate Algebra @ OpenStax

Ch.7 Rational Expressions and Functions - Intermediate Algebra @ OpenStax

Algebra

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