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Try It 7.59: Simplifying 3x+25x−2−3x2−4\frac{\frac{3}{x+2}}{\frac{5}{x-2}-\frac{3}{x^2-4}} Using the LCD

To simplify the complex rational expression 3x+25x−2−3x2−4\frac{\frac{3}{x+2}}{\frac{5}{x-2}-\frac{3}{x^2-4}} using the least common denominator (LCD) method:

Step 1. Find the LCD of all inner fractions. The inner denominators are (x+2)(x+2), (x−2)(x-2), and x2−4x^2-4. Recognize that x2−4x^2-4 is a difference of squares: x2−4=(x+2)(x−2)x^2-4 = (x+2)(x-2). Since this already contains the other two denominators as factors, the LCD is (x+2)(x−2)(x+2)(x-2).

Step 2. Multiply the numerator and denominator by the LCD, (x+2)(x−2)(x+2)(x-2). (x+2)(x−2)⋅3x+2(x+2)(x−2)⋅5x−2−(x+2)(x−2)⋅3(x+2)(x−2)\frac{(x+2)(x-2) \cdot \frac{3}{x+2}}{(x+2)(x-2) \cdot \frac{5}{x-2} - (x+2)(x-2) \cdot \frac{3}{(x+2)(x-2)}}

Step 3. Simplify by canceling common factors in each term. 3(x−2)5(x+2)−3\frac{3(x-2)}{5(x+2) - 3}

Step 4. Distribute and combine like terms in the denominator. 3(x−2)5x+10−3=3(x−2)5x+7\frac{3(x-2)}{5x + 10 - 3} = \frac{3(x-2)}{5x + 7}

The simplified expression is 3(x−2)5x+7\frac{3(x-2)}{5x + 7}.

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Updated 2026-04-30

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References

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OpenStax

Intermediate Algebra @ OpenStax

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

Algebra

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