1Cademy - Try It 7.53: Simplifying $$\frac{b-\frac{3b}{b+5}}{\frac{2}{b+5}+\frac{1}{b-5}}$$ by Writing it as Division

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Procedure to Simplify a Complex Rational Expression by Writing it as Division

Example

Try It 7.53: Simplifying $\frac{b-\frac{3b}{b+5}}{\frac{2}{b+5}+\frac{1}{b-5}}$ by Writing it as Division

To simplify the complex rational expression $\frac{b-\frac{3b}{b+5}}{\frac{2}{b+5}+\frac{1}{b-5}}$ by writing it as division, follow these steps:

Step 1. Simplify the numerator and denominator. In the numerator, the common denominator is $b+5$ . Rewrite $b$ as $\frac{b(b+5)}{b+5}$ . Subtracting gives $\frac{b(b+5)-3b}{b+5} = \frac{b^2+5b-3b}{b+5} = \frac{b^2+2b}{b+5}$ . In the denominator, the common denominator is $(b+5)(b-5)$ . Rewrite as $\frac{2(b-5)}{(b+5)(b-5)} + \frac{1(b+5)}{(b+5)(b-5)} = \frac{2b-10+b+5}{(b+5)(b-5)} = \frac{3b-5}{(b+5)(b-5)}$ . The complex fraction becomes: $\frac{\frac{b^2+2b}{b+5}}{\frac{3b-5}{(b+5)(b-5)}}$ .

Step 2. Rewrite as fraction division. Replace the main fraction bar with a division symbol: $\frac{b^2+2b}{b+5} \div \frac{3b-5}{(b+5)(b-5)}$

Step 3. Multiply by the reciprocal and simplify. Change the division to multiplication by the reciprocal: $\frac{b^2+2b}{b+5} \cdot \frac{(b+5)(b-5)}{3b-5}$ Factor the numerator $b^2+2b$ to $b(b+2)$ . $\frac{b(b+2)(b+5)(b-5)}{(b+5)(3b-5)}$ Divide out the common factor of $(b+5)$ . The simplified result is: $\frac{b(b+2)(b-5)}{3b-5}$

Updated 2026-04-30

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OpenStax Intermediate Algebra

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