Math Problem Statement

Simplify the complex rational expression: (5/(x+5) - 7) / (3 + 4/(x+5))

Solution

The expression in the image needs to be simplified. It can be written as:

5x+573+4x+5\frac{\frac{5}{x+5} - 7}{3 + \frac{4}{x+5}}

Steps to simplify:

  1. Combine terms in the numerator: 5x+57=57(x+5)x+5=57x35x+5=7x30x+5\frac{5}{x+5} - 7 = \frac{5 - 7(x+5)}{x+5} = \frac{5 - 7x - 35}{x+5} = \frac{-7x - 30}{x+5}

  2. Combine terms in the denominator: 3+4x+5=3(x+5)+4x+5=3x+15+4x+5=3x+19x+53 + \frac{4}{x+5} = \frac{3(x+5) + 4}{x+5} = \frac{3x + 15 + 4}{x+5} = \frac{3x + 19}{x+5}

  3. Simplify the whole fraction: 7x30x+53x+19x+5=7x303x+19\frac{\frac{-7x - 30}{x+5}}{\frac{3x + 19}{x+5}} = \frac{-7x - 30}{3x + 19}

So, the simplified form is:

7x303x+19\frac{-7x - 30}{3x + 19}

Would you like more details on any step or a further explanation?

Here are five relative questions to expand your understanding:

  1. How do you simplify rational expressions with variables in both the numerator and denominator?
  2. What happens when you have common denominators in complex fractions?
  3. Can this simplified form be factored further?
  4. How would the result change if additional constraints were placed on xx?
  5. Can you apply this approach to solve similar complex fraction problems?

Tip: Always look for common denominators when simplifying expressions, especially with complex fractions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Complex Fractions

Formulas

Simplification of complex fractions
Combining fractions

Theorems

Distributive Property
Fraction Addition/Subtraction

Suitable Grade Level

Grades 9-11