To simplify the expression \(\frac{\frac{2}{x} - \frac{3}{x+1}}{\frac{4}{x+1} + \frac{5}{x}} \), follow these steps:

### Step 1: Simplify the numerator
The numerator is \(\frac{2}{x} - \frac{3}{x+1}\). To combine these fractions, find a common denominator:

\[
\text{Common denominator} = x(x + 1)
\]

Thus, the numerator becomes:

\[
\frac{2(x + 1)}{x(x + 1)} - \frac{3x}{x(x + 1)} = \frac{2x + 2 - 3x}{x(x + 1)} = \frac{-x + 2}{x(x + 1)}
\]

### Step 2: Simplify the denominator
The denominator is \(\frac{4}{x+1} + \frac{5}{x}\). Again, find a common denominator:

\[
\text{Common denominator} = x(x + 1)
\]

Thus, the denominator becomes:

\[
\frac{4x}{x(x + 1)} + \frac{5(x + 1)}{x(x + 1)} = \frac{4x + 5x + 5}{x(x + 1)} = \frac{9x + 5}{x(x + 1)}
\]

### Step 3: Divide the numerator by the denominator
Now, divide the simplified numerator by the simplified denominator:

\[
\frac{\frac{-x + 2}{x(x + 1)}}{\frac{9x + 5}{x(x + 1)}}
\]

When dividing fractions, multiply by the reciprocal:

\[
\frac{-x + 2}{x(x + 1)} \times \frac{x(x + 1)}{9x + 5} = \frac{-x + 2}{9x + 5}
\]

### Final Answer:
The simplified expression is:

\[
\frac{-x + 2}{9x + 5}
\]

Would you like further details on any of these steps?

---

Here are 5 related questions:

1. How do you find the least common denominator for complex fractions?
2. How can you simplify complex rational expressions?
3. What are the steps to multiply and divide rational expressions?
4. How do you handle negative terms in fractions?
5. How do you factor polynomials in the numerator or denominator of a fraction?

**Tip:** When simplifying complex fractions, always look for opportunities to cancel out common factors after finding a common denominator.

Learn how to simplify the complex fraction (2/x - 3/(x+1)) / (4/(x+1) + 5/x) step-by-step with detailed explanations. This problem involves fraction simplification, finding common denominators, and working with rational expressions. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation