https://file.aiquickdraw.com/mathbot/file/172635780056087jwe69d.jpg

The equation in the image involves simplifying a complex algebraic expression. Let's break it down step by step to solve and simplify it.

The given equation is:

\[
\frac{\frac{1}{x+7} - \frac{1}{x}}{7} \cdot \frac{x(x+7)}{x(x+7)} = \frac{x(x+7) \cdot \left( \frac{1}{x+7} - \frac{1}{x} \right)}{7x(x+7)}
\]

We need to simplify the expression step by step:

### Step 1: Simplify the difference inside the parentheses.
We are given \( \frac{1}{x+7} - \frac{1}{x} \).

To subtract these, we need a common denominator:

\[
\frac{1}{x+7} - \frac{1}{x} = \frac{x - (x+7)}{x(x+7)} = \frac{x - x - 7}{x(x+7)} = \frac{-7}{x(x+7)}
\]

### Step 2: Substitute this result back into the equation.
The equation now becomes:

\[
\frac{\frac{-7}{x(x+7)}}{7} \cdot \frac{x(x+7)}{x(x+7)}
\]

### Step 3: Simplify the expression.
The first part is:

\[
\frac{\frac{-7}{x(x+7)}}{7} = \frac{-7}{7x(x+7)} = \frac{-1}{x(x+7)}
\]

Now multiplying by \( \frac{x(x+7)}{x(x+7)} \):

\[
\frac{-1}{x(x+7)} \cdot \frac{x(x+7)}{x(x+7)} = \frac{-1}{x(x+7)}
\]

Thus, the simplified result is:

\[
\boxed{\frac{-1}{x(x+7)}}
\]

This is the final simplified expression.

Would you like further clarification or any more details on this problem?

Here are 5 related questions for further practice:
1. How would you subtract \( \frac{1}{x+5} - \frac{1}{x} \)?
2. Can you solve \( \frac{1}{a+3} + \frac{1}{a} \)?
3. Simplify \( \frac{2}{y+4} - \frac{3}{y} \).
4. How do you simplify \( \frac{x^2}{x+2} - \frac{1}{x+2} \)?
5. What happens when you multiply \( \frac{1}{x+3} \cdot \frac{1}{x-3} \)?

**Tip:** Always find a common denominator when adding or subtracting fractions with different denominators.

Fill in the blank(s) and simplify the given algebraic expression: \( \frac{\frac{1}{x+7} - \frac{1}{x}}{7} \cdot \frac{x(x+7)}{x(x+7)} = \frac{x(x+7) \cdot \left( \frac{1}{x+7} - \frac{1}{x} \right)}{7x(x+7)} \)

Learn how to simplify complex rational expressions involving fractions. This step-by-step solution covers key algebraic concepts such as simplifying differences of fractions and rational expressions. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation