https://file.aiquickdraw.com/mathbot/file/17273613303469wuw5tyi.jpg

Let's simplify the given expression step by step. The expression in the image is:

\[
\frac{\frac{1}{x+7} + \frac{1}{x}}{\frac{2}{x}}
\]

### Step 1: Simplify the numerator.
The numerator is:
\[
\frac{1}{x+7} + \frac{1}{x}
\]
We need to combine the two terms by finding a common denominator. The common denominator of \(x+7\) and \(x\) is \((x+7)(x)\).

So, we rewrite the terms:
\[
\frac{1}{x+7} = \frac{x}{x(x+7)}, \quad \frac{1}{x} = \frac{x+7}{x(x+7)}
\]
Now, adding these:
\[
\frac{x}{x(x+7)} + \frac{x+7}{x(x+7)} = \frac{x + (x+7)}{x(x+7)} = \frac{2x+7}{x(x+7)}
\]

### Step 2: Simplify the denominator.
The denominator is:
\[
\frac{2}{x}
\]

### Step 3: Write the expression as a division.
We now have:
\[
\frac{\frac{2x+7}{x(x+7)}}{\frac{2}{x}}
\]
When dividing fractions, multiply by the reciprocal of the denominator:
\[
\frac{2x+7}{x(x+7)} \times \frac{x}{2} = \frac{(2x+7) \cdot x}{2 \cdot x(x+7)}
\]

### Step 4: Simplify the expression.
Cancel out the \(x\) terms:
\[
\frac{(2x+7)}{2(x+7)}
\]

### Final answer:
The simplified expression is:
\[
\frac{2x+7}{2(x+7)}
\]

Let me know if you need further clarifications or details!

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Here are 5 related questions you might find interesting:
1. How do you simplify more complex rational expressions?
2. How can you handle addition and subtraction of rational expressions with different denominators?
3. What is the process for multiplying and dividing fractions in algebraic expressions?
4. How does factorization help in simplifying expressions?
5. What are some common mistakes to avoid when simplifying complex fractions?

**Tip:** Always look for opportunities to factor expressions when simplifying, as it can help reduce the complexity.

Learn how to simplify the rational expression (1/(x+7) + 1/x) / (2/x) with a step-by-step guide. This example covers key algebraic concepts, including finding a common denominator and applying division of fractions. Suitable for students in Grades 9-12.

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