( x/x+1+1)*1+x/2x-1
To simplify the expression \( \left( \frac{x}{x+1} + 1 \right) \cdot 1 + \frac{x}{2x - 1} \), let's break this down step by step.
Simplify the first part: \( \frac{x}{x+1} + 1 \)
\[ \frac{x}{x+1} + 1 = \frac{x}{x+1} + \frac{x+1}{x+1} = \frac{x + (x + 1)}{x + 1} = \frac{2x + 1}{x + 1} \]
So, the expression becomes:
\[ \left( \frac{2x + 1}{x + 1} \right) + \frac{x}{2x - 1} \]
Now, find a common denominator to combine these two fractions. The common denominator will be \((x + 1)(2x - 1)\).
Rewrite each fraction with the common denominator:
The first fraction:
\[ \frac{2x + 1}{x + 1} \cdot \frac{(2x - 1)}{(2x - 1)} = \frac{(2x + 1)(2x - 1)}{(x + 1)(2x - 1)} \]
The second fraction:
\[ \frac{x}{2x - 1} \cdot \frac{(x + 1)}{(x + 1)} = \frac{x(x + 1)}{(2x - 1)(x + 1)} \]
Combine the two fractions:
\[ \frac{(2x + 1)(2x - 1) + x(x + 1)}{(x + 1)(2x - 1)} \]
Now, simplify the numerator:
Expand \( (2x + 1)(2x - 1) \):
\[ (2x + 1)(2x - 1) = 4x^2 - 2x + 2x - 1 = 4x^2 - 1 \]
And for \( x(x + 1) \), we have:
\[ x(x + 1) = x^2 + x \]
So combining everything gives:
\[ 4x^2 - 1 + x^2 + x = 5x^2 + x - 1 \]
As a result, our simplified expression is:
\[ \frac{5x^2 + x - 1}{(x + 1)(2x - 1)} \]
This is the simplified form of the original expression. If you need to evaluate it or substitute specific values, let me know!
