`(x)/(2x-1)=(2x+1)/(x+2)`
If m and n represents the solutions of the equations above, what is the value of m+n?

Options

A. `-(4)/(3)`
B. `-(3)/(4)`
C. `(2)/(3)`
D. `(5)/(4)`

Answer

To solve the equation \(\frac{x}{2x - 1} = \frac{2x + 1}{x + 2}\), we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying the fractions to eliminate the denominators:
\[ x \cdot (x + 2) = (2x + 1) \cdot (2x - 1) \]

### Step 2: Expand Both Sides
Now, we will expand both sides of the equation:
\[ x^2 + 2x = (2x + 1)(2x - 1) \]
Using the difference of squares on the right side:
\[ x^2 + 2x = 4x^2 - 1 \]

### Step 3: Rearrange the Equation
Next, we will move all terms to one side of the equation:
\[ x^2 + 2x - 4x^2 + 1 = 0 \]
This simplifies to:
\[ -3x^2 + 2x + 1 = 0 \]
To make it easier to work with, we can multiply through by -1:
\[ 3x^2 - 2x - 1 = 0 \]

### Step 4: Factor the Quadratic Equation
Now, we will factor the quadratic equation \(3x^2 - 2x - 1 = 0\). We look for two numbers that multiply to \(3 \cdot -1 = -3\) and add to \(-2\). These numbers are \(-3\) and \(1\):
\[ 3x^2 - 3x + x - 1 = 0 \]
Grouping the terms:
\[ 3x(x - 1) + 1(x - 1) = 0 \]
Factoring out \((x - 1)\):
\[ (3x + 1)(x - 1) = 0 \]

### Step 5: Solve for \(x\)
Setting each factor to zero gives us the solutions:
1. \(3x + 1 = 0 \Rightarrow x = -\frac{1}{3}\)
2. \(x - 1 = 0 \Rightarrow x = 1\)

Let \(m = 1\) and \(n = -\frac{1}{3}\).

### Step 6: Calculate \(m + n\)
Now we calculate \(m + n\):
\[ m + n = 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \]

Thus, the value of \(m + n\) is:
\[ \boxed{\frac{2}{3}} \]