If `(3x+2)/(2x+3)=(4x+3)/(3x+4)`, then x =

Options

A. `+-2`
B. `+-4`
C. `+-1`
D. `+-3`

Answer

To solve the equation \(\frac{3x + 2}{2x + 3} = \frac{4x + 3}{3x + 4}\), we will follow these steps:

### Step 1: Cross Multiply
We start by cross-multiplying the fractions to eliminate the denominators.

\[ (3x + 2)(3x + 4) = (4x + 3)(2x + 3) \]

### Step 2: Expand Both Sides
Next, we will expand both sides of the equation.

Left side:
\[ (3x + 2)(3x + 4) = 3x \cdot 3x + 3x \cdot 4 + 2 \cdot 3x + 2 \cdot 4 = 9x^2 + 12x + 6x + 8 = 9x^2 + 18x + 8 \]

Right side:
\[ (4x + 3)(2x + 3) = 4x \cdot 2x + 4x \cdot 3 + 3 \cdot 2x + 3 \cdot 3 = 8x^2 + 12x + 6x + 9 = 8x^2 + 18x + 9 \]

### Step 3: Set the Equation
Now we set the two expanded forms equal to each other:

\[ 9x^2 + 18x + 8 = 8x^2 + 18x + 9 \]

### Step 4: Move All Terms to One Side
To simplify, we will move all terms to one side of the equation:

\[ 9x^2 + 18x + 8 - 8x^2 - 18x - 9 = 0 \]

This simplifies to:

\[ x^2 - 1 = 0 \]

### Step 5: Factor the Quadratic
Next, we can factor the quadratic equation:

\[ (x - 1)(x + 1) = 0 \]

### Step 6: Solve for x
Setting each factor equal to zero gives us the solutions:

\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

### Final Answer
Thus, the values of \(x\) are:

\[ x = \pm 1 \]