The solution of the equation $\frac{2}{3x - 4} + \frac{2}{2x - 6}$ = 0 is:

Q: The solution of the equation $\frac{2}{3x - 4} + \frac{2}{2x - 6}$ = 0 is:

To solve the equation $\frac{2}{3x - 4} + \frac{2}{2x - 6} = 0$, we need to find the value of $x$ that satisfies this equation. ### Step-by-Step Solution: 1. **Combine the Fractions:** The equation is $\frac{2}{3x - 4} + \frac{2}{2x - 6} = 0$. To combine these fractions, we need a common denominator. The denominators are $3x - 4$ and $2x - 6$. 2. **Simplify the Denominators:** Notice that $2x - 6$ can be factored as $2(x - 3)$. 3. **Find the Common Denominator:** The common denominator of $3x - 4$ and $2(x - 3)$ is $(3x - 4)(2(x - 3))$. 4. **Rewrite the Equation:** \[ \frac{2(2(x - 3)) + 2(3x - 4)}{(3x - 4)(2(x - 3))} = 0 \] 5. **Set the Numerator to Zero:** Since the fraction equals zero, the numerator must be zero: \[ 2(2(x - 3)) + 2(3x - 4) = 0 \] 6. **Expand and Simplify:** \[ 4(x - 3) + 2(3x - 4) = 0 \] \[ 4x - 12 + 6x - 8 = 0 \] \[ 10x - 20 = 0 \] 7. **Solve for $x$:** \[ 10x = 20 \] \[ x = 2 \] ### Explanation of Options: - **Option 1 (0):** If $x = 0$, substituting into the original equation does not satisfy the equation, as it does not make the numerator zero. - **Option 2 (1):** If $x = 1$, substituting into the original equation does not satisfy the equation, as it does not make the numerator zero. - **Option 3 (3):** If $x = 3$, the denominator $2(x - 3)$ becomes zero, which makes the expression undefined. - **Option 4 (2):** As shown in the solution, substituting $x = 2$ satisfies the equation, making the numerator zero and thus the entire expression zero. Therefore, the correct answer is Option 4, which is $x = 2$.

Question

Accepted Answer

To solve the equation $\frac{2}{3x - 4} + \frac{2}{2x - 6} = 0$, we need to find the value of $x$ that satisfies this equation.

### Step-by-Step Solution:

1. **Combine the Fractions:**
   The equation is $\frac{2}{3x - 4} + \frac{2}{2x - 6} = 0$. To combine these fractions, we need a common denominator. The denominators are $3x - 4$ and $2x - 6$.

2. **Simplify the Denominators:**
   Notice that $2x - 6$ can be factored as $2(x - 3)$.

3. **Find the Common Denominator:**
   The common denominator of $3x - 4$ and $2(x - 3)$ is $(3x - 4)(2(x - 3))$.

4. **Rewrite the Equation:**
   $$
   \frac{2(2(x - 3)) + 2(3x - 4)}{(3x - 4)(2(x - 3))} = 0
   $$

5. **Set the Numerator to Zero:**
   Since the fraction equals zero, the numerator must be zero:
   $$
   2(2(x - 3)) + 2(3x - 4) = 0
   $$

6. **Expand and Simplify:**
   $$
   4(x - 3) + 2(3x - 4) = 0
   $$
   $$
   4x - 12 + 6x - 8 = 0
   $$
   $$
   10x - 20 = 0
   $$

7. **Solve for $x$:**
   $$
   10x = 20
   $$
   $$
   x = 2
   $$

### Explanation of Options:

- **Option 1 (0):** If $x = 0$, substituting into the original equation does not satisfy the equation, as it does not make the numerator zero.
- **Option 2 (1):** If $x = 1$, substituting into the original equation does not satisfy the equation, as it does not make the numerator zero.
- **Option 3 (3):** If $x = 3$, the denominator $2(x - 3)$ becomes zero, which makes the expression undefined.
- **Option 4 (2):** As shown in the solution, substituting $x = 2$ satisfies the equation, making the numerator zero and thus the entire expression zero.

Therefore, the correct answer is Option 4, which is $x = 2$.