`(3x)/(2x-6)+(9)/(6-2x)`
Which of the following is equivalent to the expression above?

Options

A. `-(3)/(2)`
B. `(-6x^(2)+36x-54)/((2x-6)(6-2x))`
C. `(3x+9)/(2x-6)`
D. `(3)/(2)`

Answer

To solve the inequality \(\frac{3x}{2x-6} + \frac{9}{6-2x}\), we will first rewrite the second term to have a common denominator.

### Step 1: Rewrite the second term
The expression can be rewritten as:
\[ \frac{3x}{2x-6} + \frac{9}{6-2x} = \frac{3x}{2x-6} + \frac{9}{-(2x-6)} = \frac{3x}{2x-6} - \frac{9}{2x-6} \]
This is because \(6 - 2x\) is equivalent to \(-(2x - 6)\).

### Step 2: Combine the fractions
Now that both fractions have the same denominator, we can combine them:
\[ \frac{3x - 9}{2x - 6} \]

### Step 3: Factor the numerator
Next, we can factor the numerator:
\[ 3x - 9 = 3(x - 3) \]
Thus, we can rewrite the expression as:
\[ \frac{3(x - 3)}{2x - 6} \]

### Step 4: Factor the denominator
Now, let's factor the denominator:
\[ 2x - 6 = 2(x - 3) \]
So the expression becomes:
\[ \frac{3(x - 3)}{2(x - 3)} \]

### Step 5: Cancel the common factors
Since \(x - 3\) is common in both the numerator and the denominator, we can cancel it (as long as \(x \neq 3\)):
\[ \frac{3}{2} \]

### Conclusion
Thus, the expression \(\frac{3x}{2x-6} + \frac{9}{6-2x}\) simplifies to:
\[ \frac{3}{2} \]

### Final Answer
The equivalent expression is \(\frac{3}{2}\), which corresponds to option D.