The expression `(3x -1)/(x-4)` is equivalent to which of the following ?

Options

A. `1/2`
B. `3x - (1)/(x -4)`
C. `3- (11)/(x -4)`
D. `3 + (11)/(x -4)`

Answer

To solve the expression \((3x - 1)/(x - 4)\) and find its equivalent form, we can follow these steps:

### Step 1: Rewrite the Numerator
We start with the expression:
\[ \frac{3x - 1}{x - 4} \]
We can manipulate the numerator \(3x - 1\) to make it easier to factor. We notice that we can add and subtract 12 to the numerator:
\[ 3x - 1 = 3x - 12 + 11 \]
Thus, we can rewrite the expression as:
\[ \frac{3x - 12 + 11}{x - 4} \]

### Step 2: Factor the Numerator
Now, we can group the terms in the numerator:
\[ \frac{(3x - 12) + 11}{x - 4} \]
We can factor out a 3 from \(3x - 12\):
\[ \frac{3(x - 4) + 11}{x - 4} \]

### Step 3: Split the Fraction
Now, we can split the fraction into two parts:
\[ \frac{3(x - 4)}{x - 4} + \frac{11}{x - 4} \]

### Step 4: Simplify the Expression
The first term simplifies as follows:
\[ \frac{3(x - 4)}{x - 4} = 3 \]
Thus, we have:
\[ 3 + \frac{11}{x - 4} \]

### Final Expression
So, the expression \(\frac{3x - 1}{x - 4}\) is equivalent to:
\[ 3 + \frac{11}{x - 4} \]

### Conclusion
The equivalent expression is \(3 + \frac{11}{x - 4}\).

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