Let's simplify the given expression step-by-step. The expression is:
[tex]\[ 1 + \frac{3}{2x - 1} - \frac{x - 1}{6x^2 - 3x} \][/tex]
First, let's break down the components:
1. Consider the term \(\frac{x - 1}{6x^2 - 3x}\):
The denominator \(6x^2 - 3x\) can be factored out:
[tex]\[ 6x^2 - 3x = 3x(2x - 1) \][/tex]
So, [tex]\[ \frac{x - 1}{6x^2 - 3x} = \frac{x - 1}{3x(2x - 1)} \][/tex]
2. Now substitute this back into the original expression:
[tex]\[ 1 + \frac{3}{2x - 1} - \frac{x - 1}{3x (2x - 1)} \][/tex]
3. To combine these fractions, we need a common denominator. The least common multiple of the denominators \(1\), \(2x - 1\), and \(3x(2x - 1)\) is \(3x(2x - 1)\). Rewrite each term with this common denominator:
[tex]\[ \frac{3x(2x - 1)}{3x(2x - 1)} + \frac{3 \cdot 3x}{(2x - 1) \cdot 3x} - \frac{(x - 1)}{3x (2x - 1)} \][/tex]
Simplify each term:
- The first term:
[tex]\[ 1 = \frac{3x(2x - 1)}{3x(2x - 1)} \][/tex]
- The second term:
[tex]\[ \frac{3}{2x - 1} = \frac{3 \cdot 3x}{(2x - 1) \cdot 3x} = \frac{9x}{3x(2x - 1)} \][/tex]
- The third term is already in its simplest form:
[tex]\[ - \frac{x - 1}{3x(2x - 1)} \][/tex]
4. Combine the fractions:
[tex]\[
\frac{3x(2x - 1) + 9x - (x - 1)}{3x(2x - 1)}
\][/tex]
5. Simplify the numerator:
[tex]\[
\frac{3x(2x - 1) + 9x - x + 1}{3x(2x - 1)}
\][/tex]
Distribute in the first term:
[tex]\[
\frac{6x^2 - 3x + 9x - x + 1}{3x(2x - 1)}
\][/tex]
Combine like terms:
[tex]\[
\frac{6x^2 + 5x + 1}{3x(2x - 1)}
\][/tex]
Thus, the simplified form of the expression \(1 + \frac{3}{2x - 1} - \frac{x - 1}{6x^2 - 3x}\) is:
[tex]\[
\boxed{\frac{6x^2 + 5x + 1}{3x(2x - 1)}}
\][/tex]
Refering to the options given:
[tex]\[\frac{(3 x+1)(2 x-1)}{3 x(2 x+1)}\][/tex]
[tex]\[\frac{(3 x+1)(2 x+1)}{3 x(2 x-1)}\][/tex]
[tex]\[\frac{3 x+1}{3 x}\][/tex]
[tex]\[\frac{2 x+1}{2 x-1}\][/tex]
The exact match is within the options:
[tex]\[
\boxed{\frac{(3 x+1)(2 x+1)}{3 x(2 x-1)}}
\][/tex]
