Solve each of the following equatins :
`((2x-3)/(x-1))-4((x-1)/(2x-3))=3,xne1,(3)/(2)`

Answer

To solve the equation

\[ \frac{2x-3}{x-1} - 4\frac{x-1}{2x-3} = 3 \]

where \( x \neq 1 \) and \( x \neq \frac{3}{2} \), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the fractions on the left side is \((x-1)(2x-3)\).

### Step 2: Rewrite the equation
We can rewrite the equation by multiplying both sides by the common denominator:

\[ (2x-3)(2x-3) - 4(x-1)(x-1) = 3(x-1)(2x-3) \]

### Step 3: Expand both sides
Now we will expand each term:

1. Left side:
\[ (2x-3)(2x-3) = (2x-3)^2 = 4x^2 - 12x + 9 \]
\[ -4(x-1)(x-1) = -4(x^2 - 2x + 1) = -4x^2 + 8x - 4 \]

Combining these gives:
\[ 4x^2 - 12x + 9 - 4x^2 + 8x - 4 = -4x + 5 \]

2. Right side:
\[ 3(x-1)(2x-3) = 3(2x^2 - 3x - 2x + 3) = 3(2x^2 - 5x + 3) = 6x^2 - 15x + 9 \]

### Step 4: Set the equation to zero
Now we set the equation to zero by moving all terms to one side:

\[ -4x + 5 - (6x^2 - 15x + 9) = 0 \]

This simplifies to:

\[ -6x^2 + 11x - 4 = 0 \]

Multiplying through by -1 gives:

\[ 6x^2 - 11x + 4 = 0 \]

### Step 5: Use the quadratic formula
Now we will use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 6, b = -11, c = 4 \).

1. Calculate the discriminant:
\[ b^2 - 4ac = (-11)^2 - 4 \cdot 6 \cdot 4 = 121 - 96 = 25 \]

2. Substitute into the quadratic formula:
\[ x = \frac{11 \pm \sqrt{25}}{2 \cdot 6} = \frac{11 \pm 5}{12} \]

### Step 6: Calculate the two possible values for x
1. First solution:
\[ x = \frac{11 + 5}{12} = \frac{16}{12} = \frac{4}{3} \]

2. Second solution:
\[ x = \frac{11 - 5}{12} = \frac{6}{12} = \frac{1}{2} \]

### Final Solutions
Thus, the solutions to the equation are:

\[ x = \frac{4}{3} \quad \text{and} \quad x = \frac{1}{2} \]