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

Answer

To solve the equation

\[ \frac{2x-3}{x-1} - 4\left(\frac{x-1}{2x-3}\right) = 3, \quad x \neq 1, \left(\frac{3}{2}\right) \]

we will follow these steps:

### Step 1: Substitute for \( t \)
Let
\[ t = \frac{2x-3}{x-1} \]
Then, the equation can be rewritten as:
\[ t - 4 \cdot \frac{1}{t} = 3 \]

### Step 2: Clear the fraction
Multiply through by \( t \) (assuming \( t \neq 0 \)):
\[ t^2 - 4 = 3t \]

### Step 3: Rearrange the equation
Rearranging gives:
\[ t^2 - 3t - 4 = 0 \]

### Step 4: Factor the quadratic equation
We can factor the quadratic:
\[ (t - 4)(t + 1) = 0 \]

### Step 5: Solve for \( t \)
Setting each factor to zero gives:
\[ t - 4 = 0 \quad \Rightarrow \quad t = 4 \]
\[ t + 1 = 0 \quad \Rightarrow \quad t = -1 \]

### Step 6: Substitute back to find \( x \)
Now we substitute back for \( t \):

1. For \( t = 4 \):
\[ \frac{2x - 3}{x - 1} = 4 \]
Cross-multiplying gives:
\[ 2x - 3 = 4(x - 1) \]
Expanding and rearranging:
\[ 2x - 3 = 4x - 4 \quad \Rightarrow \quad -2x = -1 \quad \Rightarrow \quad x = \frac{1}{2} \]

2. For \( t = -1 \):
\[ \frac{2x - 3}{x - 1} = -1 \]
Cross-multiplying gives:
\[ 2x - 3 = -1(x - 1) \]
Expanding and rearranging:
\[ 2x - 3 = -x + 1 \quad \Rightarrow \quad 3x = 4 \quad \Rightarrow \quad x = \frac{4}{3} \]

### Final Solutions
The solutions to the equation are:
\[ x = \frac{1}{2} \quad \text{and} \quad x = \frac{4}{3} \]

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