`|(2x+1)/(x-1)| gt 2`

Answer

To solve the inequality \(\left|\frac{2x+1}{x-1}\right| > 2\), we will break it down into two cases based on the properties of absolute values.

### Step 1: Break down the absolute value
The inequality \(\left|\frac{2x+1}{x-1}\right| > 2\) can be expressed as two separate inequalities:
1. \(\frac{2x+1}{x-1} > 2\)
2. \(\frac{2x+1}{x-1} < -2\)

### Step 2: Solve the first inequality
For the first inequality \(\frac{2x+1}{x-1} > 2\):
- Rearranging gives:
\[ \frac{2x+1}{x-1} - 2 > 0 \]
- Finding a common denominator:
\[ \frac{2x+1 - 2(x-1)}{x-1} > 0 \]
- Simplifying the numerator:
\[ \frac{2x + 1 - 2x + 2}{x-1} > 0 \implies \frac{3}{x-1} > 0 \]
- This inequality holds when the denominator is positive:
\[ x - 1 > 0 \implies x > 1 \]

### Step 3: Solve the second inequality
For the second inequality \(\frac{2x+1}{x-1} < -2\):
- Rearranging gives:
\[ \frac{2x+1}{x-1} + 2 < 0 \]
- Finding a common denominator:
\[ \frac{2x+1 + 2(x-1)}{x-1} < 0 \]
- Simplifying the numerator:
\[ \frac{2x + 1 + 2x - 2}{x-1} < 0 \implies \frac{4x - 1}{x-1} < 0 \]
- To find when this is negative, we need to consider the signs of the numerator and denominator:
- The numerator \(4x - 1 = 0\) gives \(x = \frac{1}{4}\).
- The denominator \(x - 1 = 0\) gives \(x = 1\).

### Step 4: Analyze the sign changes
We will analyze the sign of \(\frac{4x - 1}{x - 1}\) across the intervals:
1. \(x < \frac{1}{4}\)
2. \(\frac{1}{4} < x < 1\)
3. \(x > 1\)

- For \(x < \frac{1}{4}\): Both \(4x - 1 < 0\) and \(x - 1 < 0\) → Positive.
- For \(\frac{1}{4} < x < 1\): \(4x - 1 > 0\) and \(x - 1 < 0\) → Negative.
- For \(x > 1\): Both \(4x - 1 > 0\) and \(x - 1 > 0\) → Positive.

### Step 5: Combine the results
From the analysis:
- The first inequality gives \(x > 1\).
- The second inequality gives \(\frac{1}{4} < x < 1\).

Thus, the solution to the original inequality \(\left|\frac{2x+1}{x-1}\right| > 2\) is:
\[ \left(\frac{1}{4}, 1\right) \cup (1, \infty) \]

### Final Answer
The solution is:
\[ \left(\frac{1}{4}, 1\right) \cup (1, \infty) \]