Solve the following inequations :
`(2x + 4)/(x-1) ge 5`

Answer

To solve the inequality \((2x + 4)/(x - 1) \geq 5\), we will follow these steps:

### Step 1: Rearrange the Inequality
We start by moving \(5\) to the left-hand side of the inequality:

\[ \frac{2x + 4}{x - 1} - 5 \geq 0 \]

### Step 2: Combine the Terms
Next, we need to combine the terms on the left-hand side. To do this, we express \(5\) with a common denominator:

\[ \frac{2x + 4 - 5(x - 1)}{x - 1} \geq 0 \]

This simplifies to:

\[ \frac{2x + 4 - 5x + 5}{x - 1} \geq 0 \]

### Step 3: Simplify the Numerator
Now, we simplify the numerator:

\[ \frac{-3x + 9}{x - 1} \geq 0 \]

### Step 4: Factor the Numerator
We can factor out \(-3\) from the numerator:

\[ \frac{-3(x - 3)}{x - 1} \geq 0 \]

### Step 5: Analyze the Inequality
To solve the inequality, we need to determine where the expression is greater than or equal to zero. This occurs when the numerator and denominator have the same sign or when the numerator is zero.

### Step 6: Identify Critical Points
The critical points occur when the numerator and denominator are zero:

- Numerator: \(-3(x - 3) = 0 \Rightarrow x = 3\)
- Denominator: \(x - 1 = 0 \Rightarrow x = 1\)

### Step 7: Test Intervals
We will test the intervals determined by the critical points \(x = 1\) and \(x = 3\):

1. **Interval \( (-\infty, 1) \)**: Choose \(x = 0\):
\[ \frac{-3(0 - 3)}{0 - 1} = \frac{9}{-1} < 0 \]

2. **Interval \( (1, 3) \)**: Choose \(x = 2\):
\[ \frac{-3(2 - 3)}{2 - 1} = \frac{3}{1} > 0 \]

3. **Interval \( (3, \infty) \)**: Choose \(x = 4\):
\[ \frac{-3(4 - 3)}{4 - 1} = \frac{-3}{3} < 0 \]

### Step 8: Include Critical Points
Now we check the critical points:
- At \(x = 3\), the expression equals zero.
- At \(x = 1\), the expression is undefined.

### Final Solution
The solution to the inequality is:

\[ x \in [3, \infty) \quad \text{and} \quad x \neq 1 \]

### Summary
Thus, the final solution can be expressed as:

\[ x \in (1, 3] \cup (3, \infty) \]