solve the inequation `(2x+4)/(x-1) ge 5` and represent this solution on the number line.

Options

A. `(1, 3)`
B. `(1, 3]`
C. `(-oo, 1) cup [3, oo)`
D. none of these

Answer

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

### Step 1: Rewrite the Inequality
We start with the original inequality:
\[ \frac{2x + 4}{x - 1} \geq 5 \]
We can rearrange it by moving 5 to the left side:
\[ \frac{2x + 4}{x - 1} - 5 \geq 0 \]

### Step 2: Combine the Terms
To combine the terms, we need a common denominator. The common denominator here is \(x - 1\):
\[ \frac{2x + 4 - 5(x - 1)}{x - 1} \geq 0 \]
Now, distribute the -5:
\[ \frac{2x + 4 - 5x + 5}{x - 1} \geq 0 \]
Combine like terms:
\[ \frac{-3x + 9}{x - 1} \geq 0 \]

### Step 3: Factor the Numerator
We can factor out a -3 from the numerator:
\[ \frac{-3(x - 3)}{x - 1} \geq 0 \]
This can be rewritten as:
\[ \frac{3(x - 3)}{x - 1} \leq 0 \]
(Note that the inequality sign flips because we factored out a negative number.)

### Step 4: Identify Critical Points
The critical points occur when the numerator or denominator is zero:
1. \(x - 3 = 0 \Rightarrow x = 3\)
2. \(x - 1 = 0 \Rightarrow x = 1\)

### Step 5: Test Intervals
We will test the intervals defined by these critical points: \((- \infty, 1)\), \((1, 3)\), and \((3, \infty)\).

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

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

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

### Step 6: Determine the Solution Set
The inequality \(\frac{3(x - 3)}{x - 1} \leq 0\) is satisfied in the interval \((1, 3)\). Since \(x = 1\) makes the denominator zero, it cannot be included in the solution set. However, \(x = 3\) can be included since it makes the numerator zero.

Thus, the solution set is:
\[ x \in (1, 3] \]

### Step 7: Represent on the Number Line
On the number line, we represent the solution as an open interval at 1 and a closed interval at 3.

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