solve the inequation `(x-2)/(x +5) gt2.`

Options

A. `x in ( -12, -5)`
B. `x in ( -12, 5)`
C. `x in ( 5, 12)`
D. None of these

Explanation

To solve the inequality \(\frac{x-2}{x+5} > 2\), we can follow these steps:

### Step 1: Rewrite the Inequality
Start by moving \(2\) to the left side of the inequality:
\[ \frac{x-2}{x+5} - 2 > 0 \]

### Step 2: Combine the Terms
To combine the terms, we need a common denominator:
\[ \frac{x-2}{x+5} - \frac{2(x+5)}{x+5} > 0 \]
This simplifies to:
\[ \frac{x-2 - 2(x+5)}{x+5} > 0 \]

### Step 3: Simplify the Numerator
Now, simplify the numerator:
\[ x - 2 - 2x - 10 = -x - 12 \]
So the inequality becomes:
\[ \frac{-x - 12}{x + 5} > 0 \]

### Step 4: Multiply by -1
To make the numerator positive, multiply both sides of the inequality by -1. Remember that this reverses the inequality sign:
\[ \frac{x + 12}{x + 5} < 0 \]

### Step 5: Analyze the Critical Points
Now, we need to find the critical points where the expression is zero or undefined:
1. \(x + 12 = 0 \Rightarrow x = -12\)
2. \(x + 5 = 0 \Rightarrow x = -5\)

### Step 6: Test Intervals
We will test the intervals determined by the critical points \(-12\) and \(-5\):
- Interval 1: \(x < -12\)
- Interval 2: \(-12 < x < -5\)
- Interval 3: \(x > -5\)

**Testing Interval 1:** Choose \(x = -13\):
\[ \frac{-13 + 12}{-13 + 5} = \frac{-1}{-8} > 0 \quad \text{(not valid)} \]

**Testing Interval 2:** Choose \(x = -10\):
\[ \frac{-10 + 12}{-10 + 5} = \frac{2}{-5} < 0 \quad \text{(valid)} \]

**Testing Interval 3:** Choose \(x = 0\):
\[ \frac{0 + 12}{0 + 5} = \frac{12}{5} > 0 \quad \text{(not valid)} \]

### Step 7: Conclusion
The solution to the inequality is:
\[ -12 < x < -5 \]
In interval notation, the solution is:
\[ x \in (-12, -5) \]