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

Explanation

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

### Step 1: Rearrange the Inequation
Start by moving all terms to one 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. The common denominator here is \(x + 5\):

\[ \frac{x-2 - 2(x + 5)}{x + 5} > 0 \]

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

\[ x - 2 - 2x - 10 = -x - 12 \]

So, we rewrite the inequality:

\[ \frac{-x - 12}{x + 5} > 0 \]

### Step 4: Factor Out the Negative Sign
We can factor out \(-1\) from the numerator:

\[ \frac{-(x + 12)}{x + 5} > 0 \]

This can be rewritten as:

\[ \frac{x + 12}{x + 5} < 0 \]

### Step 5: Find Critical Points
Now, we need to find the critical points where the expression is equal to zero or undefined. Set the numerator and denominator to zero:

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 these critical points: \((-∞, -12)\), \((-12, -5)\), and \((-5, ∞)\).

1. **Interval \((-∞, -12)\)**: Choose \(x = -13\)
\[ \frac{-13 + 12}{-13 + 5} = \frac{-1}{-8} > 0 \quad \text{(Positive)} \]

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

3. **Interval \((-5, ∞)\)**: Choose \(x = 0\)
\[ \frac{0 + 12}{0 + 5} = \frac{12}{5} > 0 \quad \text{(Positive)} \]

### Step 7: Determine the Solution
We are looking for where the expression is less than zero. From our tests, we found that the expression is negative in the interval \((-12, -5)\).

### Step 8: Write the Final Answer
Since the inequality does not include equality, we exclude the endpoints:

\[ x \in (-12, -5) \]