Solve ` log_(0.2). (x+2)/x le 1`.

Explanation

To solve the inequality \( \log_{0.2} \left( \frac{x+2}{x} \right) \leq 1 \), we will follow these steps:

### Step 1: Rewrite the Inequality
We start by rewriting the inequality in exponential form. Since the base \( 0.2 \) is less than 1, we can rewrite the logarithmic inequality:

\[ \frac{x+2}{x} \geq 0.2^1 \]

This simplifies to:

\[ \frac{x+2}{x} \geq 0.2 \]

### Step 2: Simplify the Inequality
Next, we can express \( 0.2 \) as a fraction:

\[ 0.2 = \frac{1}{5} \]

Thus, we have:

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

### Step 3: Cross-Multiply
To eliminate the fraction, we cross-multiply (keeping in mind that \( x \) must be positive to avoid division by zero):

\[ 5(x + 2) \geq x \]

### Step 4: Expand and Rearrange
Expanding the left side gives:

\[ 5x + 10 \geq x \]

Now, we rearrange the terms:

\[ 5x - x + 10 \geq 0 \]

This simplifies to:

\[ 4x + 10 \geq 0 \]

### Step 5: Solve for \( x \)
Now, we isolate \( x \):

\[ 4x \geq -10 \]

Dividing both sides by 4 gives:

\[ x \geq -\frac{10}{4} \]

This simplifies to:

\[ x \geq -\frac{5}{2} \]

### Step 6: Consider the Domain
Since we are dealing with a logarithm, we must also ensure that the argument of the logarithm is positive:

\[ \frac{x+2}{x} > 0 \]

This requires \( x > 0 \) or \( x < -2 \).

### Step 7: Combine the Results
Now we have two conditions:
1. \( x \geq -\frac{5}{2} \)
2. \( x > 0 \) or \( x < -2 \)

The critical points are \( -2 \) and \( -\frac{5}{2} \).

### Step 8: Analyze the Intervals
- For \( x < -2 \): The inequality \( x \geq -\frac{5}{2} \) is not satisfied.
- For \( x = -2 \): The inequality holds.
- For \( x > 0 \): The inequality \( x \geq -\frac{5}{2} \) holds.

### Final Solution
Thus, the solution set is:

\[ x \in \left[-\frac{5}{2}, -2\right) \cup (0, \infty) \]