To solve the inequality \( \frac{2}{x} - 1 \leq 3x + 2 \), follow these steps:

1. **Move all terms to one side of the inequality**:
\[ \frac{2}{x} - 1 - 3x - 2 \leq 0 \]
\[ \frac{2}{x} - 3x - 3 \leq 0 \]

2. **Find a common denominator for the terms**:
\[ \frac{2 - 3x^2 - 3x}{x} \leq 0 \]
\[ \frac{-3x^2 - 3x + 2}{x} \leq 0 \]

3. **Factor the numerator if possible**:
To factor \( -3x^2 - 3x + 2 \), rewrite it as:
\[ -3x^2 - 3x + 2 = 0 \]
This factors as:
\[ -3(x^2 + x - \frac{2}{3}) = 0 \]
\[ x^2 + x - \frac{2}{3} = 0 \]

Solving this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = 1, c = -\frac{2}{3} \):
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot -\frac{2}{3}}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 + \frac{8}{3}}}{2} \]
\[ x = \frac{-1 \pm \sqrt{\frac{11}{3}}}{2} \]
\[ x = \frac{-1 \pm \frac{\sqrt{11}}{\sqrt{3}}}{2} \]
\[ x = \frac{-1 \pm \frac{\sqrt{11}}{\sqrt{3}}}{2} \]

Now, the critical points are:
\[ x = \frac{-1 + \sqrt{\frac{11}{3}}}{2} \text{ and } x = \frac{-1 - \sqrt{\frac{11}{3}}}{2} \]

4. **Test the intervals** formed by these critical points in the inequality:
\[ \frac{-3x^2 - 3x + 2}{x} \leq 0 \]

The critical points divide the real number line into intervals. Test points in each interval to determine where the inequality holds.

5. **Include critical points and determine the solution set**:

Combining the above steps, we have:
\[ x \in \left( \frac{-1 - \sqrt{\frac{11}{3}}}{2}, 0 \right) \cup \left( 0, \frac{-1 + \sqrt{\frac{11}{3}}}{2} \right) \]

Ensure to verify the inequality at the boundaries and within the intervals.

**Summary of solution**: The solution set is all \( x \) such that:
\[ x \in \left( \frac{-1 - \sqrt{\frac{11}{3}}}{2}, 0 \right) \cup \left( 0, \frac{-1 + \sqrt{\frac{11}{3}}}{2} \right) \]

Would you like details on any specific step or have any questions?

**Relative Questions**:
1. How do you factor a quadratic equation?
2. What is the quadratic formula and how is

Explore how to solve the rational inequality (2/x - 1) <= (3x + 2) step-by-step. This problem involves rational functions, inequalities, and the quadratic formula. Suitable for high school students (Grades 10-12).

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