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The given inequality is:

\[
\frac{x}{x - 1} \leq \frac{x - 2}{x + 1}
\]

To solve this inequality, follow these steps:

### Step 1: Find the critical points by determining when the expressions are undefined.
The expressions are undefined when the denominators are zero:
- \( x - 1 = 0 \) leads to \( x = 1 \)
- \( x + 1 = 0 \) leads to \( x = -1 \)

These values are critical points where the inequality might change signs.

### Step 2: Simplify the inequality.
To simplify the inequality, we can cross-multiply since the denominators are positive or negative, but only if we keep track of the sign changes:

\[
x(x

Solve the inequality: \( \frac{x}{x-1} \leq \frac{x-2}{x+1} \).

This detailed guide provides a step-by-step solution to the rational inequality \( \frac{x}{x-1} \leq \frac{x-2}{x+1} \). Explore how to find critical points, apply cross multiplication, and understand the properties of inequalities. Suitable for high school students in Grades 10-12.

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