To solve the inequality:

\[
\frac{x - 2}{x + 2} < \frac{x + 1}{x^2}
\]

we need to find the values of \(x\) for which the inequality holds. Here is a step-by-step solution:

### Step 1: Bring the Inequality to a Common Denominator

To simplify, let's bring both sides to a common denominator:

\[
\frac{(x - 2)x^2}{(x + 2)x^2} < \frac{(x + 1)(x + 2)}{(x + 2)x^2}
\]

### Step 2: Combine the Fractions

Now, we combine the fractions:

\[
\frac{(x - 2)x^2 - (x + 1)(x + 2)}{(x + 2)x^2} < 0
\]

### Step 3: Expand and Simplify the Numerator

Expand both terms in the numerator:

\[
(x - 2)x^2 = x^3 - 2x^2
\]

\[
(x + 1)(x + 2) = x^2 + 3x + 2
\]

Substitute these into the numerator:

\[
x^3 - 2x^2 - (x^2 + 3x + 2) = x^3 - 3x^2 - 3x - 2
\]

### Step 4: Rewrite the Inequality

Now the inequality becomes:

\[
\frac{x^3 - 3x^2 - 3x - 2}{(x + 2)x^2} < 0
\]

### Step 5: Determine the Critical Points

The critical points occur when the numerator or denominator is zero:

1. **Numerator is zero**: \(x^3 - 3x^2 - 3x - 2 = 0\)
   
   To solve this cubic equation, you may try possible rational roots or use numerical methods to approximate the roots.

2. **Denominator is zero**: \((x + 2)x^2 = 0\)

   The solutions are \(x = 0\) and \(x = -2\).

### Step 6: Analyze the Intervals

With the critical points, analyze the intervals defined by these points and test the sign of the expression \(\frac{x^3 - 3x^2 - 3x - 2}{(x + 2)x^2}\) within each interval.

### Conclusion

The intervals where the inequality is true will be those where the expression is negative. After determining the exact intervals by testing values or using the roots obtained from solving the cubic equation, we can write the solution set.

Would you like more details on any of the steps or a more in-depth interval analysis?

### Related Questions to Explore Further:

1. How do you solve a cubic equation?
2. What methods can be used to analyze inequalities involving polynomials?
3. What are rational roots, and how do you find them?
4. How can graphing help in understanding inequalities?
5. What are the properties of polynomials that help determine their roots?

**Tip:** Always check for the critical points where the denominator is zero, as these points can indicate discontinuities or vertical asymptotes in the function.

This detailed solution explores the steps to solve the inequality (x-2)/(x+2) < (x+1)/x^2. Learn how to find a common denominator, expand polynomials, and analyze critical points to determine the solution set. Suitable for Grades 10-12 students studying inequalities and rational functions.

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