https://file.aiquickdraw.com/mathbot/file/17284157486536r2w04ex.jpg

Let's start by solving the mathematical problems shown in the image.

### 2b. Solve the inequality:
\[
x^3 + 2x^2 + x + 2 > 0
\]

This is a cubic inequality. To solve it:

1. Find the roots of the cubic equation \( x^3 + 2x^2 + x + 2 = 0 \).
2. Analyze the sign changes on the intervals determined by the roots.

The equation factors as:
\[
x^3 + 2x^2 + x + 2 = (x+1)(x^2+x+2)
\]
Thus, the inequality becomes:
\[
(x+1)(x^2 + x + 2) > 0
\]
The quadratic \( x^2 + x + 2 \) has no real roots since its discriminant \( 1^2 - 4(1)(2) = -7 \) is negative. Hence, \( x^2 + x + 2 \) is always positive.

So, we only need to solve \( x + 1 > 0 \), which gives \( x > -1 \).

Thus, the solution to the inequality is:
\[
x > -1
\]

### 3. Solve the inequality:
\[
\frac{2}{x} \geq x + 1
\]

Rearranging, we get:
\[
\frac{2}{x} - x \geq 1
\]
Multiplying both sides by \( x \) (assuming \( x \neq 0 \)), we get two cases based on the sign of \( x \).

For \( x > 0 \):
\[
2 - x^2 \geq x
\]
This simplifies to \( x^2 + x - 2 \leq 0 \). Factoring, we get:
\[
(x - 1)(x + 2) \leq 0
\]
The solution is \( -2 \leq x \leq 1 \).

For \( x < 0 \):
\[
2 - x^2 \leq x
\]
Simplifies to \( x^2 + x - 2 \geq 0 \), which has no real solutions in this case. So, we discard this.

Thus, the solution is:
\[
-2 \leq x \leq 1, \quad x \neq 0
\]

### 4. Solve the inequality:
\[
\frac{x-2}{x^2} \leq \frac{x+1}{x+3}
\]

First, cross-multiply to avoid fractions:
\[
(x-2)(x+3) \leq x^2(x+1)
\]

Expanding both sides:
\[
x^2 + x - 6 \leq x^3 + x^2
\]

Simplifying:
\[
0 \leq x^3
\]

This inequality is satisfied for \( x \geq 0 \).

Thus, the solution is:
\[
x \geq 0
\]

---

Would you like more details on any of the solutions or further explanation? 

Here are 5 related questions:
1. How do we determine the roots of cubic polynomials in general?
2. Can you explain why multiplying by \( x \) requires splitting into cases for inequalities?
3. How does the discriminant determine whether a quadratic has real roots?
4. What is the significance of analyzing sign changes in inequalities?
5. How do we solve inequalities with rational expressions?

**Tip:** Always consider whether multiplying by a variable might change the inequality's direction, especially when the variable can be positive or negative.

Solve the inequalities given in the image.

Learn how to solve cubic and rational inequalities like x^3 + 2x^2 + x + 2 > 0 and 2/x ≥ x + 1 with detailed step-by-step solutions. Ideal for Grade 11-12 students, this guide covers quadratic discriminants, sign changes, and cross-multiplication techniques.

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