https://r2.erweima.ai/mathbot/image/1726184240267gy2j1dkw.jpg

The image contains two mathematical problems. Let’s break them down:

### Left Side: Equation
\[
(x - 3)(x + 2)(2x - 5) = 0
\]
This is a product of three factors set equal to zero. To solve, we apply the **zero product property**, which states that if the product of several terms is zero, then at least one of the terms must be zero.

The factors are:
1. \( x - 3 = 0 \)
   - Solution: \( x = 3 \)
2. \( x + 2 = 0 \)
   - Solution: \( x = -2 \)
3. \( 2x - 5 = 0 \)
   - Solution: \( x = \frac{5}{2} \)

Thus, the solutions to this equation are:
\[
x = 3, \quad x = -2, \quad x = \frac{5}{2}
\]

### Right Side: Inequality
\[
\frac{2 - x}{x + 1} > \frac{3}{2}
\]
To solve this inequality:

1. Cross-multiply to eliminate the denominators:
\[
(2 - x)(2) > 3(x + 1)
\]
2. Expand both sides:
\[
4 - 2x > 3x + 3
\]
3. Rearrange to get all terms involving \( x \) on one side:
\[
4 - 3 > 3x + 2x
\]
\[
1 > 5x
\]
4. Solve for \( x \):
\[
x < \frac{1}{5}
\]

Now we must consider the domain restriction, since \( x + 1 \neq 0 \), meaning \( x \neq -1 \).

### Final Solution:
\[
x < \frac{1}{5}, \quad x \neq -1
\]

---

Would you like more detailed steps or clarification? Here are some related questions:

1. How is the zero product property applied to factorized equations?
2. What are the steps to solve rational inequalities?
3. How does cross-multiplication work in inequalities?
4. What is the importance of considering the domain in inequalities?
5. Can you explain how to handle cases where division by zero might occur?

**Tip**: Always check for restrictions on the variable, especially when dealing with rational expressions!

(x - 3)(x + 2)(2x - 5) = 0 and (2 - x) / (x + 1) > 3/2

Learn how to solve the equation (x - 3)(x + 2)(2x - 5) = 0 using the zero product property and the inequality (2 - x) / (x + 1) > 3/2 with cross-multiplication and consideration of domain restrictions. Suitable for students in Grades 8-10.

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