To solve the equation \( |3x + 5| + |2x - 3| = 25 \), we need to consider different cases based on the values of \( x \) that affect the signs inside the absolute values.

### Step 1: Break down the cases

1. **Case 1: \( x \geq \frac{3}{2} \)**
   - In this case, both expressions inside the absolute values are non-negative. Thus, we can drop the absolute value symbols:
   \[
   (3x + 5) + (2x - 3) = 25
   \]
   Simplifying:
   \[
   5x + 2 = 25
   \]
   Solving for \( x \):
   \[
   5x = 23 \quad \Rightarrow \quad x = \frac{23}{5}
   \]
   Since \( \frac{23}{5} \approx 4.6 \geq \frac{3}{2} \), this solution is valid.

2. **Case 2: \( -\frac{5}{3} \leq x < \frac{3}{2} \)**
   - In this case, \( 3x + 5 \geq 0 \) but \( 2x - 3 < 0 \), so we change the sign of the second term:
   \[
   (3x + 5) - (2x - 3) = 25
   \]
   Simplifying:
   \[
   x + 8 = 25
   \]
   Solving for \( x \):
   \[
   x = 17
   \]
   But \( x = 17 \) does not lie in the range \( -\frac{5}{3} \leq x < \frac{3}{2} \), so there is no solution in this case.

3. **Case 3: \( x < -\frac{5}{3} \)**
   - In this case, both expressions inside the absolute values are negative, so we change the signs of both terms:
   \[
   -(3x + 5) - (2x - 3) = 25
   \]
   Simplifying:
   \[
   -5x - 2 = 25
   \]
   Solving for \( x \):
   \[
   -5x = 27 \quad \Rightarrow \quad x = -\frac{27}{5}
   \]
   Since \( -\frac{27}{5} \approx -5.4 < -\frac{5}{3} \), this solution is valid.

### Step 2: Final solutions

The valid solutions for \( x \) are:
\[
x = \frac{23}{5} \quad \text{and} \quad x = -\frac{27}{5}
\]

### Summary:
The solutions to the equation are:
\[
x = \frac{23}{5} \quad \text{or} \quad x = -\frac{27}{5}
\]

Would you like more details on any of the steps?

### Follow-up questions:
1. How do absolute values impact the number of cases in an equation?
2. Can you explain why case 2 has no valid solution?
3. How can we verify solutions when solving absolute value equations?
4. What are some applications of absolute value in real-life scenarios?
5. Can absolute value equations have infinitely many solutions?

**Tip:** When solving absolute value equations, always check that each solution fits within the constraints of the case it was derived from.

Explore how to solve the absolute value equation |3x + 5| + |2x - 3| = 25 with a step-by-step breakdown. Learn about handling different cases based on the signs inside absolute values. Suitable for students in Grades 9-11.

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