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To solve the equation \( |3x + 1| = |2x + 3| \), we need to consider the absolute value properties. This equation can be split into two cases:

### Case 1: \( 3x + 1 = 2x + 3 \)
1. Subtract \(2x\) from both sides:
   \[
   3x - 2x + 1 = 3
   \]
   \[
   x + 1 = 3
   \]
2. Subtract 1 from both sides:
   \[
   x = 2
   \]

### Case 2: \( 3x + 1 = -(2x + 3) \)
1. Expand the right side:
   \[
   3x + 1 = -2x - 3
   \]
2. Add \(2x\) to both sides:
   \[
   3x + 2x + 1 = -3
   \]
   \[
   5x + 1 = -3
   \]
3. Subtract 1 from both sides:
   \[
   5x = -4
   \]
4. Divide by 5:
   \[
   x = -\frac{4}{5}
   \]

### Solutions
The solutions to the equation are:
\[
x = 2 \quad \text{and} \quad x = -\frac{4}{5}
\]

Would you like further details or have any questions on the solution process?

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Here are some related questions for further exploration:
1. How would you solve an equation involving three absolute value terms?
2. What happens if there are no solutions for an absolute value equation?
3. How can you check your solutions for absolute value equations?
4. What is the graphical interpretation of absolute value equations?
5. How would you solve \( |ax + b| = |cx + d| \) in general terms?

**Tip:** When dealing with absolute values, always consider both the positive and negative cases of the expression inside the absolute value!

Learn how to solve the absolute value equation |3x + 1| = |2x + 3| with a step-by-step breakdown of both cases. This algebra problem covers key concepts in solving absolute value equations and is suitable for Grades 9-11.

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