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To solve the equation \(|5x - 3| = |7x + 3|\), we need to consider two cases, since absolute values can yield both positive and negative results.

### Case 1: 
The expressions inside the absolute values are equal:

\[
5x - 3 = 7x + 3
\]

Solve for \(x\):
\[
5x - 7x = 3 + 3
\]
\[
-2x = 6
\]
\[
x = -3
\]

### Case 2: 
The expressions inside the absolute values are opposites:

\[
5x - 3 = -(7x + 3)
\]

Simplify:
\[
5x - 3 = -7x - 3
\]
\[
5x + 7x = -3 + 3
\]
\[
12x = 0
\]
\[
x = 0
\]

### Solutions:
The solutions are \(x = -3\) and \(x = 0\).

Would you like any additional details or have any questions about this solution?

### Related Questions:
1. What is the definition of absolute value, and how does it affect solving equations?
2. How do we approach cases when dealing with absolute value equations?
3. Can absolute value equations have no solution? If so, when?
4. How can we verify that the solutions for this equation are correct?
5. What happens when there are more than two terms in an absolute value equation?

### Tip:
Always check both cases when solving absolute value equations—one for equality and one for opposites.

Learn how to solve the absolute value equation |5x - 3| = |7x + 3| step by step. This guide walks through solving the two cases where the absolute values are either equal or opposites. Suitable for Grades 8-10.

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