Solve the equation. \( \quad|7 x|=|6 x-3| \)

To solve the equation \( |7x| = |6x - 3| \), we can consider two cases based on the properties of absolute values. **Case 1:** Both expressions are non-negative. This leads to the equation: \( 7x = 6x - 3 \). Solving for \( x \), we get: \( 7x - 6x = -3 \) \( x = -3 \). **Case 2:** The first expression is non-negative and the second is negative. This leads to: \( 7x = -(6x - 3) \). Solving this gives: \( 7x = -6x + 3 \) \( 7x + 6x = 3 \) \( 13x = 3 \) \( x = \frac{3}{13} \). **Case 3:** Both expressions negative. This leads to: \( -7x = -(6x - 3) \), which simplifies to \( -7x = -6x + 3 \) \( -7x + 6x = 3 \) \( -x = 3 \) \( x = -3 \). **Case 4:** The first expression is negative and the second is non-negative. This leads us to: \( -7x = 6x - 3 \). Solving gives us: \( -7x - 6x = -3 \) \( -13x = -3 \) \( x = \frac{3}{13} \). Collecting all our solutions, we find \( x = -3 \) and \( x = \frac{3}{13} \). Thus, the complete solution set for the equation \( |7x| = |6x - 3| \) is: \[ x = \frac{3}{13}, \quad x = -3 \]