\( 3|3-4 x|=15 x \) What is the solution to the given equation

To solve the equation: \[ 3|3 - 4x| = 15x \] **Step 1: Simplify the Equation** Divide both sides by 3 to simplify: \[ |3 - 4x| = 5x \] Since the absolute value is always non-negative, we must have: \[ 5x \geq 0 \quad \Rightarrow \quad x \geq 0 \] **Step 2: Consider the Two Cases for the Absolute Value** 1. **Case 1: \( 3 - 4x \geq 0 \)** \[ 3 - 4x = 5x \] Combine like terms: \[ 3 = 9x \] Solve for \( x \): \[ x = \frac{1}{3} \] **Verification:** \( \frac{1}{3} \geq 0 \) and \( 3 - 4\left(\frac{1}{3}\right) = 3 - \frac{4}{3} = \frac{5}{3} \geq 0 \). This solution is valid. 2. **Case 2: \( 3 - 4x < 0 \)** \[ -(3 - 4x) = 5x \quad \Rightarrow \quad 4x - 3 = 5x \] Solve for \( x \): \[ -3 = x \] **Verification:** \( -3 < 0 \), which does not satisfy \( x \geq 0 \). Therefore, this solution is **invalid**. **Conclusion:** The only valid solution to the equation is: \[ x = \frac{1}{3} \] **Answer:** \( \boxed{\dfrac{1}{3}} \)