1. **State the problem:** Solve the equation $ (3x - 8)(4x - 5) = 0 $ for $x$. 2. **Formula and rule:** The zero product property states that if a product of two factors equals zero, then at least one of the factors must be zero. So, set each factor equal to zero: $$ 3x - 8 = 0 \quad \text{or} \quad 4x - 5 = 0 $$ 3. **Solve the first equation:** $$ 3x - 8 = 0 $$ Add 8 to both sides: $$ 3x = 8 $$ Divide both sides by 3: $$ x = \frac{\cancel{3}x}{\cancel{3}} = \frac{8}{3} $$ 4. **Solve the second equation:** $$ 4x - 5 = 0 $$ Add 5 to both sides: $$ 4x = 5 $$ Divide both sides by 4: $$ x = \frac{\cancel{4}x}{\cancel{4}} = \frac{5}{4} $$ 5. **Final answer:** The solutions to the equation are $$ x = \frac{8}{3} \quad \text{or} \quad x = \frac{5}{4} $$