1. We are given the system of inequalities: $$3(2x-5) > 4x+7$$ $$5x - 2(3-x) \leq 12$$ 2. Solve the first inequality: $$3(2x - 5) > 4x + 7$$ Distribute 3: $$6x - 15 > 4x + 7$$ Subtract $4x$ from both sides: $$6x - 4x - 15 > 7$$ Simplify: $$2x - 15 > 7$$ Add 15 to both sides: $$2x > 22$$ Divide both sides by 2: $$x > 11$$ 3. Solve the second inequality: $$5x - 2(3 - x) \leq 12$$ Distribute $-2$: $$5x - 6 + 2x \leq 12$$ Combine like terms: $$7x - 6 \leq 12$$ Add 6 to both sides: $$7x \leq 18$$ Divide both sides by 7: $$x \leq \frac{18}{7}$$ 4. Combine the results: The solution to the system is values of $x$ that satisfy both inequalities: $$x > 11$$ and $$x \leq \frac{18}{7}$$ Since $$11 > \frac{18}{7} \approx 2.57$$, there is no overlap in solutions. 5. Therefore, the system has no solution where both inequalities are true simultaneously.