1. **State the problem:** We need to find which two expressions correctly represent the inequality $$-3(2x - 5) < 5(2 - x)$$. 2. **Expand both sides:** $$-3(2x - 5) = -6x + 15$$ $$5(2 - x) = 10 - 5x$$ 3. **Rewrite the inequality with expanded terms:** $$-6x + 15 < 10 - 5x$$ 4. **Check each given option:** - Option 1: $$x < 5$$ (we will verify if this is equivalent) - Option 2: $$-6x - 5 < 10 - x$$ (check if equivalent) - Option 3: $$-6x + 15 < 10 - 5x$$ (matches our expanded inequality) 5. **Solve the inequality $$-6x + 15 < 10 - 5x$$:** Subtract 10 from both sides: $$-6x + 15 - 10 < 10 - 5x - 10$$ $$-6x + 5 < -5x$$ Add $$6x$$ to both sides: $$-6x + 5 + 6x < -5x + 6x$$ $$5 < x$$ Rewrite as: $$x > 5$$ 6. **Check option 1: $$x < 5$$ is the opposite of the solution, so it is incorrect.** 7. **Check option 2: $$-6x - 5 < 10 - x$$** Rewrite as: $$-6x - 5 < 10 - x$$ Add $$x$$ to both sides: $$-6x + x - 5 < 10$$ $$-5x - 5 < 10$$ Add 5 to both sides: $$-5x < 15$$ Divide both sides by $$-5$$ (remember to reverse inequality): $$\cancel{-5}x > \cancel{-5} \frac{15}{-5}$$ $$x > -3$$ This is not equivalent to the original inequality, so option 2 is incorrect. 8. **Option 3 matches the expanded inequality exactly, so it is correct.** 9. **Check the solution $$x > 5$$ against the number lines:** - The number line with an open circle at 5 and shading to the right represents $$x > 5$$ (correct). - The number line with an open circle at 5 and shading to the left represents $$x < 5$$ (incorrect). **Final correct representations:** - $$-6x + 15 < 10 - 5x$$ - Number line with open circle at 5 and shading to the right (representing $$x > 5$$) **Note:** The inequality $$x < 5$$ is incorrect.