1. The problem asks to solve the compound inequality: $$5x - 19 \leq 1 \quad \text{OR} \quad -4x + 3 < -6$$ for $x$. 2. We solve each inequality separately and then combine the solutions using the OR condition. 3. Solve the first inequality: $$5x - 19 \leq 1$$ Add 19 to both sides: $$5x \leq 20$$ Divide both sides by 5: $$x \leq 4$$ 4. Solve the second inequality: $$-4x + 3 < -6$$ Subtract 3 from both sides: $$-4x < -9$$ Divide both sides by -4$ (remember to reverse the inequality sign when dividing by a negative number): $$x > \frac{9}{4}$$ 5. Combine the two solutions with OR: $$x \leq 4 \quad \text{OR} \quad x > \frac{9}{4}$$ 6. The OR means $x$ can be any value that satisfies either inequality. So the solution is all $x$ such that: $$x \leq 4$$ or $$x > \frac{9}{4}$$ 7. The answer choices show intervals. The compound inequality solution is the union of these intervals: $$(-\infty, 4] \cup \left(\frac{9}{4}, \infty\right)$$ 8. The problem's graph shows the interval $$\frac{9}{4} < x \leq 4$$ which is the intersection, not the union. Since the problem states OR, the correct solution is $$x \leq 4$$ or $$x > \frac{9}{4}$$. 9. Therefore, the correct answer is A) $$x \geq 4$$ is incorrect because it only includes values greater than or equal to 4. 10. The correct solution is $$x \leq 4$$ or $$x > \frac{9}{4}$$ which covers all $x$ except those between $$-\infty$$ and $$\frac{9}{4}$$. Final answer: $$x \leq 4 \quad \text{OR} \quad x > \frac{9}{4}$$