1. **State the problem:** We need to solve the system of inequalities graphically: $$y \geq \frac{1}{3}x + 4$$ $$y \leq -\frac{5}{3}x - 2$$ and find a point in the solution set. 2. **Understand the inequalities:** - The first inequality means the solution includes the line $y = \frac{1}{3}x + 4$ and all points above it. - The second inequality means the solution includes the line $y = -\frac{5}{3}x - 2$ and all points below it. 3. **Graph the lines:** - For $y = \frac{1}{3}x + 4$, the y-intercept is 4 and the slope is $\frac{1}{3}$. - For $y = -\frac{5}{3}x - 2$, the y-intercept is -2 and the slope is $-\frac{5}{3}$. 4. **Find the intersection point of the two lines:** Set the right sides equal: $$\frac{1}{3}x + 4 = -\frac{5}{3}x - 2$$ Multiply both sides by 3 to clear denominators: $$3 \times \left(\frac{1}{3}x + 4\right) = 3 \times \left(-\frac{5}{3}x - 2\right)$$ $$x + 12 = -5x - 6$$ Add $5x$ to both sides: $$x + 5x + 12 = -5x + 5x - 6$$ $$6x + 12 = -6$$ Subtract 12 from both sides: $$6x + \cancel{12} - \cancel{12} = -6 - 12$$ $$6x = -18$$ Divide both sides by 6: $$\frac{6x}{\cancel{6}} = \frac{-18}{\cancel{6}}$$ $$x = -3$$ Substitute $x = -3$ into one of the lines to find $y$: $$y = \frac{1}{3}(-3) + 4 = -1 + 4 = 3$$ So the intersection point is $(-3, 3)$. 5. **Determine the solution region:** - The solution set is where the shaded regions overlap: above the first line and below the second line. - This overlap lies to the left of the intersection point $(-3, 3)$. 6. **Choose a test point in the solution set:** Try $(-4, 4)$: Check first inequality: $$4 \geq \frac{1}{3}(-4) + 4 = -\frac{4}{3} + 4 = \frac{8}{3} \approx 2.67$$ True. Check second inequality: $$4 \leq -\frac{5}{3}(-4) - 2 = \frac{20}{3} - 2 = \frac{20}{3} - \frac{6}{3} = \frac{14}{3} \approx 4.67$$ True. Therefore, $(-4, 4)$ is in the solution set. **Final answer:** The solution set is the region above $y = \frac{1}{3}x + 4$ and below $y = -\frac{5}{3}x - 2$, overlapping to the left of $(-3, 3)$. One point in the solution set is $\boxed{(-4, 4)}$.