1. The problem is to analyze the system of inequalities: $$y > -x - 2$$ $$y < -5x + 2$$ 2. These inequalities represent regions above and below two lines on the Cartesian plane. 3. The first inequality $y > -x - 2$ means the solution region is above the line $y = -x - 2$. 4. The second inequality $y < -5x + 2$ means the solution region is below the line $y = -5x + 2$. 5. To find the intersection of these regions, we first find the intersection point of the two lines by setting: $$-x - 2 = -5x + 2$$ 6. Solve for $x$: $$-x - 2 = -5x + 2$$ $$-x + 5x = 2 + 2$$ $$4x = 4$$ $$x = \frac{4}{4} = 1$$ 7. Substitute $x=1$ into one of the line equations to find $y$: $$y = -1 - 2 = -3$$ 8. The lines intersect at the point $(1, -3)$. 9. The solution to the system is the set of points where $y$ is greater than $-x - 2$ and less than $-5x + 2$ simultaneously. 10. This region lies between the two lines, above the first and below the second, intersecting at $(1, -3)$. Final answer: The solution region is the set of points satisfying $$-x - 2 < y < -5x + 2$$ with the lines intersecting at $(1, -3)$.