To solve the given system of inequalities graphically:

1. Inequalities:
   • $y < 2x - 5$
   • $y \leq -x - 2$

Step 1: Understanding Each Inequality

1. For $y < 2x - 5$:

   • The equation $y = 2x - 5$ represents a straight line with slope $m = 2$ and $y$-intercept at $-5$.
   • The inequality $y < 2x - 5$ means the solution region is below this line. The line will be dashed to indicate that points on the line are not included.

2. For $y \leq -x - 2$:

   • The equation $y = -x - 2$ represents a straight line with slope $m = -1$ and $y$-intercept at $-2$.
   • The inequality $y \leq -x - 2$ means the solution region is below or on this line. The line will be solid to indicate that points on the line are included.

Step 2: Graphing Both Lines

• Plot $y = 2x - 5$: Start at $(0, -5)$, use the slope $2$ (rise 2, run 1).
• Plot $y = -x - 2$: Start at $(0, -2)$, use the slope $-1$ (rise -1, run 1).
• Shade the regions:
  • Below the dashed line $y = 2x - 5$.
  • Below or on the solid line $y = -x - 2$.

Step 3: Identifying the Solution Region

The solution region is where both shaded regions overlap. It lies:

• Below the dashed line $y = 2x - 5$.
• Below or on the solid line $y = -x - 2$.

Step 4: Point in the Solution Set

Choose a test point within the overlapping region. For example:

• Point $(0, -4)$:
  • Check against $y < 2x - 5$: $-4 < 2(0) - 5 \implies -4 < -5$ → False.