Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.

Steps to Solve

1. Graph the first inequality Start with the inequality $y \geq -x + 7$.

   • First, identify the line $y = -x + 7$.
   • The y-intercept is $(0, 7)$ and the slope is -1.
   • Plot the line and use a solid line since the inequality is "greater than or equal to" (includes equality).
   • Shade the area above this line to represent $y \geq -x + 7$.

2. Graph the second inequality Next, graph the inequality $y > 2x - 5$.

   • Identify the line $y = 2x - 5$.
   • The y-intercept is $(0, -5)$ and the slope is 2.
   • Plot the line using a dashed line since the inequality is "greater than" (does not include equality).
   • Shade the area above this line to represent $y > 2x - 5$.

3. Find the intersection Determine where the shaded regions of the two inequalities overlap. This area represents the solution set for the system of inequalities.

4. Select a point from the solution set Choose a point within the intersection area. One possible point to consider is $(3, 5)$.

   • Check to ensure it satisfies both inequalities:
     • For $y \geq -x + 7$: $5 \geq -3 + 7$ → $5 \geq 4$ (True).
     • For $y > 2x - 5$: $5 > 2(3) - 5$ → $5 > 1$ (True).
   • Since both inequalities are satisfied, the point $(3, 5)$ is part of the solution set.