From the provided image, you are tasked with solving a system of inequalities graphically. Below is the solution process for the inequalities:

Inequalities:

1. $y \geq x + 2$
2. $y \geq -2x - 7$

Steps to Graph:

1. Graph the first inequality ($y \geq x + 2$):

   • The boundary line is $y = x + 2$. This line is solid because of the "greater than or equal to" ($\geq$) condition.
   • To plot:
     • Start with the y-intercept at $(0, 2)$.
     • Use the slope of $1$ (rise over run) to plot other points: move up 1 and right 1 repeatedly.
   • Shade above the line, as $y$ is greater than or equal to the expression.

2. Graph the second inequality ($y \geq -2x - 7$):

   • The boundary line is $y = -2x - 7$. This line is also solid due to the "greater than or equal to" ($\geq$) condition.
   • To plot:
     • Start with the y-intercept at $(0, -7)$.
     • Use the slope of $-2$ (rise over run): move down 2 and right 1, or up 2 and left 1.
   • Shade above this line.

Solution Set:

The solution set is the region where the shaded areas of both inequalities overlap.

Finding a Point in the Solution:

• Look for an easy-to-identify point within the overlapping region. For instance, a valid point may be $(0, 2)$, $(1, 4)$, or others depending on the region.

Would you like me to simulate the graph for better visualization or clarify further?

Related Questions:

1. How do we identify the solution region for inequalities?
2. What is the role of the boundary line in graphing inequalities?
3. How can you check if a point lies within the solution region?
4. What changes when an inequality uses $<$ or $>$ instead of $\leq$ or $\geq$?
5. How do slopes and intercepts help in graphing lines?

Tip:

Always test a point (like $(0, 0)$) not on the boundary to verify which side of the line should be shaded.