The task involves solving a system of linear inequalities graphically and identifying a point in the solution set. Here's the approach:

1. Rewrite the Inequalities:

• First inequality: $y < \frac{1}{2}x + 3$
• Second inequality: $y < -2x - 7$

These are already in slope-intercept form ($y = mx + b$).

2. Graph the Lines:

• For $y = \frac{1}{2}x + 3$:

  • Slope ($m$) is $\frac{1}{2}$ and y-intercept ($b$) is $3$.
  • Plot the y-intercept at $(0, 3)$.
  • Use the slope $\frac{1}{2}$ to find another point: from $(0, 3)$, go up 1 and right 2 to $(2, 4)$.
  • Draw a dashed line (inequality is strict, $<$).

• For $y = -2x - 7$:

  • Slope ($m$) is $-2$ and y-intercept ($b$) is $-7$.
  • Plot the y-intercept at $(0, -7)$.
  • Use the slope $-2$: from $(0, -7)$, go down 2 and right 1 to $(1, -9)$.
  • Draw a dashed line.

3. Shade the Regions:

• For $y < \frac{1}{2}x + 3$: Shade below the line $y = \frac{1}{2}x + 3$.
• For $y < -2x - 7$: Shade below the line $y = -2x - 7$.

The solution set is the overlapping shaded region below both lines.

4. Identify a Point:

From the graph:

• A point in the overlapping region could be $(-6, -10)$.

Would you like detailed guidance on graphing or solving similar problems?

Here are related questions for practice:

1. How do you determine the solution region for linear inequalities graphically?
2. What does the slope of a line represent, and how is it used in graphing?
3. Why are dashed lines used for inequalities with $<$ or $>$?
4. How can you verify if a point lies in the solution set of a system of inequalities?
5. What happens when two inequality regions do not overlap?

Tip: Always test a point (like $(0, 0)$) in each inequality to verify the correct shading!