To solve the given system of inequalities graphically, we need to analyze each inequality separately, plot them, and identify the region where both inequalities hold true. Here’s a step-by-step approach:

Inequality 1:

$y < x + 2$

• This represents a line with a slope of 1 and a y-intercept of 2.
• To graph it, plot the line $y = x + 2$ with a dashed line (indicating that points on the line itself are not included in the solution because the inequality is strict, "<").
• Shade the region below this line because $y$ values are less than $x + 2$.

Inequality 2:

$y < -\frac{1}{2}x - 4$

• This line has a slope of $-\frac{1}{2}$ and a y-intercept of $-4$.
• Plot the line $y = -\frac{1}{2}x - 4$ with a dashed line.
• Shade the region below this line as well.

Finding the Solution Region:

The solution region is where the shaded regions for both inequalities overlap.

Determining a Solution Point:

Once you've graphed the inequalities and found the overlapping region, any point in this region is part of the solution set. A point such as $(0, -5)$ may be in the solution region if it satisfies both inequalities:

1. For $y < x + 2$, substituting $(0, -5)$: $-5 < 0 + 2 \Rightarrow -5 < 2$ (True)
2. For $y < -\frac{1}{2}x - 4$, substituting $(0, -5)$: $-5 < -4$ (True)

Since both inequalities are satisfied, $(0, -5)$ is a point in the solution set.

Would you like me to explain any part in more detail, or do you have any further questions?

Here are 5 related questions that expand on this concept:

1. How do we determine if a point lies in the solution region for any system of inequalities?
2. What changes if the inequalities are "greater than" instead of "less than"?
3. How would the solution change if one of the inequalities included "≤" or "≥"?
4. How do you find the intersection points of the boundary lines in such problems?
5. What role does the slope of each line play in determining the solution region?

Tip: Always use a dashed line for strict inequalities (like "<" or ">") and a solid line for inclusive inequalities (like "≤" or "≥").