To determine which ordered pair is in the solution set of the given system of inequalities, we need to test each ordered pair against the two inequalities:

Inequalities:

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

Let's test each option:

A. $(-5, 3)$

1. Check $y < \frac{1}{2}x + 4$: $3 < \frac{1}{2}(-5) + 4 = -2.5 + 4 = 1.5 \quad (\text{False})$

Since the first inequality is false, this pair does not satisfy the system.

B. $(0, 4)$

1. Check $y < \frac{1}{2}x + 4$: $4 < \frac{1}{2}(0) + 4 = 4 \quad (\text{False})$

Since the first inequality is false, this pair does not satisfy the system.

C. $(3, -5)$

1. Check $y < \frac{1}{2}x + 4$: $-5 < \frac{1}{2}(3) + 4 = 1.5 + 4 = 5.5 \quad (\text{True})$

2. Check $y \geq -x + 1$: $-5 \geq -(3) + 1 = -3 + 1 = -2 \quad (\text{False})$

Since the second inequality is false, this pair does not satisfy the system.

D. $(4, 0)$

1. Check $y < \frac{1}{2}x + 4$: $0 < \frac{1}{2}(4) + 4 = 2 + 4 = 6 \quad (\text{True})$

2. Check $y \geq -x + 1$: $0 \geq -(4) + 1 = -4 + 1 = -3 \quad (\text{True})$

Since both inequalities are true for this pair, the correct answer is:

Answer: D. (4, 0)

Would you like further details or explanations?

Related Questions:

1. How do you graph a system of inequalities?
2. What does the solution set of a system of inequalities represent?
3. How can you determine if a point lies within the solution set of an inequality?
4. What strategies can be used to solve systems of linear inequalities?
5. What are the real-world applications of systems of inequalities?

Tip: Always check each point in the system of inequalities separately to ensure it satisfies all conditions.