Graph the system of inequalities. Determine all possible soliutions from the given ordered pair

y  >  -x -4

y < 1/2x + 1

(-1, -3),  (2,2),  (4, 1),  (-4, -2)

 a
(-1, -3)
 b
(2, 2)
 c
(4, 1)
 d
(-4, -2)


To graph the system of inequalities and determine which of the given ordered pairs are solutions, we need to:

1. **Graph the two inequalities**:
   - \( y > -x - 4 \)
   - \( y < \frac{1}{2}x + 1 \)

### Steps to solve:
1. **Graph the first inequality:**
   - Convert the inequality to an equation to find the boundary line: 
     \[
     y = -x - 4
     \]
   This is a straight line with a slope of \(-1\) and a y-intercept of \(-4\). The boundary line will be dashed because the inequality is strict (\(>\), not \(\geq\)). The region above the line will be shaded, as the inequality is \(y > -x - 4\).

2. **Graph the second inequality:**
   - Convert the inequality to an equation to find the boundary line: 
     \[
     y = \frac{1}{2}x + 1
     \]
   This is a straight line with a slope of \(\frac{1}{2}\) and a y-intercept of \(1\). The boundary line will also be dashed, and the region below the line will be shaded, as the inequality is \(y < \frac{1}{2}x + 1\).

### Determining which points are solutions:
For each point, we'll check whether it satisfies both inequalities.

### 1. Checking point \((-1, -3)\):
   - Substitute into \( y > -x - 4 \):
     \[
     -3 > -(-1) - 4 \quad \Rightarrow \quad -3 > 1 - 4 \quad \Rightarrow \quad -3 > -3 \quad \text{(False)}
     \]
   Since the first inequality is false, \((-1, -3)\) is **not** a solution.

### 2. Checking point \((2, 2)\):
   - For \( y > -x - 4 \):
     \[
     2 > -(2) - 4 \quad \Rightarrow \quad 2 > -2 - 4 \quad \Rightarrow \quad 2 > -6 \quad \text{(True)}
     \]
   - For \( y < \frac{1}{2}x + 1 \):
     \[
     2 < \frac{1}{2}(2) + 1 \quad \Rightarrow \quad 2 < 1 + 1 \quad \Rightarrow \quad 2 < 2 \quad \text{(False)}
     \]
   Since the second inequality is false, \((2, 2)\) is **not** a solution.

### 3. Checking point \((4, 1)\):
   - For \( y > -x - 4 \):
     \[
     1 > -(4) - 4 \quad \Rightarrow \quad 1 > -4 - 4 \quad \Rightarrow \quad 1 > -8 \quad \text{(True)}
     \]
   - For \( y < \frac{1}{2}x + 1 \):
     \[
     1 < \frac{1}{2}(4) + 1 \quad \Rightarrow \quad 1 < 2 + 1 \quad \Rightarrow \quad 1 < 3 \quad \text{(True)}
     \]
   Since both inequalities are true, \((4, 1)\) is a **solution**.

### 4. Checking point \((-4, -2)\):
   - For \( y > -x - 4 \):
     \[
     -2 > -(-4) - 4 \quad \Rightarrow \quad -2 > 4 - 4 \quad \Rightarrow \quad -2 > 0 \quad \text{(False)}
     \]
   Since the first inequality is false, \((-4, -2)\) is **not** a solution.

### Conclusion:
The only solution among the given points is **(4, 1)**.

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Would you like me to visualize the graph for you or provide more details? Here are 5 questions you can explore next:

1. What are the boundary lines for the system of inequalities?
2. How do you determine which side of the line to shade?
3. Can a point on a dashed boundary line be a solution?
4. How can you verify graphically if a point is a solution?
5. What happens when inequalities have the same boundary line?

**Tip:** Always check each inequality separately to verify if a point is a solution to the system of inequalities!

Graph the system of inequalities. Determine all possible solutions from the given ordered pairs: y > -x - 4, y < 1/2x + 1, and points (-1, -3), (2, 2), (4, 1), (-4, -2).

Explore how to graph the system of inequalities y > -x - 4 and y < 1/2x + 1. Determine which of the points (-1, -3), (2, 2), (4, 1), (-4, -2) are solutions to the system using key algebraic methods. Suitable for students in Grades 8-10.

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