To determine which ordered pairs are in the solution set of the system of linear inequalities:
[tex]\[
\begin{array}{l}
y \geq -\frac{1}{2} x \\
y < \frac{1}{2} x + 1
\end{array}
\][/tex]
we need to verify each pair to see if it satisfies both inequalities.
Let's check each ordered pair:
1. Pair (5, -2):
- First inequality: \( -2 \geq -\frac{1}{2} \cdot 5 \Rightarrow -2 \geq -2.5 \) (True)
- Second inequality: \( -2 < \frac{1}{2} \cdot 5 + 1 \Rightarrow -2 < 3.5 \) (True)
- Both conditions are satisfied, so (5, -2) is in the solution set.
2. Pair (3, 1):
- First inequality: \( 1 \geq -\frac{1}{2} \cdot 3 \Rightarrow 1 \geq -1.5 \) (True)
- Second inequality: \( 1 < \frac{1}{2} \cdot 3 + 1 \Rightarrow 1 < 2.5 \) (True)
- Both conditions are satisfied, so (3, 1) is in the solution set.
3. Pair (-4, 2):
- First inequality: \( 2 \geq -\frac{1}{2} \cdot (-4) \Rightarrow 2 \geq 2 \) (True)
- Second inequality: \( 2 < \frac{1}{2} \cdot (-4) + 1 \Rightarrow 2 < -1 + 1 \Rightarrow 2 < 0 \) (False)
- The second condition is not satisfied, so (-4, 2) is not in the solution set.
4. Pair (3, -1):
- First inequality: \( -1 \geq -\frac{1}{2} \cdot 3 \Rightarrow -1 \geq -1.5 \) (True)
- Second inequality: \( -1 < \frac{1}{2} \cdot 3 + 1 \Rightarrow -1 < 2.5 \) (True)
- Both conditions are satisfied, so (3, -1) is in the solution set.
5. Pair (4, -3):
- First inequality: \( -3 \geq -\frac{1}{2} \cdot 4 \Rightarrow -3 \geq -2 \) (False)
- Second inequality: \( -3 < \frac{1}{2} \cdot 4 + 1 \Rightarrow -3 < 3 \) (True)
- The first condition is not satisfied, so (4, -3) is not in the solution set.
6. Pair (4, 2):
- First inequality: \( 2 \geq -\frac{1}{2} \cdot 4 \Rightarrow 2 \geq -2 \) (True)
- Second inequality: \( 2 < \frac{1}{2} \cdot 4 + 1 \Rightarrow 2 < 3 \) (True)
- Both conditions are satisfied, so (4, 2) is in the solution set.
7. Pair (-3, 1):
- First inequality: \( 1 \geq -\frac{1}{2} \cdot (-3) \Rightarrow 1 \geq 1.5 \) (False)
- Second inequality: \( 1 < \frac{1}{2} \cdot (-3) + 1 \Rightarrow 1 < -1.5 + 1 \Rightarrow 1 < -0.5 \) (False)
- Both conditions are not satisfied, so (-3, 1) is not in the solution set.
Based on the above, the ordered pairs that are in the solution set are:
- (5, -2)
- (3, 1)
- (3, -1)
- (4, 2)
Therefore, the ordered pairs in the solution set are:
- (5, -2), (3, 1), (3, -1), (4, 2).
