Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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).
[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).
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
