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To determine which ordered pairs make both inequalities true, we will analyze each provided pair and test them against the given conditions:
1. Pair (0, 3):
- For [tex]\( y < 5x + 2 \)[/tex]:
[tex]\( 3 < 5(0) + 2 \)[/tex]
[tex]\( \Rightarrow 3 < 2 \)[/tex]
- This is false.
- Since the first condition fails, this pair does not satisfy both inequalities.
2. Pair (4, 3):
- For [tex]\( y < 5x + 2 \)[/tex]:
[tex]\( 3 < 5(4) + 2 \)[/tex]
[tex]\( \Rightarrow 3 < 20 + 2 \)[/tex]
[tex]\( \Rightarrow 3 < 22 \)[/tex]
- This is true.
- For [tex]\( y \geq \frac{1}{2}x + 1 \)[/tex]:
[tex]\( 3 \geq \frac{1}{2}(4) + 1 \)[/tex]
[tex]\( \Rightarrow 3 \geq 2 + 1 \)[/tex]
[tex]\( \Rightarrow 3 \geq 3 \)[/tex]
- This is true.
- Since both conditions are met, this pair satisfies both inequalities.
3. Pair (2, 9):
- For [tex]\( y < 5x + 2 \)[/tex]:
[tex]\( 9 < 5(2) + 2 \)[/tex]
[tex]\( \Rightarrow 9 < 10 + 2 \)[/tex]
[tex]\( \Rightarrow 9 < 12 \)[/tex]
- This is true.
- For [tex]\( y \geq \frac{1}{2}x + 1 \)[/tex]:
[tex]\( 9 \geq \frac{1}{2}(2) + 1 \)[/tex]
[tex]\( \Rightarrow 9 \geq 1 + 1 \)[/tex]
[tex]\( \Rightarrow 9 \geq 2 \)[/tex]
- This is true.
- Since both conditions are met, this pair satisfies both inequalities.
4. Pair (8, -2):
- For [tex]\( y < 5x + 2 \)[/tex]:
[tex]\( -2 < 5(8) + 2 \)[/tex]
[tex]\( \Rightarrow -2 < 40 + 2 \)[/tex]
[tex]\( \Rightarrow -2 < 42 \)[/tex]
- This is true.
- For [tex]\( y \geq \frac{1}{2}x + 1 \)[/tex]:
[tex]\( -2 \geq \frac{1}{2}(8) + 1 \)[/tex]
[tex]\( \Rightarrow -2 \geq 4 + 1 \)[/tex]
[tex]\( \Rightarrow -2 \geq 5 \)[/tex]
- This is false.
- Since the second condition fails, this pair does not satisfy both inequalities.
5. Pair (6, 10):
- For [tex]\( y < 5x + 2 \)[/tex]:
[tex]\( 10 < 5(6) + 2 \)[/tex]
[tex]\( \Rightarrow 10 < 30 + 2 \)[/tex]
[tex]\( \Rightarrow 10 < 32 \)[/tex]
- This is true.
- For [tex]\( y \geq \frac{1}{2}x + 1 \)[/tex]:
[tex]\( 10 \geq \frac{1}{2}(6) + 1 \)[/tex]
[tex]\( \Rightarrow 10 \geq 3 + 1 \)[/tex]
[tex]\( \Rightarrow 10 \geq 4 \)[/tex]
- This is true.
- Since both conditions are met, this pair satisfies both inequalities.
Therefore, the ordered pairs that satisfy both inequalities are:
- [tex]\((4, 3)\)[/tex]
- [tex]\((2, 9)\)[/tex]
- [tex]\((6, 10)\)[/tex]
Since the task is to select two options, valid selections from the pairs above can be [tex]\((4, 3)\)[/tex] and [tex]\((2, 9)\)[/tex], or any other combination of two pairs from the valid options.
1. Pair (0, 3):
- For [tex]\( y < 5x + 2 \)[/tex]:
[tex]\( 3 < 5(0) + 2 \)[/tex]
[tex]\( \Rightarrow 3 < 2 \)[/tex]
- This is false.
- Since the first condition fails, this pair does not satisfy both inequalities.
2. Pair (4, 3):
- For [tex]\( y < 5x + 2 \)[/tex]:
[tex]\( 3 < 5(4) + 2 \)[/tex]
[tex]\( \Rightarrow 3 < 20 + 2 \)[/tex]
[tex]\( \Rightarrow 3 < 22 \)[/tex]
- This is true.
- For [tex]\( y \geq \frac{1}{2}x + 1 \)[/tex]:
[tex]\( 3 \geq \frac{1}{2}(4) + 1 \)[/tex]
[tex]\( \Rightarrow 3 \geq 2 + 1 \)[/tex]
[tex]\( \Rightarrow 3 \geq 3 \)[/tex]
- This is true.
- Since both conditions are met, this pair satisfies both inequalities.
3. Pair (2, 9):
- For [tex]\( y < 5x + 2 \)[/tex]:
[tex]\( 9 < 5(2) + 2 \)[/tex]
[tex]\( \Rightarrow 9 < 10 + 2 \)[/tex]
[tex]\( \Rightarrow 9 < 12 \)[/tex]
- This is true.
- For [tex]\( y \geq \frac{1}{2}x + 1 \)[/tex]:
[tex]\( 9 \geq \frac{1}{2}(2) + 1 \)[/tex]
[tex]\( \Rightarrow 9 \geq 1 + 1 \)[/tex]
[tex]\( \Rightarrow 9 \geq 2 \)[/tex]
- This is true.
- Since both conditions are met, this pair satisfies both inequalities.
4. Pair (8, -2):
- For [tex]\( y < 5x + 2 \)[/tex]:
[tex]\( -2 < 5(8) + 2 \)[/tex]
[tex]\( \Rightarrow -2 < 40 + 2 \)[/tex]
[tex]\( \Rightarrow -2 < 42 \)[/tex]
- This is true.
- For [tex]\( y \geq \frac{1}{2}x + 1 \)[/tex]:
[tex]\( -2 \geq \frac{1}{2}(8) + 1 \)[/tex]
[tex]\( \Rightarrow -2 \geq 4 + 1 \)[/tex]
[tex]\( \Rightarrow -2 \geq 5 \)[/tex]
- This is false.
- Since the second condition fails, this pair does not satisfy both inequalities.
5. Pair (6, 10):
- For [tex]\( y < 5x + 2 \)[/tex]:
[tex]\( 10 < 5(6) + 2 \)[/tex]
[tex]\( \Rightarrow 10 < 30 + 2 \)[/tex]
[tex]\( \Rightarrow 10 < 32 \)[/tex]
- This is true.
- For [tex]\( y \geq \frac{1}{2}x + 1 \)[/tex]:
[tex]\( 10 \geq \frac{1}{2}(6) + 1 \)[/tex]
[tex]\( \Rightarrow 10 \geq 3 + 1 \)[/tex]
[tex]\( \Rightarrow 10 \geq 4 \)[/tex]
- This is true.
- Since both conditions are met, this pair satisfies both inequalities.
Therefore, the ordered pairs that satisfy both inequalities are:
- [tex]\((4, 3)\)[/tex]
- [tex]\((2, 9)\)[/tex]
- [tex]\((6, 10)\)[/tex]
Since the task is to select two options, valid selections from the pairs above can be [tex]\((4, 3)\)[/tex] and [tex]\((2, 9)\)[/tex], or any other combination of two pairs from the valid options.
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