Answer :
To determine which ordered pairs satisfy both inequalities:
1. [tex]\( y \leq -x + 1 \)[/tex]
2. [tex]\( y > x \)[/tex]
Let's evaluate the given pairs:
1. Pair [tex]\((-1, 2)\)[/tex]:
- Check the first inequality: [tex]\( y \leq -x + 1 \)[/tex]
[tex]\[ 2 \leq -(-1) + 1 \implies 2 \leq 2 \][/tex]
This is true.
- Check the second inequality: [tex]\( y > x \)[/tex]
[tex]\[ 2 > -1 \][/tex]
This is true.
- Thus, [tex]\((-1, 2)\)[/tex] satisfies both inequalities.
2. Pair [tex]\((0, 1)\)[/tex]:
- Check the first inequality: [tex]\( y \leq -x + 1 \)[/tex]
[tex]\[ 1 \leq -(0) + 1 \implies 1 \leq 1 \][/tex]
This is true.
- Check the second inequality: [tex]\( y > x \)[/tex]
[tex]\[ 1 > 0 \][/tex]
This is true.
- Thus, [tex]\((0, 1)\)[/tex] satisfies both inequalities.
3. Pair [tex]\((1, 0)\)[/tex]:
- Check the first inequality: [tex]\( y \leq -x + 1 \)[/tex]
[tex]\[ 0 \leq -1 + 1 \implies 0 \leq 0 \][/tex]
This is true.
- Check the second inequality: [tex]\( y > x \)[/tex]
[tex]\[ 0 > 1 \][/tex]
This is false.
- Thus, [tex]\((1, 0)\)[/tex] does not satisfy both inequalities.
4. Pair [tex]\((2, -1)\)[/tex]:
- Check the first inequality: [tex]\( y \leq -x + 1 \)[/tex]
[tex]\[ -1 \leq -2 + 1 \implies -1 \leq -1 \][/tex]
This is true.
- Check the second inequality: [tex]\( y > x \)[/tex]
[tex]\[ -1 > 2 \][/tex]
This is false.
- Thus, [tex]\((2, -1)\)[/tex] does not satisfy both inequalities.
5. Pair [tex]\((0.5, 0.75)\)[/tex]:
- Check the first inequality: [tex]\( y \leq -x + 1 \)[/tex]
[tex]\[ 0.75 \leq -0.5 + 1 \implies 0.75 \leq 0.5 \][/tex]
This is false.
- Therefore, no need to check the second inequality.
- Thus, [tex]\((0.5, 0.75)\)[/tex] does not satisfy both inequalities.
In summary, the ordered pairs that make both inequalities true are:
[tex]\[ (-1, 2) \text{ and } (0, 1) \][/tex]
1. [tex]\( y \leq -x + 1 \)[/tex]
2. [tex]\( y > x \)[/tex]
Let's evaluate the given pairs:
1. Pair [tex]\((-1, 2)\)[/tex]:
- Check the first inequality: [tex]\( y \leq -x + 1 \)[/tex]
[tex]\[ 2 \leq -(-1) + 1 \implies 2 \leq 2 \][/tex]
This is true.
- Check the second inequality: [tex]\( y > x \)[/tex]
[tex]\[ 2 > -1 \][/tex]
This is true.
- Thus, [tex]\((-1, 2)\)[/tex] satisfies both inequalities.
2. Pair [tex]\((0, 1)\)[/tex]:
- Check the first inequality: [tex]\( y \leq -x + 1 \)[/tex]
[tex]\[ 1 \leq -(0) + 1 \implies 1 \leq 1 \][/tex]
This is true.
- Check the second inequality: [tex]\( y > x \)[/tex]
[tex]\[ 1 > 0 \][/tex]
This is true.
- Thus, [tex]\((0, 1)\)[/tex] satisfies both inequalities.
3. Pair [tex]\((1, 0)\)[/tex]:
- Check the first inequality: [tex]\( y \leq -x + 1 \)[/tex]
[tex]\[ 0 \leq -1 + 1 \implies 0 \leq 0 \][/tex]
This is true.
- Check the second inequality: [tex]\( y > x \)[/tex]
[tex]\[ 0 > 1 \][/tex]
This is false.
- Thus, [tex]\((1, 0)\)[/tex] does not satisfy both inequalities.
4. Pair [tex]\((2, -1)\)[/tex]:
- Check the first inequality: [tex]\( y \leq -x + 1 \)[/tex]
[tex]\[ -1 \leq -2 + 1 \implies -1 \leq -1 \][/tex]
This is true.
- Check the second inequality: [tex]\( y > x \)[/tex]
[tex]\[ -1 > 2 \][/tex]
This is false.
- Thus, [tex]\((2, -1)\)[/tex] does not satisfy both inequalities.
5. Pair [tex]\((0.5, 0.75)\)[/tex]:
- Check the first inequality: [tex]\( y \leq -x + 1 \)[/tex]
[tex]\[ 0.75 \leq -0.5 + 1 \implies 0.75 \leq 0.5 \][/tex]
This is false.
- Therefore, no need to check the second inequality.
- Thus, [tex]\((0.5, 0.75)\)[/tex] does not satisfy both inequalities.
In summary, the ordered pairs that make both inequalities true are:
[tex]\[ (-1, 2) \text{ and } (0, 1) \][/tex]
