Answer :
To determine which of the given points satisfy the inequality [tex]\( y < 2x + 1 \)[/tex], we will evaluate each point against the inequality step-by-step.
### Checking the points against the inequality:
Point [tex]\((-2, -5)\)[/tex]:
- Here, [tex]\(x = -2\)[/tex] and [tex]\(y = -5\)[/tex].
- Substitute [tex]\(x\)[/tex] into the inequality:
[tex]\[ y < 2(-2) + 1 \][/tex]
[tex]\[ y < -4 + 1 \][/tex]
[tex]\[ y < -3 \][/tex]
- Now check if [tex]\(-5 < -3\)[/tex]:
[tex]\[ -5 < -3 \text{ (True)} \][/tex]
- So, [tex]\((-2, -5)\)[/tex] satisfies the inequality.
Point [tex]\((0, -4)\)[/tex]:
- Here, [tex]\(x = 0\)[/tex] and [tex]\(y = -4\)[/tex].
- Substitute [tex]\(x\)[/tex] into the inequality:
[tex]\[ y < 2(0) + 1 \][/tex]
[tex]\[ y < 1 \][/tex]
- Now check if [tex]\(-4 < 1\)[/tex]:
[tex]\[ -4 < 1 \text{ (True)} \][/tex]
- So, [tex]\((0, -4)\)[/tex] satisfies the inequality.
Point [tex]\((1, 1)\)[/tex]:
- Here, [tex]\(x = 1\)[/tex] and [tex]\(y = 1\)[/tex].
- Substitute [tex]\(x\)[/tex] into the inequality:
[tex]\[ y < 2(1) + 1 \][/tex]
[tex]\[ y < 3 \][/tex]
- Now check if [tex]\(1 < 3\)[/tex]:
[tex]\[ 1 < 3 \text{ (True)} \][/tex]
- So, [tex]\((1, 1)\)[/tex] satisfies the inequality.
Point [tex]\((3, 5)\)[/tex]:
- Here, [tex]\(x = 3\)[/tex] and [tex]\(y = 5\)[/tex].
- Substitute [tex]\(x\)[/tex] into the inequality:
[tex]\[ y < 2(3) + 1 \][/tex]
[tex]\[ y < 6 + 1 \][/tex]
[tex]\[ y < 7 \][/tex]
- Now check if [tex]\(5 < 7\)[/tex]:
[tex]\[ 5 < 7 \text{ (True)} \][/tex]
- So, [tex]\((3, 5)\)[/tex] satisfies the inequality.
Point [tex]\((5, 5)\)[/tex]:
- Here, [tex]\(x = 5\)[/tex] and [tex]\(y = 5\)[/tex].
- Substitute [tex]\(x\)[/tex] into the inequality:
[tex]\[ y < 2(5) + 1 \][/tex]
[tex]\[ y < 10 + 1 \][/tex]
[tex]\[ y < 11 \][/tex]
- Now check if [tex]\(5 < 11\)[/tex]:
[tex]\[ 5 < 11 \text{ (True)} \][/tex]
- So, [tex]\((5, 5)\)[/tex] satisfies the inequality.
### Conclusion:
All the given points satisfy the inequality [tex]\( y < 2x + 1 \)[/tex]:
- [tex]\((-2, -5)\)[/tex]
- [tex]\((0, -4)\)[/tex]
- [tex]\((1, 1)\)[/tex]
- [tex]\((3, 5)\)[/tex]
- [tex]\((5, 5)\)[/tex]
Thus, all the points mentioned are solutions to the inequality.
### Checking the points against the inequality:
Point [tex]\((-2, -5)\)[/tex]:
- Here, [tex]\(x = -2\)[/tex] and [tex]\(y = -5\)[/tex].
- Substitute [tex]\(x\)[/tex] into the inequality:
[tex]\[ y < 2(-2) + 1 \][/tex]
[tex]\[ y < -4 + 1 \][/tex]
[tex]\[ y < -3 \][/tex]
- Now check if [tex]\(-5 < -3\)[/tex]:
[tex]\[ -5 < -3 \text{ (True)} \][/tex]
- So, [tex]\((-2, -5)\)[/tex] satisfies the inequality.
Point [tex]\((0, -4)\)[/tex]:
- Here, [tex]\(x = 0\)[/tex] and [tex]\(y = -4\)[/tex].
- Substitute [tex]\(x\)[/tex] into the inequality:
[tex]\[ y < 2(0) + 1 \][/tex]
[tex]\[ y < 1 \][/tex]
- Now check if [tex]\(-4 < 1\)[/tex]:
[tex]\[ -4 < 1 \text{ (True)} \][/tex]
- So, [tex]\((0, -4)\)[/tex] satisfies the inequality.
Point [tex]\((1, 1)\)[/tex]:
- Here, [tex]\(x = 1\)[/tex] and [tex]\(y = 1\)[/tex].
- Substitute [tex]\(x\)[/tex] into the inequality:
[tex]\[ y < 2(1) + 1 \][/tex]
[tex]\[ y < 3 \][/tex]
- Now check if [tex]\(1 < 3\)[/tex]:
[tex]\[ 1 < 3 \text{ (True)} \][/tex]
- So, [tex]\((1, 1)\)[/tex] satisfies the inequality.
Point [tex]\((3, 5)\)[/tex]:
- Here, [tex]\(x = 3\)[/tex] and [tex]\(y = 5\)[/tex].
- Substitute [tex]\(x\)[/tex] into the inequality:
[tex]\[ y < 2(3) + 1 \][/tex]
[tex]\[ y < 6 + 1 \][/tex]
[tex]\[ y < 7 \][/tex]
- Now check if [tex]\(5 < 7\)[/tex]:
[tex]\[ 5 < 7 \text{ (True)} \][/tex]
- So, [tex]\((3, 5)\)[/tex] satisfies the inequality.
Point [tex]\((5, 5)\)[/tex]:
- Here, [tex]\(x = 5\)[/tex] and [tex]\(y = 5\)[/tex].
- Substitute [tex]\(x\)[/tex] into the inequality:
[tex]\[ y < 2(5) + 1 \][/tex]
[tex]\[ y < 10 + 1 \][/tex]
[tex]\[ y < 11 \][/tex]
- Now check if [tex]\(5 < 11\)[/tex]:
[tex]\[ 5 < 11 \text{ (True)} \][/tex]
- So, [tex]\((5, 5)\)[/tex] satisfies the inequality.
### Conclusion:
All the given points satisfy the inequality [tex]\( y < 2x + 1 \)[/tex]:
- [tex]\((-2, -5)\)[/tex]
- [tex]\((0, -4)\)[/tex]
- [tex]\((1, 1)\)[/tex]
- [tex]\((3, 5)\)[/tex]
- [tex]\((5, 5)\)[/tex]
Thus, all the points mentioned are solutions to the inequality.
