Answer :
To determine which points satisfy the inequality [tex]\( y < 0.5x + 2 \)[/tex], we'll check each point individually by substituting the [tex]\(x\)[/tex] and [tex]\( y\)[/tex] values into the inequality:
1. Point [tex]\((-3, -2)\)[/tex]:
- Substitute [tex]\( x = -3 \)[/tex] into the inequality: [tex]\( y < 0.5(-3) + 2 \)[/tex]
- Calculate the right-hand side: [tex]\( 0.5(-3) + 2 = -1.5 + 2 = 0.5 \)[/tex]
- So, the inequality becomes: [tex]\( -2 < 0.5 \)[/tex]
- This is true, so [tex]\((-3, -2)\)[/tex] satisfies the inequality.
2. Point [tex]\((-2, 1)\)[/tex]:
- Substitute [tex]\( x = -2 \)[/tex] into the inequality: [tex]\( y < 0.5(-2) + 2 \)[/tex]
- Calculate the right-hand side: [tex]\( 0.5(-2) + 2 = -1 + 2 = 1 \)[/tex]
- So, the inequality becomes: [tex]\( 1 < 1 \)[/tex]
- This is false, so [tex]\((-2, 1)\)[/tex] does not satisfy the inequality.
3. Point [tex]\((-1, -2)\)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] into the inequality: [tex]\( y < 0.5(-1) + 2 \)[/tex]
- Calculate the right-hand side: [tex]\( 0.5(-1) + 2 = -0.5 + 2 = 1.5 \)[/tex]
- So, the inequality becomes: [tex]\( -2 < 1.5 \)[/tex]
- This is true, so [tex]\((-1, -2)\)[/tex] satisfies the inequality.
4. Point [tex]\((-1, 2)\)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] into the inequality: [tex]\( y < 0.5(-1) + 2 \)[/tex]
- Calculate the right-hand side: [tex]\( 0.5(-1) + 2 = -0.5 + 2 = 1.5 \)[/tex]
- So, the inequality becomes: [tex]\( 2 < 1.5 \)[/tex]
- This is false, so [tex]\((-1, 2)\)[/tex] does not satisfy the inequality.
5. Point [tex]\((1, -2)\)[/tex]:
- Substitute [tex]\( x = 1 \)[/tex] into the inequality: [tex]\( y < 0.5(1) + 2 \)[/tex]
- Calculate the right-hand side: [tex]\( 0.5(1) + 2 = 0.5 + 2 = 2.5 \)[/tex]
- So, the inequality becomes: [tex]\( -2 < 2.5 \)[/tex]
- This is true, so [tex]\((1, -2)\)[/tex] satisfies the inequality.
The points that satisfy the inequality [tex]\( y < 0.5x + 2 \)[/tex] are:
- [tex]\((-3, -2)\)[/tex]
- [tex]\((-1, -2)\)[/tex]
- [tex]\((1, -2)\)[/tex]
These are the three options that are solutions to the inequality.
1. Point [tex]\((-3, -2)\)[/tex]:
- Substitute [tex]\( x = -3 \)[/tex] into the inequality: [tex]\( y < 0.5(-3) + 2 \)[/tex]
- Calculate the right-hand side: [tex]\( 0.5(-3) + 2 = -1.5 + 2 = 0.5 \)[/tex]
- So, the inequality becomes: [tex]\( -2 < 0.5 \)[/tex]
- This is true, so [tex]\((-3, -2)\)[/tex] satisfies the inequality.
2. Point [tex]\((-2, 1)\)[/tex]:
- Substitute [tex]\( x = -2 \)[/tex] into the inequality: [tex]\( y < 0.5(-2) + 2 \)[/tex]
- Calculate the right-hand side: [tex]\( 0.5(-2) + 2 = -1 + 2 = 1 \)[/tex]
- So, the inequality becomes: [tex]\( 1 < 1 \)[/tex]
- This is false, so [tex]\((-2, 1)\)[/tex] does not satisfy the inequality.
3. Point [tex]\((-1, -2)\)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] into the inequality: [tex]\( y < 0.5(-1) + 2 \)[/tex]
- Calculate the right-hand side: [tex]\( 0.5(-1) + 2 = -0.5 + 2 = 1.5 \)[/tex]
- So, the inequality becomes: [tex]\( -2 < 1.5 \)[/tex]
- This is true, so [tex]\((-1, -2)\)[/tex] satisfies the inequality.
4. Point [tex]\((-1, 2)\)[/tex]:
- Substitute [tex]\( x = -1 \)[/tex] into the inequality: [tex]\( y < 0.5(-1) + 2 \)[/tex]
- Calculate the right-hand side: [tex]\( 0.5(-1) + 2 = -0.5 + 2 = 1.5 \)[/tex]
- So, the inequality becomes: [tex]\( 2 < 1.5 \)[/tex]
- This is false, so [tex]\((-1, 2)\)[/tex] does not satisfy the inequality.
5. Point [tex]\((1, -2)\)[/tex]:
- Substitute [tex]\( x = 1 \)[/tex] into the inequality: [tex]\( y < 0.5(1) + 2 \)[/tex]
- Calculate the right-hand side: [tex]\( 0.5(1) + 2 = 0.5 + 2 = 2.5 \)[/tex]
- So, the inequality becomes: [tex]\( -2 < 2.5 \)[/tex]
- This is true, so [tex]\((1, -2)\)[/tex] satisfies the inequality.
The points that satisfy the inequality [tex]\( y < 0.5x + 2 \)[/tex] are:
- [tex]\((-3, -2)\)[/tex]
- [tex]\((-1, -2)\)[/tex]
- [tex]\((1, -2)\)[/tex]
These are the three options that are solutions to the inequality.
