Answer :
To determine which points satisfy the linear inequality \( y < 0.5x + 2 \), we need to test each given point against the inequality. We will substitute the \(x\) and \(y\) values from each point into the inequality and check if the inequality holds true.
Let's evaluate each point step-by-step:
1. Point \((-3, -2)\):
- Substitute \(x = -3\) and \(y = -2\) into the inequality.
- Compute the right-hand side: \( 0.5 \times (-3) + 2 = -1.5 + 2 = 0.5 \).
- Compare with \( y \): \(-2 < 0.5\).
- Since \(-2\) is less than \(0.5\), the point \((-3, -2)\) satisfies the inequality.
2. Point \((-2, 1)\):
- Substitute \(x = -2\) and \(y = 1\) into the inequality.
- Compute the right-hand side: \( 0.5 \times (-2) + 2 = -1 + 2 = 1 \).
- Compare with \( y \): \(1 < 1\).
- Since \(1\) is not less than \(1\), the point \((-2, 1)\) does not satisfy the inequality.
3. Point \((-1, -2)\):
- Substitute \(x = -1\) and \(y = -2\) into the inequality.
- Compute the right-hand side: \( 0.5 \times (-1) + 2 = -0.5 + 2 = 1.5 \).
- Compare with \( y \): \(-2 < 1.5\).
- Since \(-2\) is less than \(1.5\), the point \((-1, -2)\) satisfies the inequality.
4. Point \((-1, 2)\):
- Substitute \(x = -1\) and \(y = 2\) into the inequality.
- Compute the right-hand side: \( 0.5 \times (-1) + 2 = -0.5 + 2 = 1.5 \).
- Compare with \( y \): \(2 < 1.5\).
- Since \(2\) is not less than \(1.5\), the point \((-1, 2)\) does not satisfy the inequality.
5. Point \((1, -2)\):
- Substitute \(x = 1\) and \(y = -2\) into the inequality.
- Compute the right-hand side: \( 0.5 \times (1) + 2 = 0.5 + 2 = 2.5 \).
- Compare with \( y \): \(-2 < 2.5\).
- Since \(-2\) is less than \(2.5\), the point \((1, -2)\) satisfies the inequality.
Based on these evaluations, the points that satisfy the inequality \( y < 0.5 x + 2 \) are:
- \((-3, -2)\)
- \((-1, -2)\)
- \((1, -2)\)
So, the three options that are solutions to the inequality are:
- \((-3, -2)\)
- \((-1, -2)\)
- [tex]\((1, -2)\)[/tex]
Let's evaluate each point step-by-step:
1. Point \((-3, -2)\):
- Substitute \(x = -3\) and \(y = -2\) into the inequality.
- Compute the right-hand side: \( 0.5 \times (-3) + 2 = -1.5 + 2 = 0.5 \).
- Compare with \( y \): \(-2 < 0.5\).
- Since \(-2\) is less than \(0.5\), the point \((-3, -2)\) satisfies the inequality.
2. Point \((-2, 1)\):
- Substitute \(x = -2\) and \(y = 1\) into the inequality.
- Compute the right-hand side: \( 0.5 \times (-2) + 2 = -1 + 2 = 1 \).
- Compare with \( y \): \(1 < 1\).
- Since \(1\) is not less than \(1\), the point \((-2, 1)\) does not satisfy the inequality.
3. Point \((-1, -2)\):
- Substitute \(x = -1\) and \(y = -2\) into the inequality.
- Compute the right-hand side: \( 0.5 \times (-1) + 2 = -0.5 + 2 = 1.5 \).
- Compare with \( y \): \(-2 < 1.5\).
- Since \(-2\) is less than \(1.5\), the point \((-1, -2)\) satisfies the inequality.
4. Point \((-1, 2)\):
- Substitute \(x = -1\) and \(y = 2\) into the inequality.
- Compute the right-hand side: \( 0.5 \times (-1) + 2 = -0.5 + 2 = 1.5 \).
- Compare with \( y \): \(2 < 1.5\).
- Since \(2\) is not less than \(1.5\), the point \((-1, 2)\) does not satisfy the inequality.
5. Point \((1, -2)\):
- Substitute \(x = 1\) and \(y = -2\) into the inequality.
- Compute the right-hand side: \( 0.5 \times (1) + 2 = 0.5 + 2 = 2.5 \).
- Compare with \( y \): \(-2 < 2.5\).
- Since \(-2\) is less than \(2.5\), the point \((1, -2)\) satisfies the inequality.
Based on these evaluations, the points that satisfy the inequality \( y < 0.5 x + 2 \) are:
- \((-3, -2)\)
- \((-1, -2)\)
- \((1, -2)\)
So, the three options that are solutions to the inequality are:
- \((-3, -2)\)
- \((-1, -2)\)
- [tex]\((1, -2)\)[/tex]
