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Sagot :
To solve the inequality [tex]\(\frac{3}{2} y - 2 x \geq 1\)[/tex] and determine which test points hold true and where the shaded area lies, we will analyze the results.
First, we need to evaluate whether the given test points satisfy the inequality. The test points provided for evaluation are:
1. [tex]\((0, 0)\)[/tex]
2. [tex]\((2, 0)\)[/tex]
3. [tex]\((0, 2)\)[/tex]
4. [tex]\((2, 2)\)[/tex]
5. [tex]\((-2, -2)\)[/tex]
### Evaluation of Test Points:
1. Test Point: (0, 0)
[tex]\[ \frac{3}{2}(0) - 2(0) = 0 \geq 1 \quad \text{(False, does not satisfy the inequality)} \][/tex]
2. Test Point: (2, 0)
[tex]\[ \frac{3}{2}(0) - 2(2) = -4 \geq 1 \quad \text{(False, does not satisfy the inequality)} \][/tex]
3. Test Point: (0, 2)
[tex]\[ \frac{3}{2}(2) - 2(0) = 3 \geq 1 \quad \text{(True, satisfies the inequality)} \][/tex]
4. Test Point: (2, 2)
[tex]\[ \frac{3}{2}(2) - 2(2) = 3 - 4 = -1 \geq 1 \quad \text{(False, does not satisfy the inequality)} \][/tex]
5. Test Point: (-2, -2)
[tex]\[ \frac{3}{2}(-2) - 2(-2) = -3 + 4 = 1 \geq 1 \quad \text{(True, satisfies the inequality)} \][/tex]
From these evaluations, the test points [tex]\((0, 2)\)[/tex] and [tex]\((-2, -2)\)[/tex] satisfy the inequality [tex]\(\frac{3}{2} y - 2 x \geq 1\)[/tex].
### Shaded Area Determination:
Now, we need to determine where the shaded area lies for this inequality. When solving inequalities of the form [tex]\(ay - bx \geq c\)[/tex], the shaded region typically lies above or towards the region where the inequality is satisfied when substituting points.
In the case of [tex]\(\frac{3}{2} y - 2 x \geq 1\)[/tex], the inequality will be satisfied by points that lie above the boundary line formed by the equality [tex]\(\frac{3}{2} y - 2 x = 1\)[/tex].
Thus, the shaded area for this inequality lies "above the boundary line."
### Final Answers:
- Which test point holds true for the inequality?
The test points [tex]\((0, 2)\)[/tex] and [tex]\((-2, -2)\)[/tex] hold true for the inequality [tex]\(\frac{3}{2} y - 2 x \geq 1\)[/tex].
- Where does the shaded area lie for this inequality?
The shaded area for the inequality lies above the boundary line.
Now, let’s select the correct answers from the drop-down menu based on the analysis above:
- The test point [tex]\((-2, -2)\)[/tex] holds true for this inequality.
- The shaded area for the inequality lies above the boundary line.
So, you can select [tex]\((-2, -2)\)[/tex] from the test points and "above" for the position of the shaded area in the drop-down menus.
First, we need to evaluate whether the given test points satisfy the inequality. The test points provided for evaluation are:
1. [tex]\((0, 0)\)[/tex]
2. [tex]\((2, 0)\)[/tex]
3. [tex]\((0, 2)\)[/tex]
4. [tex]\((2, 2)\)[/tex]
5. [tex]\((-2, -2)\)[/tex]
### Evaluation of Test Points:
1. Test Point: (0, 0)
[tex]\[ \frac{3}{2}(0) - 2(0) = 0 \geq 1 \quad \text{(False, does not satisfy the inequality)} \][/tex]
2. Test Point: (2, 0)
[tex]\[ \frac{3}{2}(0) - 2(2) = -4 \geq 1 \quad \text{(False, does not satisfy the inequality)} \][/tex]
3. Test Point: (0, 2)
[tex]\[ \frac{3}{2}(2) - 2(0) = 3 \geq 1 \quad \text{(True, satisfies the inequality)} \][/tex]
4. Test Point: (2, 2)
[tex]\[ \frac{3}{2}(2) - 2(2) = 3 - 4 = -1 \geq 1 \quad \text{(False, does not satisfy the inequality)} \][/tex]
5. Test Point: (-2, -2)
[tex]\[ \frac{3}{2}(-2) - 2(-2) = -3 + 4 = 1 \geq 1 \quad \text{(True, satisfies the inequality)} \][/tex]
From these evaluations, the test points [tex]\((0, 2)\)[/tex] and [tex]\((-2, -2)\)[/tex] satisfy the inequality [tex]\(\frac{3}{2} y - 2 x \geq 1\)[/tex].
### Shaded Area Determination:
Now, we need to determine where the shaded area lies for this inequality. When solving inequalities of the form [tex]\(ay - bx \geq c\)[/tex], the shaded region typically lies above or towards the region where the inequality is satisfied when substituting points.
In the case of [tex]\(\frac{3}{2} y - 2 x \geq 1\)[/tex], the inequality will be satisfied by points that lie above the boundary line formed by the equality [tex]\(\frac{3}{2} y - 2 x = 1\)[/tex].
Thus, the shaded area for this inequality lies "above the boundary line."
### Final Answers:
- Which test point holds true for the inequality?
The test points [tex]\((0, 2)\)[/tex] and [tex]\((-2, -2)\)[/tex] hold true for the inequality [tex]\(\frac{3}{2} y - 2 x \geq 1\)[/tex].
- Where does the shaded area lie for this inequality?
The shaded area for the inequality lies above the boundary line.
Now, let’s select the correct answers from the drop-down menu based on the analysis above:
- The test point [tex]\((-2, -2)\)[/tex] holds true for this inequality.
- The shaded area for the inequality lies above the boundary line.
So, you can select [tex]\((-2, -2)\)[/tex] from the test points and "above" for the position of the shaded area in the drop-down menus.
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