Find the best answers to your questions at Westonci.ca, where experts and enthusiasts provide accurate, reliable information. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To solve the problem, follow these steps:
### Step 1: Identify a test point on the coordinate plane
A common test point that can be used to check the inequality is [tex]\((0, 0)\)[/tex] because it's the origin and usually simplifies calculations.
### Step 2: Plug the test point into the inequality
The given inequality is [tex]\(\frac{2}{3} x - 2 y \geq 1\)[/tex]. Substituting [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality:
[tex]\[ \frac{2}{3} (0) - 2 (0) \geq 1 \][/tex]
This simplifies to:
[tex]\[ 0 \geq 1 \][/tex]
### Step 3: Determine the truth of the statement
The inequality [tex]\(0 \geq 1\)[/tex] is false. Therefore, the point [tex]\((0, 0)\)[/tex] does not satisfy the inequality.
### Step 4: Choose another test point
Since the point [tex]\((0, 0)\)[/tex] is not a solution, consider picking another point, preferably from the half-plane not including (0,0), such as [tex]\((3, 0)\)[/tex].
### Step 5: Plug in the new point
Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality:
[tex]\[ \frac{2}{3} (3) - 2 (0) \geq 1 \][/tex]
This simplifies to:
[tex]\[ 2 \geq 1 \][/tex]
### Step 6: Determine the truth of the statement
The inequality [tex]\(2 \geq 1\)[/tex] holds true. Therefore, the test point [tex]\((3, 0)\)[/tex] satisfies the inequality.
### Step 7: Identify the shaded region based on test points
Since the point [tex]\((0, 0)\)[/tex] does not satisfy the inequality but the point [tex]\((3, 0)\)[/tex] does, the shaded area for the inequality [tex]\(\frac{2}{3} x - 2 y \geq 1\)[/tex] lies on the side of the boundary line that includes the point [tex]\((3, 0)\)[/tex].
### Final Answer:
The test point [tex]\((3, 0)\)[/tex] holds true for this inequality. The shaded area for the inequality lies above or on the boundary line.
### Step 1: Identify a test point on the coordinate plane
A common test point that can be used to check the inequality is [tex]\((0, 0)\)[/tex] because it's the origin and usually simplifies calculations.
### Step 2: Plug the test point into the inequality
The given inequality is [tex]\(\frac{2}{3} x - 2 y \geq 1\)[/tex]. Substituting [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality:
[tex]\[ \frac{2}{3} (0) - 2 (0) \geq 1 \][/tex]
This simplifies to:
[tex]\[ 0 \geq 1 \][/tex]
### Step 3: Determine the truth of the statement
The inequality [tex]\(0 \geq 1\)[/tex] is false. Therefore, the point [tex]\((0, 0)\)[/tex] does not satisfy the inequality.
### Step 4: Choose another test point
Since the point [tex]\((0, 0)\)[/tex] is not a solution, consider picking another point, preferably from the half-plane not including (0,0), such as [tex]\((3, 0)\)[/tex].
### Step 5: Plug in the new point
Substitute [tex]\(x = 3\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality:
[tex]\[ \frac{2}{3} (3) - 2 (0) \geq 1 \][/tex]
This simplifies to:
[tex]\[ 2 \geq 1 \][/tex]
### Step 6: Determine the truth of the statement
The inequality [tex]\(2 \geq 1\)[/tex] holds true. Therefore, the test point [tex]\((3, 0)\)[/tex] satisfies the inequality.
### Step 7: Identify the shaded region based on test points
Since the point [tex]\((0, 0)\)[/tex] does not satisfy the inequality but the point [tex]\((3, 0)\)[/tex] does, the shaded area for the inequality [tex]\(\frac{2}{3} x - 2 y \geq 1\)[/tex] lies on the side of the boundary line that includes the point [tex]\((3, 0)\)[/tex].
### Final Answer:
The test point [tex]\((3, 0)\)[/tex] holds true for this inequality. The shaded area for the inequality lies above or on the boundary line.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.