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.
