The position of points O (0,0) and P `(2,-2)` in the region of graph of inequation `2x-3ylt5` , will be

Options

A. O inside and P outside
B. O and P both inside
C. O outside and P outside
D. O outside and P inside

Answer

To determine the position of points O (0,0) and P (2,-2) in the region defined by the inequality \(2x - 3y < 5\), we will follow these steps:

### Step 1: Rewrite the Inequality
We start with the given inequality:
\[ 2x - 3y < 5 \]
To analyze the boundary, we first consider the corresponding equation:
\[ 2x - 3y = 5 \]

### Step 2: Find the Intercepts
To graph the line represented by the equation, we find the x-intercept and y-intercept.

**Finding the x-intercept:**
Set \(y = 0\):
\[ 2x - 3(0) = 5 \implies 2x = 5 \implies x = \frac{5}{2} = 2.5 \]
So, the x-intercept is \((2.5, 0)\).

**Finding the y-intercept:**
Set \(x = 0\):
\[ 2(0) - 3y = 5 \implies -3y = 5 \implies y = -\frac{5}{3} \approx -1.67 \]
So, the y-intercept is \((0, -\frac{5}{3})\).

### Step 3: Plot the Line
We can now plot the line \(2x - 3y = 5\) using the intercepts:
- x-intercept: \((2.5, 0)\)
- y-intercept: \((0, -\frac{5}{3})\)

### Step 4: Determine the Region
Since the inequality is \(2x - 3y < 5\), we shade the region below the line (towards the origin).

### Step 5: Test Point O (0,0)
Now we check if point O (0,0) is in the shaded region:
\[ 2(0) - 3(0) < 5 \implies 0 < 5 \]
This is true, so point O is inside the region.

### Step 6: Test Point P (2,-2)
Next, we check if point P (2,-2) is in the shaded region:
\[ 2(2) - 3(-2) < 5 \implies 4 + 6 < 5 \implies 10 < 5 \]
This is false, so point P is outside the region.

### Conclusion
- Point O (0,0) is inside the region.
- Point P (2,-2) is outside the region.

### Final Answer
The position of points O and P in the region of the graph of the inequality \(2x - 3y < 5\) is:
- O is inside the region.
- P is outside the region.