Represent the following inequations graphically in two dimensional plane and hence solve them:
`2x-3ygt6`

Answer

To solve the inequality \(2x - 3y > 6\) graphically in a two-dimensional plane, we will follow these steps:

### Step 1: Convert the Inequality to an Equation
First, we convert the inequality into an equation to find the boundary line:
\[ 2x - 3y = 6 \]

### Step 2: Find Points on the Line
Next, we will find two points on the line by substituting values for \(x\) and \(y\).

1. **When \(x = 0\)**:
\[ 2(0) - 3y = 6 \implies -3y = 6 \implies y = -2 \]
So, one point is \((0, -2)\).

2. **When \(y = 0\)**:
\[ 2x - 3(0) = 6 \implies 2x = 6 \implies x = 3 \]
So, another point is \((3, 0)\).

### Step 3: Plot the Points and Draw the Line
Now, we plot the points \((0, -2)\) and \((3, 0)\) on a coordinate plane and draw a line through these points. Since the inequality is strict (greater than, not greater than or equal to), we will use a dashed line to indicate that points on the line are not included in the solution.

### Step 4: Determine the Shaded Region
To find which side of the line to shade, we can test a point not on the line. A convenient point to test is the origin \((0, 0)\):

Substituting \((0, 0)\) into the inequality:
\[ 2(0) - 3(0) > 6 \implies 0 > 6 \quad \text{(False)} \]
Since this is false, we shade the region that does not include the origin, which is the region above the line.

### Step 5: Finalize the Solution
The solution set for the inequality \(2x - 3y > 6\) is the region above the dashed line, excluding the line itself.

### Summary of Steps:
1. Convert the inequality to an equation.
2. Find points on the line.
3. Plot the points and draw a dashed line.
4. Test a point to determine the shaded region.
5. Shade the appropriate region.