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

Answer

To solve the inequality \(3x + 2y > 6\) graphically, we will follow these steps:

### Step 1: Convert the inequality to an equation
To graph the inequality, we first convert it into an equation:
\[ 3x + 2y = 6 \]

### Step 2: Find the intercepts
To graph the line, we need to find the x-intercept and y-intercept.

- **Finding the x-intercept**: Set \(y = 0\) in the equation.
\[ 3x + 2(0) = 6 \implies 3x = 6 \implies x = 2 \]
So, the x-intercept is \((2, 0)\).

- **Finding the y-intercept**: Set \(x = 0\) in the equation.
\[ 3(0) + 2y = 6 \implies 2y = 6 \implies y = 3 \]
So, the y-intercept is \((0, 3)\).

### Step 3: Plot the line
Now, we plot the points \((2, 0)\) and \((0, 3)\) on the graph and draw a straight line through these points. Since the inequality is strict (greater than), 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 that is not on the line. A convenient point to test is the origin \((0, 0)\).

Substituting \((0, 0)\) into the inequality:
\[ 3(0) + 2(0) > 6 \implies 0 > 6 \]
This statement is false, so the region that includes the origin is not part of the solution. Therefore, we shade the region that does not include the origin, which is the area above the line.

### Step 5: Final representation
The solution to the inequality \(3x + 2y > 6\) is the shaded region above the dashed line, excluding the line itself.