solve the inequation `2x - y ge3` by graphical method.

Explanation

To solve the inequation \(2x - y \geq 3\) using the graphical method, we will follow these steps:

### Step 1: Convert the Inequation to an Equation
First, we convert the inequality into an equation to find the boundary line. The equation corresponding to the inequality \(2x - y \geq 3\) is:
\[ 2x - y = 3 \]

### Step 2: Rearranging the Equation
Next, we rearrange the equation to express \(y\) in terms of \(x\):
\[ y = 2x - 3 \]

### Step 3: Find Intercepts
To graph the line, we can find the x-intercept and y-intercept.

- **X-intercept**: Set \(y = 0\):
\[ 0 = 2x - 3 \implies 2x = 3 \implies x = \frac{3}{2} = 1.5 \]
So, the x-intercept is \(\left(\frac{3}{2}, 0\right)\).

- **Y-intercept**: Set \(x = 0\):
\[ y = 2(0) - 3 = -3 \]
So, the y-intercept is \((0, -3)\).

### Step 4: Plot the Points
Now, we plot the points \(\left(\frac{3}{2}, 0\right)\) and \((0, -3)\) on the graph.

### Step 5: Draw the Boundary Line
Next, we draw the line that passes through these two points. Since the inequality is \(\geq\), we will use a solid line to indicate that points on the line are included in the solution set.

### Step 6: Determine the Solution Region
To find which side of the line represents the solution to the inequality \(2x - y \geq 3\), 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:
\[ 2(0) - 0 \geq 3 \implies 0 \geq 3 \]
This is false, so the origin does not satisfy the inequality. Therefore, the solution region is on the opposite side of the line from the origin.

### Step 7: Shade the Solution Region
Finally, we shade the region that satisfies the inequality \(2x - y \geq 3\). This region is above the line.

### Summary
The solution to the inequation \(2x - y \geq 3\) is the region above the line \(y = 2x - 3\), including the line itself.

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