`2x + y ge 0`

Explanation

To solve the inequality \(2x + y \geq 0\) and plot its graph, follow these steps:

### Step 1: Convert the inequality into an equation
To begin, we convert the inequality into an equation:
\[ 2x + y = 0 \]

### Step 2: Find intercepts
Next, we find the x-intercept and y-intercept to plot the line.

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

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

### Step 3: Find additional points
Since the intercepts are the same, we need to find additional points to define the line better.

- Let’s choose \(x = 2\):
\[ 2(2) + y = 0 \implies 4 + y = 0 \implies y = -4 \]
This gives us the point \((2, -4)\).

- Now, let’s choose \(y = 2\):
\[ 2x + 2 = 0 \implies 2x = -2 \implies x = -1 \]
This gives us the point \((-1, 2)\).

### Step 4: Plot the points
We have the following points to plot:
1. \((0, 0)\)
2. \((2, -4)\)
3. \((-1, 2)\)

### Step 5: Draw the line
Since the inequality includes the equal sign (\(\geq\)), we will draw a solid line through the points \((0, 0)\), \((2, -4)\), and \((-1, 2)\).

### Step 6: Determine the shading region
To find out which side of the line to shade, we can test a point not on the line. A common choice is the origin \((0, 0)\):
\[ 2(0) + 0 \geq 0 \implies 0 \geq 0 \]
This is true, so we shade the side of the line that includes the origin.

### Final Graph
The final graph will show a solid line through the points \((0, 0)\), \((2, -4)\), and \((-1, 2)\) with the region containing the origin shaded.

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