`x - 2y le 4`

Explanation

To solve the inequality \( x - 2y \leq 4 \) graphically, we will follow these steps:

### Step 1: Convert the inequality to an equation
We start by converting the inequality into an equation to find the boundary line:
\[ x - 2y = 4 \]

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

**Finding the x-intercept:**
Set \( y = 0 \):
\[ x - 2(0) = 4 \implies x = 4 \]
So, the x-intercept is \( (4, 0) \).

**Finding the y-intercept:**
Set \( x = 0 \):
\[ 0 - 2y = 4 \implies -2y = 4 \implies y = -2 \]
So, the y-intercept is \( (0, -2) \).

### Step 3: Plot the points
Now we plot the points \( (4, 0) \) and \( (0, -2) \) on the Cartesian plane.

### Step 4: Draw the boundary line
Draw a straight line through the points \( (4, 0) \) and \( (0, -2) \). Since the inequality is less than or equal to (≤), we will use a solid line to indicate that points on the line are included in the solution.

### Step 5: Determine the shading region
To determine which side of the line to shade, we can test a point that is not on the line. A common choice is the origin \( (0, 0) \).

Substituting \( (0, 0) \) into the inequality:
\[ 0 - 2(0) \leq 4 \implies 0 \leq 4 \]
This is true, so we shade the region that includes the origin.

### Step 6: Final representation
The shaded region represents all the solutions to the inequality \( x - 2y \leq 4 \). Any point in this region satisfies the inequality.

### Summary
The solution to the inequality \( x - 2y \leq 4 \) is represented graphically by the shaded region below and including the line \( x - 2y = 4 \).

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