`3x + 4y le 12`

Explanation

To solve the inequality \(3x + 4y \leq 12\) and plot it on a graph, we can follow these steps:

### Step 1: Rewrite the inequality as an equation
First, we convert the inequality into an equation to find the boundary line:
\[ 3x + 4y = 12 \]

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

**Finding the x-intercept:**
Set \(y = 0\):
\[ 3x + 4(0) = 12 \implies 3x = 12 \implies x = 4 \]
So, the x-intercept is \((4, 0)\).

**Finding the y-intercept:**
Set \(x = 0\):
\[ 3(0) + 4y = 12 \implies 4y = 12 \implies y = 3 \]
So, the y-intercept is \((0, 3)\).

### Step 3: Plot the intercepts on a graph
Now, we plot the points \((4, 0)\) and \((0, 3)\) on the graph.

### Step 4: Draw the boundary line
Draw a straight line through the points \((4, 0)\) and \((0, 3)\). Since the inequality is less than or equal to (\(\leq\)), we will use a solid line to indicate that points on the line are included in the solution set.

### Step 5: 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 common choice is the origin \((0, 0)\):
\[ 3(0) + 4(0) \leq 12 \implies 0 \leq 12 \]
This is true, so we shade the region that includes the origin.

### Step 6: Final representation
The solution to the inequality \(3x + 4y \leq 12\) is the shaded region on one side of the line, including the line itself.