To graph the linear inequality \( x - 3y \leq 6 \), follow these step-by-step instructions:
### Step 1: Understand the Inequality
The inequality \( x - 3y \leq 6 \) represents a region in the coordinate plane where the linear function \( x - 3y \) is less than or equal to 6.
### Step 2: Convert the Inequality to an Equation
First, we convert the inequality into an equation to find the boundary line:
[tex]\[ x - 3y = 6 \][/tex]
### Step 3: Find the Intercepts of the Line
To graph the boundary line, we'll find the x-intercept and y-intercept.
- X-Intercept: Set \( y = 0 \) in the equation \( x - 3y = 6 \):
[tex]\[ x - 3(0) = 6 \][/tex]
[tex]\[ x = 6 \][/tex]
The x-intercept is \( (6, 0) \).
- Y-Intercept: Set \( x = 0 \) in the equation \( x - 3y = 6 \):
[tex]\[ 0 - 3y = 6 \][/tex]
[tex]\[ -3y = 6 \][/tex]
[tex]\[ y = -2 \][/tex]
The y-intercept is \( (0, -2) \).
### Step 4: Draw the Boundary Line
Next, plot the intercepts on the coordinate plane and draw the line passing through these points:
- Point A: \( (6, 0) \)
- Point B: \( (0, -2) \)
Since the inequality is \( x - 3y \leq 6 \), the line \( x - 3y = 6 \) is solid (indicating that points on the line are included in the solution).
### Step 5: Determine the Shaded Region
We now need to determine which side of the line represents the solution to the inequality. Choose a test point not on the line. A common test point is the origin \( (0, 0) \).
- Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( x - 3y \leq 6 \):
[tex]\[ 0 - 3(0) \leq 6 \][/tex]
[tex]\[ 0 \leq 6 \][/tex]
This statement is true.
Since the origin satisfies the inequality, we shade the region that includes the origin.
### Final Graph
- Draw a solid line through points \( (6, 0) \) and \( (0, -2) \).
- Shade the entire region below and including this line.
The shaded region and the line represent the solution to the inequality \( x - 3y \leq 6 \).
Below is a sketch of the graph:
```
y
|
2 |
|
1 |
|
0____|____________ x
-6 -4 -2 0 2 4 6
-1 |
|
-2 ---------
|
```
In this graph, the line [tex]\( x - 3y = 6 \)[/tex] is represented and the area below this line (including the line) is shaded. The asterisks () represent the points [tex]\( (6,0) \)[/tex] and [tex]\( (0,-2) \)[/tex] respectively.
