Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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.
### 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.
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.