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Sagot :
To graph the solution set of the linear inequality [tex]\( 12x + 2 < -12y + 2 \)[/tex], we need to follow a few steps, as detailed below:
### Step 1: Simplify the Inequality
First, let's simplify the given inequality:
[tex]\[ 12x + 2 < -12y + 2 \][/tex]
Subtract 2 from both sides:
[tex]\[ 12x < -12y \][/tex]
Divide each term by 12:
[tex]\[ x < -y \][/tex]
This can also be rewritten as:
[tex]\[ y > -x \][/tex]
### Step 2: Find the Boundary Line
The boundary line is obtained by converting the inequality into an equation:
[tex]\[ y = -x \][/tex]
This boundary line divides the coordinate plane into two regions.
### Step 3: Determine the Type of Boundary Line
Since the inequality [tex]\( y > -x \)[/tex] does not include an equality (i.e., it is a strict inequality), we will use a dashed boundary line to show that points on the line itself are not included in the solution set.
### Step 4: Plot the Boundary Line
The boundary line [tex]\( y = -x \)[/tex] can be plotted by finding two points on the line:
1. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -0 = 0 \][/tex]
So, one point is (0,0).
2. When [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -2 = -2 \][/tex]
So, another point is (2, -2).
Using these points, we can draw a dashed line through them.
### Step 5: Identify the Shaded Region
The inequality [tex]\( y > -x \)[/tex] means that we need to shade the region above the line [tex]\( y = -x \)[/tex].
### Step 6: Final Graph
1. Draw a dashed line through the points [tex]\( (0,0) \)[/tex] and [tex]\( (2,-2) \)[/tex].
2. Shade the region above the dashed line [tex]\( y = -x \)[/tex].
Here’s a sketch of how the graph would look:
```
y
^
|
10+ |
|
|
|
|
| *
0 +------------------------------------------------------> x
|
-10|
|
|
-10
```
The dashed line represents [tex]\( y = -x \)[/tex], and the shaded area above the line is the solution set for the inequality [tex]\( y > -x \)[/tex].
### Summary
- Convert the inequality [tex]\( 12x + 2 < -12y + 2 \)[/tex] to [tex]\( y > -x \)[/tex].
- Draw the boundary line [tex]\( y = -x \)[/tex] using a dashed line since it is a strict inequality.
- Shade the region above the line to represent all the points that satisfy the inequality [tex]\( y > -x \)[/tex].
### Step 1: Simplify the Inequality
First, let's simplify the given inequality:
[tex]\[ 12x + 2 < -12y + 2 \][/tex]
Subtract 2 from both sides:
[tex]\[ 12x < -12y \][/tex]
Divide each term by 12:
[tex]\[ x < -y \][/tex]
This can also be rewritten as:
[tex]\[ y > -x \][/tex]
### Step 2: Find the Boundary Line
The boundary line is obtained by converting the inequality into an equation:
[tex]\[ y = -x \][/tex]
This boundary line divides the coordinate plane into two regions.
### Step 3: Determine the Type of Boundary Line
Since the inequality [tex]\( y > -x \)[/tex] does not include an equality (i.e., it is a strict inequality), we will use a dashed boundary line to show that points on the line itself are not included in the solution set.
### Step 4: Plot the Boundary Line
The boundary line [tex]\( y = -x \)[/tex] can be plotted by finding two points on the line:
1. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -0 = 0 \][/tex]
So, one point is (0,0).
2. When [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -2 = -2 \][/tex]
So, another point is (2, -2).
Using these points, we can draw a dashed line through them.
### Step 5: Identify the Shaded Region
The inequality [tex]\( y > -x \)[/tex] means that we need to shade the region above the line [tex]\( y = -x \)[/tex].
### Step 6: Final Graph
1. Draw a dashed line through the points [tex]\( (0,0) \)[/tex] and [tex]\( (2,-2) \)[/tex].
2. Shade the region above the dashed line [tex]\( y = -x \)[/tex].
Here’s a sketch of how the graph would look:
```
y
^
|
10+ |
|
|
|
|
| *
0 +------------------------------------------------------> x
|
-10|
|
|
-10
```
The dashed line represents [tex]\( y = -x \)[/tex], and the shaded area above the line is the solution set for the inequality [tex]\( y > -x \)[/tex].
### Summary
- Convert the inequality [tex]\( 12x + 2 < -12y + 2 \)[/tex] to [tex]\( y > -x \)[/tex].
- Draw the boundary line [tex]\( y = -x \)[/tex] using a dashed line since it is a strict inequality.
- Shade the region above the line to represent all the points that satisfy the inequality [tex]\( y > -x \)[/tex].
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