To graph the linear inequality [tex]\(2x - 3y < 12\)[/tex], let's follow these steps:
### 1. Transform the inequality into an equation.
First, we transform the inequality [tex]\(2x - 3y < 12\)[/tex] into its corresponding equation:
[tex]\[ 2x - 3y = 12 \][/tex]
### 2. Graph the line [tex]\(2x - 3y = 12\)[/tex].
To graph this line:
- Find the x-intercept:
Set [tex]\(y = 0\)[/tex]:
[tex]\[ 2x - 3(0) = 12 \implies x = 6 \][/tex]
So the x-intercept is [tex]\((6,0)\)[/tex].
- Find the y-intercept:
Set [tex]\(x = 0\)[/tex]:
[tex]\[ 2(0) - 3y = 12 \implies -3y = 12 \implies y = -4 \][/tex]
So the y-intercept is [tex]\( (0,-4) \)[/tex].
Plot these intercepts [tex]\((6,0)\)[/tex] and [tex]\((0,-4)\)[/tex] on the graph and draw a straight line through them. This line represents the equation [tex]\(2x - 3y = 12\)[/tex].
### 3. Determine which side of the line represents [tex]\(2x - 3y < 12\)[/tex].
To determine which side of the line to shade (i.e., where the inequality [tex]\(2x - 3y < 12\)[/tex] holds), test a point that is not on the line. A simple choice is the origin [tex]\((0,0)\)[/tex]:
Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 0\)[/tex] into the inequality:
[tex]\[ 2(0) - 3(0) < 12 \implies 0 < 12 \][/tex]
This statement is true. Hence, the region that contains the origin is the solution set to the inequality [tex]\(2x - 3y < 12\)[/tex].
### 4. Shade the appropriate region.
On the graph, shade the region that includes the origin (below the line) to represent [tex]\(2x - 3y < 12\)[/tex].
### 5. Use a dashed line for the inequality.
Since the inequality is strict (i.e., it does not include the boundary line where [tex]\(2x - 3y = 12\)[/tex]), we draw the line [tex]\(2x - 3y = 12\)[/tex] as a dashed line to indicate that points on the line do not satisfy the inequality.
### Final Graph:
- Plot the line [tex]\(2x - 3y = 12\)[/tex] using intercepts (6, 0) and (0, -4) as a dashed line.
- Shade the region below this dashed line to represent the solution set for [tex]\(2x - 3y < 12\)[/tex].
Here is a rough sketch of the graph:
```
y
|
|
| (6,0)
|
|
|
| * (0,-4)
|
+---------------------------- x
```
The area below the dashed line [tex]\(2x - 3y = 12\)[/tex] is the solution set for the inequality [tex]\(2x - 3y < 12\)[/tex].
