Answer :
To graph the solution to the system of inequalities \(3y > 2x + 12\) and \(2x + y \leq -5\), follow these steps:
### Step 1: Convert inequalities to equations
First, convert each inequality to an equation to help in plotting the boundary lines:
1. \(3y = 2x + 12\)
2. \(2x + y = -5\)
### Step 2: Find intercepts and plot the boundary lines
Next, find the intercepts for each line to plot them:
#### For \(3y = 2x + 12\):
- Y-intercept:
Set \(x = 0\):
[tex]\[ 3y = 12 \implies y = 4 \][/tex]
So, the y-intercept is \((0, 4)\).
- X-intercept:
Set \(y = 0\):
[tex]\[ 0 = 2x + 12 \implies 2x = -12 \implies x = -6 \][/tex]
So, the x-intercept is \((-6, 0)\).
#### For \(2x + y = -5\):
- Y-intercept:
Set \(x = 0\):
[tex]\[ y = -5 \][/tex]
So, the y-intercept is \((0, -5)\).
- X-intercept:
Set \(y = 0\):
[tex]\[ 2x = -5 \implies x = -\frac{5}{2} = -2.5 \][/tex]
So, the x-intercept is \((-2.5, 0)\).
### Step 3: Plot the lines on the coordinate plane
Draw the lines using the intercepts found:
1. \(3y = 2x + 12\):
- Plot the points \((0, 4)\) and \((-6, 0)\) and draw the line passing through these points.
- Since it's a "greater than" inequality \(3y > 2x + 12\), this line should be dashed, and you will shade above the line.
2. \(2x + y = -5\):
- Plot the points \((0, -5)\) and \((-2.5, 0)\) and draw the line passing through these points.
- Since it's a "less than or equal to" inequality \(2x + y \le -5\), this line should be solid, and you will shade below the line.
### Step 4: Determine the region of intersection
The solution set for the system of inequalities is the region where the shaded areas overlap:
- For \(3y > 2x + 12\), shade above the dashed line.
- For \(2x + y \leq -5\), shade below the solid line.
### Solution Visualization
1. Draw the dashed line: Through points \((0, 4)\) and \((-6, 0)\), labeled \(3y = 2x + 12\).
2. Draw the solid line: Through points \((0, -5)\) and \((-2.5, 0)\), labeled \(2x + y = -5\).
3. Shade the appropriate regions:
- Above the dashed line \(3y = 2x + 12\).
- Below the solid line \(2x + y = -5\).
### Graph:
```
(Note: The following is a textual representation; for actual solution, plot this on graph paper or using graphing software.)
y
^
4+
|
|
|
|______________________________________> x
-5 |
|
|
| *
```
In reality, use graph paper or a graphing tool to accurately represent the answer. The critical aspect is to ensure the regions are properly shaded and the boundary lines correctly drawn based on whether they are inclusive (solid) or exclusive (dashed). The overlapping region forms the solution to the system of inequalities.
### Step 1: Convert inequalities to equations
First, convert each inequality to an equation to help in plotting the boundary lines:
1. \(3y = 2x + 12\)
2. \(2x + y = -5\)
### Step 2: Find intercepts and plot the boundary lines
Next, find the intercepts for each line to plot them:
#### For \(3y = 2x + 12\):
- Y-intercept:
Set \(x = 0\):
[tex]\[ 3y = 12 \implies y = 4 \][/tex]
So, the y-intercept is \((0, 4)\).
- X-intercept:
Set \(y = 0\):
[tex]\[ 0 = 2x + 12 \implies 2x = -12 \implies x = -6 \][/tex]
So, the x-intercept is \((-6, 0)\).
#### For \(2x + y = -5\):
- Y-intercept:
Set \(x = 0\):
[tex]\[ y = -5 \][/tex]
So, the y-intercept is \((0, -5)\).
- X-intercept:
Set \(y = 0\):
[tex]\[ 2x = -5 \implies x = -\frac{5}{2} = -2.5 \][/tex]
So, the x-intercept is \((-2.5, 0)\).
### Step 3: Plot the lines on the coordinate plane
Draw the lines using the intercepts found:
1. \(3y = 2x + 12\):
- Plot the points \((0, 4)\) and \((-6, 0)\) and draw the line passing through these points.
- Since it's a "greater than" inequality \(3y > 2x + 12\), this line should be dashed, and you will shade above the line.
2. \(2x + y = -5\):
- Plot the points \((0, -5)\) and \((-2.5, 0)\) and draw the line passing through these points.
- Since it's a "less than or equal to" inequality \(2x + y \le -5\), this line should be solid, and you will shade below the line.
### Step 4: Determine the region of intersection
The solution set for the system of inequalities is the region where the shaded areas overlap:
- For \(3y > 2x + 12\), shade above the dashed line.
- For \(2x + y \leq -5\), shade below the solid line.
### Solution Visualization
1. Draw the dashed line: Through points \((0, 4)\) and \((-6, 0)\), labeled \(3y = 2x + 12\).
2. Draw the solid line: Through points \((0, -5)\) and \((-2.5, 0)\), labeled \(2x + y = -5\).
3. Shade the appropriate regions:
- Above the dashed line \(3y = 2x + 12\).
- Below the solid line \(2x + y = -5\).
### Graph:
```
(Note: The following is a textual representation; for actual solution, plot this on graph paper or using graphing software.)
y
^
4+
|
|
|
|______________________________________> x
-5 |
|
|
| *
```
In reality, use graph paper or a graphing tool to accurately represent the answer. The critical aspect is to ensure the regions are properly shaded and the boundary lines correctly drawn based on whether they are inclusive (solid) or exclusive (dashed). The overlapping region forms the solution to the system of inequalities.
