Answer :
To graph the solution to the system of inequalities:
[tex]\[ \begin{array}{l} y < -3x + 4 \\ y > 3x - 5 \end{array} \][/tex]
we need to follow a step-by-step approach. Let's walk through the process:
### Step 1: Graph the Boundary Lines
First, we need to graph the boundary lines for both inequalities. These boundaries are given by the equations \( y = -3x + 4 \) and \( y = 3x - 5 \).
#### Line 1: \( y = -3x + 4 \)
1. Find the y-intercept by setting \( x = 0 \):
[tex]\[ y = -3(0) + 4 = 4 \][/tex]
So, the line passes through the point \( (0, 4) \).
2. Find another point on the line by choosing \( x = 1 \):
[tex]\[ y = -3(1) + 4 = 1 \][/tex]
So, the line also passes through the point \( (1, 1) \).
#### Line 2: \( y = 3x - 5 \)
1. Find the y-intercept by setting \( x = 0 \):
[tex]\[ y = 3(0) - 5 = -5 \][/tex]
So, the line passes through the point \( (0, -5) \).
2. Find another point on the line by choosing \( x = 1 \):
[tex]\[ y = 3(1) - 5 = -2 \][/tex]
So, the line also passes through the point \( (1, -2) \).
Plot these points and draw the lines on a coordinate plane.
### Step 2: Clearly Indicate the Boundary Lines
Since these inequalities are strict (using < and >), we will draw these boundary lines as dashed lines to indicate that points on the lines are not included in the solution set.
### Step 3: Shading the Regions
Next, we will determine which side of these lines to shade based on the inequalities.
#### Inequality \( y < -3x + 4 \)
For this inequality, shade the region below the line \( y = -3x + 4 \).
#### Inequality \( y > 3x - 5 \)
For this inequality, shade the region above the line \( y = 3x - 5 \).
### Step 4: Identify the Intersection
The solution to the system of inequalities is the region where the shaded areas overlap.
### Step 5: Plotting the Graph
Draw the following:
1. The line \( y = -3x + 4 \) as a dashed line passing through \( (0, 4) \) and \( (1, 1) \).
2. The line \( y = 3x - 5 \) as a dashed line passing through \( (0, -5) \) and \( (1, -2) \).
3. Shade the area below the line \( y = -3x + 4 \).
4. Shade the area above the line \( y = 3x - 5 \).
5. The overlapping region will be the solution to the system of inequalities.
### Sketching the Final Graph
Here is an illustrative summary of what the graph should look like:
```
|
| \
| \
| \ <-- y = -3x + 4 (dashed line)
| \
y-axis | . . . . . . . . . . . . . .
| Shaded \
| Area \
| \
| Intersection x --------------> x-axis
| / region
|-----------/----------------
| /
| /
| /
| / <-- y = 3x - 5 (dashed line)
| /
| /
------------------------------------------------------
```
The overlapping shading represents the solution region.
[tex]\[ \begin{array}{l} y < -3x + 4 \\ y > 3x - 5 \end{array} \][/tex]
we need to follow a step-by-step approach. Let's walk through the process:
### Step 1: Graph the Boundary Lines
First, we need to graph the boundary lines for both inequalities. These boundaries are given by the equations \( y = -3x + 4 \) and \( y = 3x - 5 \).
#### Line 1: \( y = -3x + 4 \)
1. Find the y-intercept by setting \( x = 0 \):
[tex]\[ y = -3(0) + 4 = 4 \][/tex]
So, the line passes through the point \( (0, 4) \).
2. Find another point on the line by choosing \( x = 1 \):
[tex]\[ y = -3(1) + 4 = 1 \][/tex]
So, the line also passes through the point \( (1, 1) \).
#### Line 2: \( y = 3x - 5 \)
1. Find the y-intercept by setting \( x = 0 \):
[tex]\[ y = 3(0) - 5 = -5 \][/tex]
So, the line passes through the point \( (0, -5) \).
2. Find another point on the line by choosing \( x = 1 \):
[tex]\[ y = 3(1) - 5 = -2 \][/tex]
So, the line also passes through the point \( (1, -2) \).
Plot these points and draw the lines on a coordinate plane.
### Step 2: Clearly Indicate the Boundary Lines
Since these inequalities are strict (using < and >), we will draw these boundary lines as dashed lines to indicate that points on the lines are not included in the solution set.
### Step 3: Shading the Regions
Next, we will determine which side of these lines to shade based on the inequalities.
#### Inequality \( y < -3x + 4 \)
For this inequality, shade the region below the line \( y = -3x + 4 \).
#### Inequality \( y > 3x - 5 \)
For this inequality, shade the region above the line \( y = 3x - 5 \).
### Step 4: Identify the Intersection
The solution to the system of inequalities is the region where the shaded areas overlap.
### Step 5: Plotting the Graph
Draw the following:
1. The line \( y = -3x + 4 \) as a dashed line passing through \( (0, 4) \) and \( (1, 1) \).
2. The line \( y = 3x - 5 \) as a dashed line passing through \( (0, -5) \) and \( (1, -2) \).
3. Shade the area below the line \( y = -3x + 4 \).
4. Shade the area above the line \( y = 3x - 5 \).
5. The overlapping region will be the solution to the system of inequalities.
### Sketching the Final Graph
Here is an illustrative summary of what the graph should look like:
```
|
| \
| \
| \ <-- y = -3x + 4 (dashed line)
| \
y-axis | . . . . . . . . . . . . . .
| Shaded \
| Area \
| \
| Intersection x --------------> x-axis
| / region
|-----------/----------------
| /
| /
| /
| / <-- y = 3x - 5 (dashed line)
| /
| /
------------------------------------------------------
```
The overlapping shading represents the solution region.
