Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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.
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.