Question
Solve each set of inequalities by graphic.
1.) 3x - 2y less then or equal to 4
X + 3y less then or equal to 6
1.) 3x - 2y less then or equal to 4
X + 3y less then or equal to 6
Answers
GPT-4o mini
To solve the given set of inequalities graphically, we'll first convert them into their equivalent equations and then graph the lines corresponding to each inequality. Finally, we'll determine the feasible region that satisfies both inequalities.
### Inequalities:
1. \( 3x - 2y \leq 4 \)
2. \( x + 3y \leq 6 \)
### Step 1: Convert inequalities to equations
1. **Equation 1**: \( 3x - 2y = 4 \)
2. **Equation 2**: \( x + 3y = 6 \)
### Step 2: Find the intercepts for both lines
#### For Equation 1: \( 3x - 2y = 4 \)
- **x-intercept**: Set \( y = 0 \)
\[
3x = 4 \implies x = \frac{4}{3}
\]
- **y-intercept**: Set \( x = 0 \)
\[
-2y = 4 \implies y = -2
\]
So the intercepts are \( \left(\frac{4}{3}, 0\right) \) and \( (0, -2) \).
#### For Equation 2: \( x + 3y = 6 \)
- **x-intercept**: Set \( y = 0 \)
\[
x = 6
\]
- **y-intercept**: Set \( x = 0 \)
\[
3y = 6 \implies y = 2
\]
So the intercepts are \( (6, 0) \) and \( (0, 2) \).
### Step 3: Graph the lines
1. Plot the points \( \left(\frac{4}{3}, 0\right), (0, -2) \) for the first equation and draw a solid line since it is "less than or equal to".
2. Plot the points \( (6, 0), (0, 2) \) for the second equation and draw a solid line since it is also "less than or equal to".
### Step 4: Determine the feasible region
- For the inequality \( 3x - 2y \leq 4 \), shade the area below the line, including the line itself.
- For the inequality \( x + 3y \leq 6 \), shade the area below the line, including the line itself.
### Step 5: Identify the intersection
The feasible region is where the shaded areas from both inequalities overlap.
### Conclusion
The solution to the system of inequalities is the region where the shaded areas intersect on the graph. This region can be further analyzed to find the vertices (corners) if needed for optimization problems.
If you need a visual representation, you can graph these on graph paper or a graphing software/tool.
### Inequalities:
1. \( 3x - 2y \leq 4 \)
2. \( x + 3y \leq 6 \)
### Step 1: Convert inequalities to equations
1. **Equation 1**: \( 3x - 2y = 4 \)
2. **Equation 2**: \( x + 3y = 6 \)
### Step 2: Find the intercepts for both lines
#### For Equation 1: \( 3x - 2y = 4 \)
- **x-intercept**: Set \( y = 0 \)
\[
3x = 4 \implies x = \frac{4}{3}
\]
- **y-intercept**: Set \( x = 0 \)
\[
-2y = 4 \implies y = -2
\]
So the intercepts are \( \left(\frac{4}{3}, 0\right) \) and \( (0, -2) \).
#### For Equation 2: \( x + 3y = 6 \)
- **x-intercept**: Set \( y = 0 \)
\[
x = 6
\]
- **y-intercept**: Set \( x = 0 \)
\[
3y = 6 \implies y = 2
\]
So the intercepts are \( (6, 0) \) and \( (0, 2) \).
### Step 3: Graph the lines
1. Plot the points \( \left(\frac{4}{3}, 0\right), (0, -2) \) for the first equation and draw a solid line since it is "less than or equal to".
2. Plot the points \( (6, 0), (0, 2) \) for the second equation and draw a solid line since it is also "less than or equal to".
### Step 4: Determine the feasible region
- For the inequality \( 3x - 2y \leq 4 \), shade the area below the line, including the line itself.
- For the inequality \( x + 3y \leq 6 \), shade the area below the line, including the line itself.
### Step 5: Identify the intersection
The feasible region is where the shaded areas from both inequalities overlap.
### Conclusion
The solution to the system of inequalities is the region where the shaded areas intersect on the graph. This region can be further analyzed to find the vertices (corners) if needed for optimization problems.
If you need a visual representation, you can graph these on graph paper or a graphing software/tool.