Question
Tell me where to plot the points for the following linear inequalities: y > 1/2x + 2, y ≤ -1/3x - 2
Answers
GPT-4o mini
To plot the points for the given linear inequalities, you'll first need to understand the lines represented by the equations and then figure out the appropriate regions that satisfy the inequalities.
### Inequality 1: \( y > \frac{1}{2}x + 2 \)
1. **Graph the boundary line**: Start by graphing the equation \( y = \frac{1}{2}x + 2 \).
- The y-intercept is at (0, 2).
- The slope is \( \frac{1}{2} \), meaning for every 2 units you move to the right along the x-axis, you move 1 unit up on the y-axis.
- Plot another point, for example, when \( x = 2 \):
\[
y = \frac{1}{2}(2) + 2 = 3
\]
So another point is (2, 3).
2. **Draw the line**: Since the inequality is \( y > \), use a dashed line to indicate that points on the line are not included.
3. **Shade the region**: For \( y > \frac{1}{2}x + 2\), shade the area above the line, as this represents all the points where \( y \) is greater than the line.
### Inequality 2: \( y \leq -\frac{1}{3}x - 2 \)
1. **Graph the boundary line**: Now, graph the equation \( y = -\frac{1}{3}x - 2 \).
- The y-intercept is at (0, -2).
- The slope is \( -\frac{1}{3} \), meaning for every 3 units you move to the right, you move 1 unit down.
- Plot another point, for example, when \( x = 3 \):
\[
y = -\frac{1}{3}(3) - 2 = -3
\]
So another point is (3, -3).
2. **Draw the line**: Since the inequality is \( y \leq \), draw a solid line to show that points on the line are included.
3. **Shade the region**: For \( y \leq -\frac{1}{3}x - 2\), shade the area below the line, which includes all points where \( y \) is less than or equal to the line.
### Summary of Graphing Steps
- For the line \( y = \frac{1}{2}x + 2 \):
- Use a dashed line.
- Shade above the line for inequality \( y > \).
- For the line \( y = -\frac{1}{3}x - 2 \):
- Use a solid line.
- Shade below the line for inequality \( y \leq \).
### Where Both Inequalities Hold
The solution to the system of inequalities will be the region where both shaded areas overlap. This is commonly referred to as the feasible region.
Ensure that when plotting these points and shading regions, you pay attention to the direction of the inequalities to determine where the solutions lie!
### Inequality 1: \( y > \frac{1}{2}x + 2 \)
1. **Graph the boundary line**: Start by graphing the equation \( y = \frac{1}{2}x + 2 \).
- The y-intercept is at (0, 2).
- The slope is \( \frac{1}{2} \), meaning for every 2 units you move to the right along the x-axis, you move 1 unit up on the y-axis.
- Plot another point, for example, when \( x = 2 \):
\[
y = \frac{1}{2}(2) + 2 = 3
\]
So another point is (2, 3).
2. **Draw the line**: Since the inequality is \( y > \), use a dashed line to indicate that points on the line are not included.
3. **Shade the region**: For \( y > \frac{1}{2}x + 2\), shade the area above the line, as this represents all the points where \( y \) is greater than the line.
### Inequality 2: \( y \leq -\frac{1}{3}x - 2 \)
1. **Graph the boundary line**: Now, graph the equation \( y = -\frac{1}{3}x - 2 \).
- The y-intercept is at (0, -2).
- The slope is \( -\frac{1}{3} \), meaning for every 3 units you move to the right, you move 1 unit down.
- Plot another point, for example, when \( x = 3 \):
\[
y = -\frac{1}{3}(3) - 2 = -3
\]
So another point is (3, -3).
2. **Draw the line**: Since the inequality is \( y \leq \), draw a solid line to show that points on the line are included.
3. **Shade the region**: For \( y \leq -\frac{1}{3}x - 2\), shade the area below the line, which includes all points where \( y \) is less than or equal to the line.
### Summary of Graphing Steps
- For the line \( y = \frac{1}{2}x + 2 \):
- Use a dashed line.
- Shade above the line for inequality \( y > \).
- For the line \( y = -\frac{1}{3}x - 2 \):
- Use a solid line.
- Shade below the line for inequality \( y \leq \).
### Where Both Inequalities Hold
The solution to the system of inequalities will be the region where both shaded areas overlap. This is commonly referred to as the feasible region.
Ensure that when plotting these points and shading regions, you pay attention to the direction of the inequalities to determine where the solutions lie!