Question
Graph the inequality on the axes below. y, is less than or equal to, minus, one half, x, minus, 1 y≤− 2 1 x−1
Answers
GPT-4o mini
To graph the inequality \( y \leq -\frac{1}{2}x - 1 \), follow these steps:
1. **Identify the Boundary Line:** First, we need to graph the equation of the line that corresponds to the inequality. This would be the equation:
\[
y = -\frac{1}{2}x - 1
\]
Here, the slope \( m = -\frac{1}{2} \) and the y-intercept \( b = -1 \).
2. **Plot the Line:**
- Start by plotting the y-intercept: point (0, -1).
- From the y-intercept, use the slope to determine another point. Since the slope is -1/2, for every 2 units you move right on the x-axis, you move down 1 unit on the y-axis.
- From (0, -1), moving 2 units right gives you (2, -1.5) (since \(-1 - 0.5 = -1.5\)), and moving 2 more units to (4, -2).
3. **Draw the Line:**
- Since the inequality is "less than or equal to," you should draw a solid line through the points you plotted. This indicates that points on the line are included in the solution set.
4. **Shade the Region:**
- Since the inequality is \( y \leq -\frac{1}{2}x - 1 \), you will shade below the line. This is because you want to show all the points for which \( y \) is less than or equal to the value on the line.
The resulting graph will have a solid line representing \( y = -\frac{1}{2}x - 1 \) and the area below the line will be shaded to represent the solution to the inequality.
If you would like to visualize this, use a graphing tool or software to achieve a more precise graphical representation.
1. **Identify the Boundary Line:** First, we need to graph the equation of the line that corresponds to the inequality. This would be the equation:
\[
y = -\frac{1}{2}x - 1
\]
Here, the slope \( m = -\frac{1}{2} \) and the y-intercept \( b = -1 \).
2. **Plot the Line:**
- Start by plotting the y-intercept: point (0, -1).
- From the y-intercept, use the slope to determine another point. Since the slope is -1/2, for every 2 units you move right on the x-axis, you move down 1 unit on the y-axis.
- From (0, -1), moving 2 units right gives you (2, -1.5) (since \(-1 - 0.5 = -1.5\)), and moving 2 more units to (4, -2).
3. **Draw the Line:**
- Since the inequality is "less than or equal to," you should draw a solid line through the points you plotted. This indicates that points on the line are included in the solution set.
4. **Shade the Region:**
- Since the inequality is \( y \leq -\frac{1}{2}x - 1 \), you will shade below the line. This is because you want to show all the points for which \( y \) is less than or equal to the value on the line.
The resulting graph will have a solid line representing \( y = -\frac{1}{2}x - 1 \) and the area below the line will be shaded to represent the solution to the inequality.
If you would like to visualize this, use a graphing tool or software to achieve a more precise graphical representation.
try again its wrong
GPT-4o mini
I apologize for the confusion! Let's go through the steps again to graph the inequality \( y \leq -\frac{1}{2}x - 1 \) correctly.
### Steps to graph the inequality \( y \leq -\frac{1}{2}x - 1 \):
1. **Identify the Boundary Line:**
- The boundary line of the inequality is given by the equation \( y = -\frac{1}{2}x - 1 \).
- This line has a slope of \(-\frac{1}{2}\) and a y-intercept of \(-1\).
2. **Graph the Boundary Line:**
- **Plot the y-intercept**: The point where the line crosses the y-axis is (0, -1).
- **Use the slope**: From (0, -1), move right 2 units (positive x-direction) and down 1 unit (negative y-direction) to get another point. This gives the point (2, -1.5).
- Plot another point using the slope: from (0, -1), moving left 2 units (to -2) and up 1 unit gives the point (-2, 0).
3. **Draw the Line:**
- Since the inequality is "less than or equal to" (indicated by the \( \leq \)), you will draw a solid line. This means points on the line itself are included in the solution set.
4. **Shade the Region:**
- Now, since the inequality is \( y \leq -\frac{1}{2}x - 1 \), you will shade **below** the line. This indicates the region where the values of \( y \) are less than or equal to those on the line.
### Summary:
- Solid line for \( y = -\frac{1}{2}x - 1\)
- Shade the region below the line
If you have graphing paper or a graphing tool, you can plot this directly, or let me know if you need additional details or clarifications!
### Steps to graph the inequality \( y \leq -\frac{1}{2}x - 1 \):
1. **Identify the Boundary Line:**
- The boundary line of the inequality is given by the equation \( y = -\frac{1}{2}x - 1 \).
- This line has a slope of \(-\frac{1}{2}\) and a y-intercept of \(-1\).
2. **Graph the Boundary Line:**
- **Plot the y-intercept**: The point where the line crosses the y-axis is (0, -1).
- **Use the slope**: From (0, -1), move right 2 units (positive x-direction) and down 1 unit (negative y-direction) to get another point. This gives the point (2, -1.5).
- Plot another point using the slope: from (0, -1), moving left 2 units (to -2) and up 1 unit gives the point (-2, 0).
3. **Draw the Line:**
- Since the inequality is "less than or equal to" (indicated by the \( \leq \)), you will draw a solid line. This means points on the line itself are included in the solution set.
4. **Shade the Region:**
- Now, since the inequality is \( y \leq -\frac{1}{2}x - 1 \), you will shade **below** the line. This indicates the region where the values of \( y \) are less than or equal to those on the line.
### Summary:
- Solid line for \( y = -\frac{1}{2}x - 1\)
- Shade the region below the line
If you have graphing paper or a graphing tool, you can plot this directly, or let me know if you need additional details or clarifications!