Question
Graph the inequality 6x+3y>-27
Answers
GPT-4o mini
To graph the inequality \( 6x + 3y > -27 \), we can follow these steps:
1. **Rewrite the inequality as an equation**: First, replace the inequality with an equality to find the boundary line. So, we consider the equation \( 6x + 3y = -27 \).
2. **Convert to slope-intercept form**: Rearranging the equation into the slope-intercept form \( y = mx + b \):
\[
3y = -6x - 27
\]
\[
y = -2x - 9
\]
Here, the slope \( m \) is -2, and the y-intercept \( b \) is -9.
3. **Graph the boundary line**: Since the original inequality is strict (greater than, not greater than or equal to), we will use a dashed line for the boundary. To plot the line:
- Start at the y-intercept (0, -9).
- From (0, -9), use the slope to find another point. From (0, -9), going down 2 units and right 1 unit gives the point (1, -11).
- Draw a dashed line through these points, as the inequality is not inclusive.
4. **Test a point to determine the shading**: Choose a test point not on the line; (0, 0) is a good choice.
\[
6(0) + 3(0) > -27 \implies 0 > -27
\]
This statement is true, so we will shade the area that includes (0, 0).
5. **Final graph**: The region above the dashed line represents the solution to the inequality \( 6x + 3y > -27 \).
Here’s a summary of the graph:
- Draw a dashed line for the equation \( y = -2x - 9 \).
- Shade the area above the line where the inequality \( 6x + 3y > -27 \) holds true.
You can visualize this graph accurately on graphing paper or using graphing software.
1. **Rewrite the inequality as an equation**: First, replace the inequality with an equality to find the boundary line. So, we consider the equation \( 6x + 3y = -27 \).
2. **Convert to slope-intercept form**: Rearranging the equation into the slope-intercept form \( y = mx + b \):
\[
3y = -6x - 27
\]
\[
y = -2x - 9
\]
Here, the slope \( m \) is -2, and the y-intercept \( b \) is -9.
3. **Graph the boundary line**: Since the original inequality is strict (greater than, not greater than or equal to), we will use a dashed line for the boundary. To plot the line:
- Start at the y-intercept (0, -9).
- From (0, -9), use the slope to find another point. From (0, -9), going down 2 units and right 1 unit gives the point (1, -11).
- Draw a dashed line through these points, as the inequality is not inclusive.
4. **Test a point to determine the shading**: Choose a test point not on the line; (0, 0) is a good choice.
\[
6(0) + 3(0) > -27 \implies 0 > -27
\]
This statement is true, so we will shade the area that includes (0, 0).
5. **Final graph**: The region above the dashed line represents the solution to the inequality \( 6x + 3y > -27 \).
Here’s a summary of the graph:
- Draw a dashed line for the equation \( y = -2x - 9 \).
- Shade the area above the line where the inequality \( 6x + 3y > -27 \) holds true.
You can visualize this graph accurately on graphing paper or using graphing software.