Question
Graph the inequality.
6x+3y>−27
6x+3y>−27
Answers
Answer
what are the points
Answered by
GPT-4o mini
To graph the inequality \( 6x + 3y > -27 \), follow these steps:
1. **Rewrite the inequality in slope-intercept form (y = mx + b)**:
\[
3y > -6x - 27
\]
Now, divide each term by 3:
\[
y > -2x - 9
\]
2. **Graph the boundary line**:
The boundary line for the inequality is the equation \( y = -2x - 9 \).
- This line has a slope of -2 and a y-intercept of -9.
- To graph it, you can plot the y-intercept \((0, -9)\) and then use the slope to find another point. The slope of -2 means you go down 2 units for every 1 unit you move to the right. From \((0, -9)\), moving right to \(x = 1\) takes you to \(y = -11\) (point \((1, -11)\)).
- Draw the line through these points. Since the inequality is greater than (not greater than or equal to), make the line dashed, indicating that points on the line are not included in the solution.
3. **Shade the appropriate region**:
- Since the inequality is \( y > -2x - 9 \), you should shade above the line. This indicates that for any point in the shaded region, the y-value will be greater than the value given by \(-2x - 9\).
### Summary of the Graphing Steps:
- Draw a dashed line for \(y = -2x - 9\).
- Shade the region above this line.
You can plot these points on graph paper or using graphing software to create an accurate representation.
1. **Rewrite the inequality in slope-intercept form (y = mx + b)**:
\[
3y > -6x - 27
\]
Now, divide each term by 3:
\[
y > -2x - 9
\]
2. **Graph the boundary line**:
The boundary line for the inequality is the equation \( y = -2x - 9 \).
- This line has a slope of -2 and a y-intercept of -9.
- To graph it, you can plot the y-intercept \((0, -9)\) and then use the slope to find another point. The slope of -2 means you go down 2 units for every 1 unit you move to the right. From \((0, -9)\), moving right to \(x = 1\) takes you to \(y = -11\) (point \((1, -11)\)).
- Draw the line through these points. Since the inequality is greater than (not greater than or equal to), make the line dashed, indicating that points on the line are not included in the solution.
3. **Shade the appropriate region**:
- Since the inequality is \( y > -2x - 9 \), you should shade above the line. This indicates that for any point in the shaded region, the y-value will be greater than the value given by \(-2x - 9\).
### Summary of the Graphing Steps:
- Draw a dashed line for \(y = -2x - 9\).
- Shade the region above this line.
You can plot these points on graph paper or using graphing software to create an accurate representation.
Answered by
GPT-4o mini
To find specific points that satisfy the inequality \(6x + 3y > -27\) or equivalently \(y > -2x - 9\), we can pick various values for \(x\) and calculate the corresponding \(y\) values. Here are some points that satisfy the inequality:
1. **Choose \(x = 0\)**:
\[
y > -2(0) - 9 \implies y > -9
\]
A point that satisfies this condition is \((0, -8)\).
2. **Choose \(x = 1\)**:
\[
y > -2(1) - 9 \implies y > -11
\]
A point that satisfies this condition is \((1, -10)\).
3. **Choose \(x = -1\)**:
\[
y > -2(-1) - 9 \implies y > 2 - 9 \implies y > -7
\]
A point that satisfies this condition is \((-1, -6)\).
4. **Choose \(x = 2\)**:
\[
y > -2(2) - 9 \implies y > -4 - 9 \implies y > -13
\]
A point that satisfies this condition is \((2, -12)\).
5. **Choose \(x = -2\)**:
\[
y > -2(-2) - 9 \implies y > 4 - 9 \implies y > -5
\]
A point that satisfies this condition is \((-2, -4)\).
### Summary of Points:
Here are the points that satisfy the inequality \(6x + 3y > -27\):
- \((0, -8)\)
- \((1, -10)\)
- \((-1, -6)\)
- \((2, -12)\)
- \((-2, -4)\)
You can check that any of these points will yield a \(y\) value that is indeed greater than the corresponding \(y\) value on the line \(y = -2x - 9\).
1. **Choose \(x = 0\)**:
\[
y > -2(0) - 9 \implies y > -9
\]
A point that satisfies this condition is \((0, -8)\).
2. **Choose \(x = 1\)**:
\[
y > -2(1) - 9 \implies y > -11
\]
A point that satisfies this condition is \((1, -10)\).
3. **Choose \(x = -1\)**:
\[
y > -2(-1) - 9 \implies y > 2 - 9 \implies y > -7
\]
A point that satisfies this condition is \((-1, -6)\).
4. **Choose \(x = 2\)**:
\[
y > -2(2) - 9 \implies y > -4 - 9 \implies y > -13
\]
A point that satisfies this condition is \((2, -12)\).
5. **Choose \(x = -2\)**:
\[
y > -2(-2) - 9 \implies y > 4 - 9 \implies y > -5
\]
A point that satisfies this condition is \((-2, -4)\).
### Summary of Points:
Here are the points that satisfy the inequality \(6x + 3y > -27\):
- \((0, -8)\)
- \((1, -10)\)
- \((-1, -6)\)
- \((2, -12)\)
- \((-2, -4)\)
You can check that any of these points will yield a \(y\) value that is indeed greater than the corresponding \(y\) value on the line \(y = -2x - 9\).