Question
Solve each set of inequalities by graphing.
-y less then or equal to 3x +4
-3x +3y less then or equal to -9
-y less then or equal to 3x +4
-3x +3y less then or equal to -9
Answers
GPT-4o mini
To solve the given set of inequalities by graphing, we will first rewrite each inequality in slope-intercept form (if necessary), graph the lines, and then determine the regions that satisfy each inequality.
### Inequality 1:
\(-y \leq 3x + 4\)
First, we will rewrite the inequality:
1. Multiply both sides by -1 (and remember to flip the inequality sign):
\[
y \geq -3x - 4
\]
Now, this is in the form \(y = mx + b\). The slope \(m\) is -3, and the y-intercept \(b\) is -4.
### Inequality 2:
\(-3x + 3y \leq -9\)
Rearranging, we divide everything by 3 to simplify the inequality:
1. Divide by 3:
\[
-x + y \leq -3
\]
2. Rearranging gives us:
\[
y \leq x - 3
\]
This is now also in the form \(y = mx + b\) with \(m = 1\) and \(b = -3\).
### Steps to Graph:
1. **Graph the Line \(y = -3x - 4\)**:
- Start at (0, -4) (y-intercept).
- Use the slope of -3 (down 3 units for every 1 unit to the right):
- From (0, -4), you can go to (1, -7) or (-1, -1) for additional points.
- Since the inequality is \(\geq\), this will be a solid line.
2. **Graph the Line \(y = x - 3\)**:
- Start at (0, -3) (y-intercept).
- Use the slope of 1 (up 1 for every 1 unit to the right):
- From (0, -3), you can go to (1, -2) or (-1, -4) for additional points.
- Since the inequality is \(\leq\), this will also be a solid line.
### Shading:
- For the first inequality \(y \geq -3x - 4\), shade **above** the line.
- For the second inequality \(y \leq x - 3\), shade **below** the line.
### Finding the Solution Region:
The solution to the set of inequalities will be where the shaded regions overlap.
### Final Visualization:
1. Draw the lines as described.
2. Shade the appropriate regions:
- For the line \(y = -3x - 4\), shade above the line (including the line itself).
- For the line \(y = x - 3\), shade below the line (including the line itself).
3. The overlapping area represents the solution to the system of inequalities.
This area is where both conditions are satisfied. You can choose test points in this region to verify that they satisfy both inequalities.
If you need to plot this on a graphing tool, you would enter the equations and inequalities as described above and observe the solution region visually.
### Inequality 1:
\(-y \leq 3x + 4\)
First, we will rewrite the inequality:
1. Multiply both sides by -1 (and remember to flip the inequality sign):
\[
y \geq -3x - 4
\]
Now, this is in the form \(y = mx + b\). The slope \(m\) is -3, and the y-intercept \(b\) is -4.
### Inequality 2:
\(-3x + 3y \leq -9\)
Rearranging, we divide everything by 3 to simplify the inequality:
1. Divide by 3:
\[
-x + y \leq -3
\]
2. Rearranging gives us:
\[
y \leq x - 3
\]
This is now also in the form \(y = mx + b\) with \(m = 1\) and \(b = -3\).
### Steps to Graph:
1. **Graph the Line \(y = -3x - 4\)**:
- Start at (0, -4) (y-intercept).
- Use the slope of -3 (down 3 units for every 1 unit to the right):
- From (0, -4), you can go to (1, -7) or (-1, -1) for additional points.
- Since the inequality is \(\geq\), this will be a solid line.
2. **Graph the Line \(y = x - 3\)**:
- Start at (0, -3) (y-intercept).
- Use the slope of 1 (up 1 for every 1 unit to the right):
- From (0, -3), you can go to (1, -2) or (-1, -4) for additional points.
- Since the inequality is \(\leq\), this will also be a solid line.
### Shading:
- For the first inequality \(y \geq -3x - 4\), shade **above** the line.
- For the second inequality \(y \leq x - 3\), shade **below** the line.
### Finding the Solution Region:
The solution to the set of inequalities will be where the shaded regions overlap.
### Final Visualization:
1. Draw the lines as described.
2. Shade the appropriate regions:
- For the line \(y = -3x - 4\), shade above the line (including the line itself).
- For the line \(y = x - 3\), shade below the line (including the line itself).
3. The overlapping area represents the solution to the system of inequalities.
This area is where both conditions are satisfied. You can choose test points in this region to verify that they satisfy both inequalities.
If you need to plot this on a graphing tool, you would enter the equations and inequalities as described above and observe the solution region visually.