Question
Solve each set of inequalities by graphing.
x + y greater then or equal to -3
2x + 2y less then or equal to -2
x + y greater then or equal to -3
2x + 2y less then or equal to -2
Answers
GPT-4o mini
To solve the given set of inequalities by graphing, we will break down each inequality and then graph them on a coordinate plane.
### Inequality 1: \( x + y \geq -3 \)
1. **Convert to equation**: Start with the equality \( x + y = -3 \).
2. **Find intercepts**:
- If \( x = 0 \): \( y = -3 \) (point (0, -3))
- If \( y = 0 \): \( x = -3 \) (point (-3, 0))
3. **Graph the line**: Draw a solid line through the points (0, -3) and (-3, 0) because it is a "greater than or equal to" inequality.
4. **Shade the region**: Shade above the line, since we want the area where \( x + y \) is greater than or equal to -3.
### Inequality 2: \( 2x + 2y \leq -2 \)
1. **Convert to equation**: Start with the equality \( 2x + 2y = -2 \).
2. **Simplify the equation**: Divide the entire equation by 2 to get:
\[ x + y = -1 \]
3. **Find intercepts**:
- If \( x = 0 \): \( y = -1 \) (point (0, -1))
- If \( y = 0 \): \( x = -1 \) (point (-1, 0))
4. **Graph the line**: Draw a solid line through points (0, -1) and (-1, 0), because it is a "less than or equal to" inequality.
5. **Shade the region**: Shade below the line, as we want the area where \( 2x + 2y \) is less than or equal to -2.
### Final Step: Find the solution set
The solution set is where the shaded regions from both inequalities overlap.
### Summary of Steps:
- Graph the lines and shade each region based on the direction of the inequalities.
- The intersection of the shaded regions is the solution to the set of inequalities.
### Visual Representation:
- **Solid Line for** \( x + y = -3 \) and shade above.
- **Solid Line for** \( x + y = -1 \) and shade below.
- The solution region is where those shaded areas overlap.
If you have graphing paper or a graphing tool, you can plot these accurately. The solution set will generally be a polygon or an area represented in your graph.
### Inequality 1: \( x + y \geq -3 \)
1. **Convert to equation**: Start with the equality \( x + y = -3 \).
2. **Find intercepts**:
- If \( x = 0 \): \( y = -3 \) (point (0, -3))
- If \( y = 0 \): \( x = -3 \) (point (-3, 0))
3. **Graph the line**: Draw a solid line through the points (0, -3) and (-3, 0) because it is a "greater than or equal to" inequality.
4. **Shade the region**: Shade above the line, since we want the area where \( x + y \) is greater than or equal to -3.
### Inequality 2: \( 2x + 2y \leq -2 \)
1. **Convert to equation**: Start with the equality \( 2x + 2y = -2 \).
2. **Simplify the equation**: Divide the entire equation by 2 to get:
\[ x + y = -1 \]
3. **Find intercepts**:
- If \( x = 0 \): \( y = -1 \) (point (0, -1))
- If \( y = 0 \): \( x = -1 \) (point (-1, 0))
4. **Graph the line**: Draw a solid line through points (0, -1) and (-1, 0), because it is a "less than or equal to" inequality.
5. **Shade the region**: Shade below the line, as we want the area where \( 2x + 2y \) is less than or equal to -2.
### Final Step: Find the solution set
The solution set is where the shaded regions from both inequalities overlap.
### Summary of Steps:
- Graph the lines and shade each region based on the direction of the inequalities.
- The intersection of the shaded regions is the solution to the set of inequalities.
### Visual Representation:
- **Solid Line for** \( x + y = -3 \) and shade above.
- **Solid Line for** \( x + y = -1 \) and shade below.
- The solution region is where those shaded areas overlap.
If you have graphing paper or a graphing tool, you can plot these accurately. The solution set will generally be a polygon or an area represented in your graph.