To graph the inequality [tex]\( y + 2 > -3x - 3 \)[/tex] and determine which graph it corresponds to, let's break it down step-by-step:
### 1. Rewrite the Inequality in Slope-Intercept Form
First, we want to rewrite the inequality to make it easier to understand and graph. We do this by solving for [tex]\( y \)[/tex]:
[tex]\[ y + 2 > -3x - 3 \][/tex]
Subtract 2 from both sides:
[tex]\[ y > -3x - 3 - 2 \][/tex]
Simplify the expression:
[tex]\[ y > -3x - 5 \][/tex]
### 2. Identify the Slope and Y-Intercept
From the inequality [tex]\( y > -3x - 5 \)[/tex], we can identify two key components:
- The slope (m): This is the coefficient of [tex]\( x \)[/tex], which is [tex]\( -3 \)[/tex].
- The y-intercept (b): This is the constant term, which is [tex]\( -5 \)[/tex].
### 3. Draw the Boundary Line
Next, we need to draw the boundary line that will help us understand the region where the inequality holds.
The boundary line is given by the equality part of the inequality:
[tex]\[ y = -3x - 5 \][/tex]
To plot this line, we can use the y-intercept and the slope:
- Start at the y-intercept [tex]\((0, -5)\)[/tex].
- Use the slope [tex]\( -3 \)[/tex] (which means you go down 3 units for every 1 unit you move to the right).
### 4. Plot the Line
1. Start at the point [tex]\((0, -5)\)[/tex].
2. From this point, move right 1 unit to [tex]\((1, -5)\)[/tex] and then move down 3 units to [tex]\((1, -8)\)[/tex].
3. Draw a dashed line through these points because the inequality is strict ( > ) and does not include the boundary line.
### 5. Shade the Region
Since the inequality is [tex]\( y > -3x - 5 \)[/tex], we need to shade the region above the line. This indicates all the points where [tex]\( y \)[/tex] is greater than [tex]\( -3x - 5 \)[/tex].
### Conclusion
Based on the steps above, your graph should feature:
- A dashed line that goes through [tex]\((0, -5)\)[/tex] with a slope of [tex]\( -3 \)[/tex].
- Shading above this dashed line.
### Determining the Matched Answer Choice
Without seeing the graphs provided, you would determine which graph matches the one you drew as follows:
- Look for a graph that shows a dashed line with a y-intercept at [tex]\( -5 \)[/tex] and a slope of [tex]\( -3 \)[/tex].
- Ensure that the region above this line is shaded.
Based on these criteria, you should compare and decide which of the given options (Graph A or Graph B) matches this description.
