To solve the inequality [tex]\( y + 2 > -3x - 3 \)[/tex] and determine which graph matches the solution, follow these steps:
### Step 1: Rewrite the Inequality in Slope-Intercept Form
First, we need to isolate [tex]\( y \)[/tex] on one side of the inequality to express it in the form [tex]\( y > mx + b \)[/tex]:
[tex]\[
y + 2 > -3x - 3
\][/tex]
Subtract 2 from both sides:
[tex]\[
y > -3x - 3 - 2
\][/tex]
Simplify the right side:
[tex]\[
y > -3x - 5
\][/tex]
### Step 2: Graph the Boundary Line
Next, we graph the boundary line [tex]\( y = -3x - 5 \)[/tex]. Since the inequality is [tex]\( y > -3x - 5 \)[/tex], we will draw a dashed line to represent the boundary where [tex]\( y = -3x - 5 \)[/tex].
1. Y-intercept ( [tex]\( b \)[/tex] ): The y-intercept is -5. This means the line crosses the y-axis at the point (0, -5).
2. Slope ( [tex]\( m \)[/tex] ): The slope is -3. This means for each unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 3 units.
### Step 3: Plot Key Points
- Start at the y-intercept (0, -5).
- Use the slope to find another point. From (0, -5), move 1 unit to the right (increasing [tex]\( x \)[/tex] by 1), and move 3 units down (decreasing [tex]\( y \)[/tex] by 3 units). This gives us the point (1, -8).
Plot these two points and draw a dashed line through them to represent the boundary line [tex]\( y = -3x - 5 \)[/tex].
### Step 4: Shade the Correct Region
Since the inequality is [tex]\( y > -3x - 5 \)[/tex], we need to shade the region above the dashed line. This represents all the points where [tex]\( y \)[/tex] is greater than the value on the line.
### Step 5: Identify the Correct Graph
Examine the given answer choices (A, B, C, D) and identify the graph that depicts:
- A dashed boundary line passing through (0, -5) and (1, -8).
- The region above this line shaded.
After careful visual inspection, we conclude that the graph which matches this description is the correct one.
### Answer
The answer choice that correctly represents the graph of [tex]\( y + 2 > -3x - 3 \)[/tex] is:
```
Determined by visual inspection of the graphs, select A, B, C, or D.
```
