To graph the inequality [tex]\( y + 2 > -3x - 3 \)[/tex], follow these steps:
### Step 1: Rearrange the Inequality
First, let's rearrange the inequality into slope-intercept form ([tex]\(y > mx + b\)[/tex]).
Given inequality:
[tex]\[ y + 2 > -3x - 3 \][/tex]
Subtract 2 from both sides to isolate [tex]\(y\)[/tex]:
[tex]\[ y > -3x - 3 - 2 \][/tex]
[tex]\[ y > -3x - 5 \][/tex]
### Step 2: Identify the Boundary Line
Next, we identify the boundary line, which is given by the equality part of the inequality, [tex]\( y = -3x - 5 \)[/tex].
- Slope ([tex]\(m\)[/tex]): The coefficient of [tex]\(x\)[/tex] is [tex]\(-3\)[/tex].
- Y-intercept ([tex]\(b\)[/tex]): The constant term is [tex]\(-5\)[/tex].
### Step 3: Plot the Boundary Line
Plot the boundary line [tex]\( y = -3x - 5 \)[/tex]:
1. Start at the y-intercept [tex]\((0, -5)\)[/tex].
2. Use the slope to find another point on the line. Since the slope is [tex]\(-3\)[/tex], it means we go down 3 units for every 1 unit we move to the right.
- From [tex]\((0, -5)\)[/tex], move right 1 unit to [tex]\((1, -5)\)[/tex], and down 3 units to [tex]\((1, -8)\)[/tex].
3. Draw a dashed line through these points because the inequality is strict ([tex]\(>\)[/tex]), and not [tex]\(\geq\)[/tex].
### Step 4: Shade the Correct Region
The inequality [tex]\( y > -3x - 5 \)[/tex] indicates that the region above the line should be shaded. This is because we are looking for all points [tex]\( (x, y) \)[/tex] where the [tex]\( y \)[/tex]-coordinate is greater than the value of [tex]\(-3x - 5\)[/tex].
### Step 5: Determine the Correct Graph
Compare your graph with the given choices:
- Graph A: Shows a different slope or intercept.
- Graph B: Shows shading below the line ([tex]\(< \)[/tex]).
- Graph C: Shows the line [tex]\( y = -3x - 5 \)[/tex] with shading above the line.
- Graph D: Shows a different slope or intercept.
The graph that matches the line [tex]\( y = -3x - 5 \)[/tex] with shading above the line is Graph C.
### Answer:
C. Graph C
