Sure, let's go through the steps to graph the inequality [tex]\( x + y \leq -2 \)[/tex].
### Step 1: Understand the Boundary Line
First, we note that the inequality [tex]\( x + y \leq -2 \)[/tex] involves a linear boundary line. The boundary line for this inequality is [tex]\( x + y = -2 \)[/tex].
### Step 2: Rewrite in Slope-Intercept Form
To help us graph the line, we can rewrite the equation [tex]\( x + y = -2 \)[/tex] in the familiar slope-intercept form [tex]\( y = mx + b \)[/tex].
[tex]\[
y = -x - 2
\][/tex]
Here, the slope [tex]\( m \)[/tex] is -1 and the y-intercept [tex]\( b \)[/tex] is -2.
### Step 3: Plot the Boundary Line
- y-intercept: Plot the point [tex]\((0, -2)\)[/tex] on the graph.
- Slope: From the y-intercept, use the slope [tex]\(-1\)[/tex] (which means “down 1 unit, and right 1 unit”) to find another point. Starting from [tex]\((0, -2)\)[/tex], move down 1 unit and right 1 unit to get the point [tex]\((1, -3)\)[/tex].
- Plot another point similarly, if needed, and draw a straight line through these points.
### Step 4: Determine the Region to Shade
Determine whether to shade above or below this line for the inequality [tex]\( x + y \leq -2 \)[/tex].
To do this, pick a test point that is not on the line, such as [tex]\((0, 0)\)[/tex]:
- Substitute into the inequality: [tex]\( 0 + 0 \leq -2 \)[/tex]
- This yields: [tex]\( 0 \leq -2 \)[/tex], which is false.
Since the test point does not satisfy the inequality, we shade the opposite side of the test point. Therefore, we shade the region below the line [tex]\( y = -x - 2 \)[/tex].
### Step 5: Draw the Graph
- Boundary Line: Draw the line [tex]\( y = -x - 2 \)[/tex] as a solid line because the inequality includes the equal sign ([tex]\(\leq\)[/tex]).
- Shaded Region: Shade the area below the line.
### Final Graph
1. Draw the line passing through points [tex]\((0, -2)\)[/tex] and [tex]\((1, -3)\)[/tex].
2. Make the line solid.
3. Shade the area below the line.
Here is a visual representation for your reference:
```
| /
| /
| / /
| / /
|/____/______
|
```
In this sketch, the `` represents the point [tex]\((0, -2)\)[/tex], and the line is drawn through it with a slope of -1. The shaded area is below the line.
This completes the graph of the inequality [tex]\( x + y \leq -2 \)[/tex].
