Graph the inequality [tex]x + y \leq -2[/tex].

Question

06-07-2024
Mathematics

Answered

Graph the inequality [tex]x + y \leq -2[/tex].

Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-07-06T16:37:45-04:00

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].