Sure, let's graph the given linear inequality: [tex]\( y \geq -8 \)[/tex]. Here’s a step-by-step solution for how to do this:
1. Understand the Inequality:
- The inequality [tex]\( y \geq -8 \)[/tex] indicates that the values of [tex]\( y \)[/tex] must be greater than or equal to [tex]\(-8\)[/tex].
2. Graph the Boundary Line:
- First, we graph the boundary line [tex]\( y = -8 \)[/tex]. This is a horizontal line passing through the y-axis at [tex]\( y = -8 \)[/tex].
- Since the inequality is [tex]\( \geq \)[/tex] (greater than or equal to), this boundary line will be solid. A solid line indicates that points on the line [tex]\( y = -8 \)[/tex] are included in the solution set.
3. Shade the Appropriate Region:
- Next, we need to determine which side of the line [tex]\( y = -8 \)[/tex] to shade. For [tex]\( y \geq -8 \)[/tex], we shade the region above the line because [tex]\( y \)[/tex] is greater than or equal to [tex]\(-8\)[/tex].
4. Check with a Test Point (Optional):
- To ensure we have shaded the correct region, we can use a test point. Commonly, the origin [tex]\((0, 0)\)[/tex] is a good choice if it's not on the boundary line.
- Substitute [tex]\( (0, 0) \)[/tex] into the inequality:
[tex]\( y \geq -8 \)[/tex]
[tex]\( 0 \geq -8 \)[/tex] (This is true)
- Since the origin satisfies the inequality, it confirms that the correct region to shade is indeed above the line [tex]\( y = -8 \)[/tex].
Here's a visual representation of the steps:
1. Draw a horizontal solid line passing through [tex]\(-8\)[/tex] on the y-axis.
2. Shade the entire region above the line [tex]\( y = -8 \)[/tex].
Finally, here is the graph for the inequality [tex]\( y \geq -8 \)[/tex]:
```
y
↑
| Shade This Entire Region
|__________________________
|__________________________
|__________________________
|__________________________
|__________________________
|__________________________
|
|
|__________________________
−8________________________________________________________ > x
```
In summary:
- The solid boundary line [tex]\( y = -8 \)[/tex] represents the values where [tex]\( y \)[/tex] is exactly [tex]\(-8\)[/tex].
- The shaded region above the line represents all the points where [tex]\( y \)[/tex] is greater than or equal to [tex]\(-8\)[/tex]. These points satisfy the inequality [tex]\( y \geq -8 \)[/tex].
