To solve the inequality [tex]\( -\frac{1}{2} x \geq 4 \)[/tex], let's go through the steps in detail:
1. Rewrite the Inequality:
We start with the given inequality:
[tex]\[
-\frac{1}{2} x \geq 4
\][/tex]
2. Isolate the Variable:
To isolate [tex]\( x \)[/tex], we need to get rid of the coefficient [tex]\( -\frac{1}{2} \)[/tex]. This can be done by multiplying both sides of the inequality by the reciprocal of [tex]\( -\frac{1}{2} \)[/tex], which is [tex]\( -2 \)[/tex]. Remember, when we multiply or divide both sides of an inequality by a negative number, we reverse the inequality sign.
Multiplying both sides by [tex]\( -2 \)[/tex], we get:
[tex]\[
x \leq 4 \cdot -2
\][/tex]
3. Simplify:
Simplifying the right side of the inequality:
[tex]\[
x \leq -8
\][/tex]
4. Solution Set:
The inequality [tex]\( x \leq -8 \)[/tex] tells us that [tex]\( x \)[/tex] can be any number less than or equal to [tex]\(-8\)[/tex].
5. Number Line Representation:
On a number line, this solution set includes all the numbers to the left of [tex]\(-8\)[/tex] and [tex]\(-8\)[/tex] itself. Here's how it would be depicted:
- Draw a number line.
- Place a closed circle (because [tex]\(-8\)[/tex] is included in the solution set) at [tex]\( x = -8 \)[/tex].
- Shade the line extending to the left of [tex]\(-8\)[/tex], indicating all numbers less than [tex]\(-8\)[/tex].
Here is a visual representation:
```
<-----|====================================>
-8
```
In this representation, the closed circle at [tex]\(-8\)[/tex] and the shading to the left show that [tex]\( x \)[/tex] can be any number less than or equal to [tex]\(-8\)[/tex].
Thus, the number line correctly representing the solution set for the inequality [tex]\( -\frac{1}{2} x \geq 4 \)[/tex] includes all values from [tex]\(-\infty\)[/tex] to [tex]\(-8\)[/tex], inclusive of [tex]\(-8\)[/tex].
