Certainly! Let's solve the inequality step-by-step:
Given the inequality:
[tex]\[ -\frac{1}{2} x > 4 \][/tex]
### Step 1: Isolate the variable [tex]\( x \)[/tex]
To isolate [tex]\( x \)[/tex], we need to get rid of the fraction. We can start by multiplying both sides of the inequality by [tex]\(-2\)[/tex]. However, when we multiply or divide an inequality by a negative number, we must reverse the inequality sign.
So, let's proceed:
[tex]\[ -2 \cdot \left( -\frac{1}{2} x \right) < -2 \cdot 4 \][/tex]
This simplifies to:
[tex]\[ x < -8 \][/tex]
### Step 2: Identify the solution set
The solution to our inequality is:
[tex]\[ x < -8 \][/tex]
Now let's compare this with the given options:
1. [tex]\(\{x \mid x \in \mathbb{R}, x > -2\}\)[/tex]
2. [tex]\(\{x \mid x \in \mathbb{R}, x < -8\}\)[/tex]
3. [tex]\(\{x \mid x \in \mathbb{R}, x < -2\}\)[/tex]
4. [tex]\(\{x \mid x \in \mathbb{R}, x > -8\}\)[/tex]
The correct solution set is:
[tex]\(\{x \mid x \in \mathbb{R}, x < -8\}\)[/tex]
So, the appropriate choice is:
[tex]\(\{x \mid x \in \mathbb{R}, x < -8 \}\)[/tex], which corresponds to option 2.
