To solve the given inequality [tex]\(4(x + 2) < -\frac{1}{2}(4x - 4)\)[/tex], we will follow a step-by-step approach:
### Step 1: Distribute the constants on both sides of the inequality
First, distribute the 4 on the left-hand side of the inequality:
[tex]\[
4(x + 2) = 4x + 8
\][/tex]
Next, distribute [tex]\(-\frac{1}{2}\)[/tex] on the right-hand side of the inequality:
[tex]\[
-\frac{1}{2}(4x - 4) = -2x + 2
\][/tex]
This rewrites the inequality as:
[tex]\[
4x + 8 < -2x + 2
\][/tex]
### Step 2: Combine like terms by adding [tex]\(2x\)[/tex] to both sides of the inequality
To get all [tex]\(x\)[/tex]-terms on one side, add [tex]\(2x\)[/tex] to both sides:
[tex]\[
4x + 8 + 2x < -2x + 2 + 2x
\][/tex]
[tex]\[
6x + 8 < 2
\][/tex]
### Step 3: Isolate [tex]\(x\)[/tex] by subtracting 8 from both sides of the inequality
Subtract 8 from both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
6x + 8 - 8 < 2 - 8
\][/tex]
[tex]\[
6x < -6
\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex] by dividing both sides by 6
Divide both sides by 6:
[tex]\[
x < \frac{-6}{6}
\][/tex]
[tex]\[
x < -1
\][/tex]
### Step 5: Write the solution in interval notation and select the correct interval
The solution to the inequality is [tex]\(x < -1\)[/tex], which is represented in interval notation as:
[tex]\[
(-\infty, -1)
\][/tex]
Therefore, the correct interval notation to represent the solution to the given inequality is:
[tex]\[
(-\infty, -1)
\][/tex]
### Conclusion
The correct interval notation from the given options is:
[tex]\[
(-\infty, -1)
\][/tex]
