Solve the inequality below. Then, select the correct interval notation to represent the inequality.

[tex]\[ 4(x+2)\ \textless \ -\frac{1}{2}(4x-4) \][/tex]

A. [tex]\((- \infty, -1]\)[/tex]

B. [tex]\((- \infty, 4)\)[/tex]

C. [tex]\((- \infty, 4]\)[/tex]

D. [tex]\((- \infty, -1)\)[/tex]



Answer :

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]