Question

Absolute Value Inequalities Quick Check
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Question
Solve the absolute value inequality 2|x+4|≥14
. Provide the solution using interval notation.(1 point)
The solution is (
]∪
[
].

Answer 1

To solve the absolute value inequality \( 2|x+4| \geq 14 \), we first isolate the absolute value:

1. **Divide both sides by 2**:
\[
|x + 4| \geq 7
\]

2. **Set up the two cases for the absolute value**:
- Case 1: \( x + 4 \geq 7 \)
- Case 2: \( x + 4 \leq -7 \)

### Case 1: \( x + 4 \geq 7 \)
Subtract 4 from both sides:
\[
x \geq 3
\]

### Case 2: \( x + 4 \leq -7 \)
Subtract 4 from both sides:
\[
x \leq -11
\]

3. **Combine the solutions**:
From the two cases, we have:
- \( x \geq 3 \)
- \( x \leq -11 \)

In interval notation, the solution is:
\[
(-\infty, -11] \cup [3, \infty)
\]

So the final answer in interval notation is:
\[
(-\infty, -11] \cup [3, \infty)
\]