Absolute Value Inequalities Quick Check

Question

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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: \[ (-\infty, -11] \cup [3, \infty) \]