Question

Solve the absolute value inequality 2|x+4|≥14 . Provide the solution using interval notation.(1 point)

Answer 1

To solve the absolute value inequality \( 2|x+4| \geq 14 \), we start by dividing both sides by 2:

\[
|x+4| \geq 7
\]

Now, we can solve the absolute value inequality by considering the two cases:

1. \( x + 4 \geq 7 \)
2. \( x + 4 \leq -7 \)

**Case 1:**

\[
x + 4 \geq 7
\]

Subtracting 4 from both sides:

\[
x \geq 3
\]

**Case 2:**

\[
x + 4 \leq -7
\]

Subtracting 4 from both sides:

\[
x \leq -11
\]

Combining both cases, we have two intervals:

1. \( x \geq 3 \) corresponds to the interval \( [3, \infty) \)
2. \( x \leq -11 \) corresponds to the interval \( (-\infty, -11] \)

Therefore, the solution in interval notation is:

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