Question

Solve the absolute value inequality −2|x−7|≤−8
. Provide the solution in interval notation.(1 point)
The solution to −2|x−7|≤−8
is ([
]∪
[
]).
Remaining Attempts : 3

Answer 1

To solve the inequality \(-2|x-7| \leq -8\), we can start by dividing both sides by \(-2\). Remember that when you divide an inequality by a negative number, you must reverse the inequality sign:

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

Now, we will solve for \(x\) using the definition of absolute value. The inequality \(|x - 7| \geq 4\) can be rewritten as two separate inequalities:

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

### Solving the first inequality:
\[
x - 7 \geq 4
\]
\[
x \geq 11
\]

### Solving the second inequality:
\[
x - 7 \leq -4
\]
\[
x \leq 3
\]

Now we can combine the two solutions. The solution set will be:

\[
x \leq 3 \quad \text{or} \quad x \geq 11
\]

In interval notation, this is:

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

Thus, the solution to the inequality \(-2|x−7|≤−8\) is:

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