Question

Solve |x-6|≤6


State your answer as a compound inequality A ≤ x ≤ B:


State your answer in interval notation [A,B]:

Answer 1

To solve the inequality \(|x - 6| \leq 6\), we start by converting this absolute value inequality into a compound inequality.

The absolute value inequality \(|x - 6| \leq 6\) means that the expression \(x - 6\) is between \(-6\) and \(6\). Therefore, we can write:

\[
-6 \leq x - 6 \leq 6
\]

Next, we solve for \(x\):

1. Add \(6\) to all parts of the inequality:

\[
-6 + 6 \leq x - 6 + 6 \leq 6 + 6
\]

This simplifies to:

\[
0 \leq x \leq 12
\]

Thus, the compound inequality is:

\[
0 \leq x \leq 12
\]

In standard form, we can express it as:

\[
A = 0 \quad \text{and} \quad B = 12
\]

Now, in interval notation, this compound inequality can be expressed as:

\[
[0, 12]
\]

To summarize:

- The compound inequality is \(0 \leq x \leq 12\).
- In interval notation, it is \([0, 12]\).