Question

Solve the following inequality algebraically.
vertical bar, x, minus, 2, vertical bar, is less than, 12
∣x−2∣<12

Answer 1

To solve the inequality \( |x - 2| < 12 \), we need to consider what the absolute value means. The expression \( |x - 2| < 12 \) states that the distance between \( x \) and \( 2 \) is less than \( 12 \).

This leads to the following compound inequality:

\[
-12 < x - 2 < 12
\]

Now we will solve this compound inequality step by step.

1. **Add 2 to all parts of the inequality**:

\[
-12 + 2 < x - 2 + 2 < 12 + 2
\]

This simplifies to:

\[
-10 < x < 14
\]

Thus, the solution to the inequality \( |x - 2| < 12 \) is:

\[
\boxed{(-10, 14)}
\]

This means that \( x \) can take any value between \( -10 \) and \( 14 \), not including \( -10 \) and \( 14 \).