Question

Solve the following inequality algebraically.
vertical bar, x, minus, 2, vertical bar, is less than, 11
∣x−2∣<11

Answer 1

To solve the inequality \(|x - 2| < 11\), we can break it down into two separate inequalities based on the properties of absolute values.

The inequality \(|x - 2| < 11\) means that the expression inside the absolute value, \(x - 2\), lies between -11 and 11. Therefore, we can set up the following compound inequality:

\[
-11 < x - 2 < 11
\]

Now we can solve for \(x\) by isolating it in the middle:

1. To solve the left side of the inequality:
\[
-11 < x - 2
\]
Add 2 to both sides:
\[
-11 + 2 < x
\]
\[
-9 < x \quad \text{or} \quad x > -9
\]

2. To solve the right side of the inequality:
\[
x - 2 < 11
\]
Again, add 2 to both sides:
\[
x < 11 + 2
\]
\[
x < 13
\]

Combining both parts, we have:
\[
-9 < x < 13
\]

Thus, the solution to the inequality \(|x - 2| < 11\) is:
\[
\boxed{(-9, 13)}
\]
This means that \(x\) can take any value between -9 and 13, but not including -9 and 13 themselves.