Question

Solve the following inequality algebraically.
vertical bar, x, minus, 1, vertical bar, is greater than or equal to, 11
∣x−1∣≥11

Answer 1

To solve the inequality \( |x - 1| \geq 11 \), we need to consider the definition of absolute value. The expression \( |x - 1| \) represents the distance between \( x \) and \( 1 \) on the number line. The inequality states that this distance is at least \( 11 \).

This leads to two possible scenarios:

1. \( x - 1 \geq 11 \)
2. \( x - 1 \leq -11 \)

Let's solve each case:

### Case 1: \( x - 1 \geq 11 \)

Add \( 1 \) to both sides:

\[
x \geq 12
\]

### Case 2: \( x - 1 \leq -11 \)

Again, add \( 1 \) to both sides:

\[
x \leq -10
\]

### Conclusion

Combining the results from both cases, we have:

\[
x \leq -10 \quad \text{or} \quad x \geq 12
\]

Therefore, the solution to the inequality \( |x - 1| \geq 11 \) is:

\[
(-\infty, -10] \cup [12, \infty)
\]