Question

olve the following inequality algebraically.
vertical bar, x, minus, 1, vertical bar, is greater than or equal to, 15
∣x−1∣≥15

Answer 1

To solve the inequality \(|x - 1| \geq 15\), we need to consider the definition of absolute value. The absolute value \(|a|\) indicates the distance of \(a\) from 0 on the number line.

The inequality \(|x - 1| \geq 15\) can be rewritten into two separate cases:

1. \(x - 1 \geq 15\)
2. \(x - 1 \leq -15\)

Now, we will solve each case separately.

### Case 1: \(x - 1 \geq 15\)
To solve this inequality, we add 1 to both sides:
\[
x \geq 15 + 1
\]
\[
x \geq 16
\]

### Case 2: \(x - 1 \leq -15\)
To solve this inequality, we also add 1 to both sides:
\[
x \leq -15 + 1
\]
\[
x \leq -14
\]

### Combining Both Cases
From both cases, we find the solutions:
1. \(x \geq 16\)
2. \(x \leq -14\)

Thus, the solution to the inequality \(|x - 1| \geq 15\) is:
\[
x \leq -14 \quad \text{or} \quad x \geq 16
\]

### Final Answer
In interval notation, the solution is:
\[
(-\infty, -14] \cup [16, \infty)
\]