To solve the inequality \(|x - 1| \geq 6\), we need to consider two cases because absolute value inequalities can have two scenarios. 1. **Case 1:** \(x - 1 \geq 6\) \[ x - 1 \geq 6 \] Adding 1 to both sides: \[ x \geq 7 \] 2. **Case 2:** \(x - 1 \leq -6\) \[ x - 1 \leq -6 \] Adding 1 to both sides: \[ x \leq -5 \] Combining both cases, the solution to the inequality \(|x - 1| \geq 6\) is: \[ x \leq -5 \quad \text{or} \quad x \geq 7 \] In interval notation, the solution is: \[ (-\infty, -5] \cup [7, \infty) \]

Solve.

|x-1|>=6

1 answer

To solve the inequality \(|x - 1| \geq 6\), we need to consider two cases because absolute value inequalities can have two scenarios.

Case 1: \(x - 1 \geq 6\)

\[ x - 1 \geq 6 \] Adding 1 to both sides: \[ x \geq 7 \]
Case 2: \(x - 1 \leq -6\)

\[ x - 1 \leq -6 \] Adding 1 to both sides: \[ x \leq -5 \]

Combining both cases, the solution to the inequality \(|x - 1| \geq 6\) is: \[ x \leq -5 \quad \text{or} \quad x \geq 7 \]

In interval notation, the solution is: \[ (-\infty, -5] \cup [7, \infty) \]