Questions: Solve the absolute value inequality. x-1 ≥ 7 Select the correct answer below and, if necessary, fill in the answer choice. A. The solution set in interval notation is (-∞, -6) ∪ (8, ∞). B. The solution set is 6.

Solution Steps

To solve the absolute value inequality \(|x-1| \geq 7\), we need to consider the two cases that arise from the definition of absolute value. The inequality \(|x-1| \geq 7\) can be split into two separate inequalities: \(x-1 \geq 7\) and \(x-1 \leq -7\). Solving these inequalities will give us the solution set in interval notation.

To solve the absolute value inequality \( |x-1| \geq 7 \), we will break it down into two separate inequalities and solve each one. Let's go through the steps:

The inequality \( |x-1| \geq 7 \) means that the expression inside the absolute value, \( x-1 \), is either greater than or equal to 7 or less than or equal to -7. This can be expressed as two separate inequalities: \[ x - 1 \geq 7 \quad \text{or} \quad x - 1 \leq -7 \]

Solve the inequality \( x - 1 \geq 7 \): \[ x - 1 \geq 7 \] Add 1 to both sides: \[ x \geq 8 \]

Solve the inequality \( x - 1 \leq -7 \): \[ x - 1 \leq -7 \] Add 1 to both sides: \[ x \leq -6 \]

The solution to the inequality \( |x-1| \geq 7 \) is the union of the solutions to the two inequalities: \[ x \leq -6 \quad \text{or} \quad x \geq 8 \] In interval notation, this is expressed as: \[ (-\infty, -6] \cup [8, \infty) \]

Final Answer