Question

which number line represents |x - 2| ≥ 1?
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5

Explanation:

Step1: Solve the absolute - value inequality

If \(|x - 2|\geq1\), then we have two cases:
Case 1: \(x - 2\geq1\). Solving for \(x\), we add 2 to both sides of the inequality: \(x-2 + 2\geq1 + 2\), so \(x\geq3\).
Case 2: \(x - 2\leq - 1\). Solving for \(x\), we add 2 to both sides of the inequality: \(x-2+2\leq - 1 + 2\), so \(x\leq1\).

Step2: Analyze the number - line representation

The solution set \(x\leq1\) or \(x\geq3\) means that the number - line will have a closed circle at \(x = 1\) and \(x = 3\) (because of the \(\geq\) and \(\leq\) signs) and the line will be shaded to the left of \(x = 1\) and to the right of \(x = 3\).

Answer:

The number - line with closed circles at \(x = 1\) and \(x = 3\) and shading to the left of \(x = 1\) and to the right of \(x = 3\) (the last option among the ones shown in the image).