Question

How do you graph and solve `|x – 1| ≥ 2`?

Answer 1

One way is to consider the two cases for the absolute value separately.
Since this is a relationship in one variable it can be graphed on a number line as:

Explanation:

Case 1: `color(black)((x-1)>=0)`
`color(white)("XXX")abs(x-1)>=2`

`color(white)("XXX")x-1 >=2`

`color(white)("XXX")x >= 3`

Case 2: `color(black)(x-1) < 0`
`color(white)("XXX")abs(x-1) >= 2`

`color(white)("XXX")1-x >=2`

`color(white)("XXX")-1 >= x`