Question

What are the solution intervals for |x – 1| > 2?

Answer 1

x - 1 > 2 or x - 1 < -2

Answer 2

Solve to Get the answer

Answer 3

Recall that for any function f(x):\[|f(x)|=\left\{\begin{array}{lr}f(x) &\text{when } f(x)\ge 0 \\ -f(x) &\text{when } f(x)<0\end{array}\right.\]So therefore:\[|x-1|=\left\{\begin{array}{lr}x-1 &\text{when } x-1\ge 0 \\ 1-x &\text{when } x-1<0\end{array}\right.\]Or, equivalently:\[|x-1|=\left\{\begin{array}{lr}x-1 &\text{when } x\ge 1 \\ 1-x &\text{when } x<1\end{array}\right.\]Plug this back into the original equation, and solve for the pieces:\[2 <\left\{\begin{array}{lr}x-1 &\text{when } x\ge 1 \\ 1-x &\text{when } x<1\end{array}\right.\]\[2<x-1 \text{ when } x \ge 1 \text{ OR }2<1-x \text{ when } x > 1\]\[x>3 \text{ when } x \ge 1 \text{ OR }x<-1 \text{ when } x > 1\]To find the solution sets, find the intersections of intervals of each pair. Take note that the right hand side pair doesn't have an intersection of intervals.\[\text{Solution: }x>3\]