Solve and graph one-variable absolute-value equations and inequalities; interpret solutions in context.
Solve an absolute-value inequality with greater than or greater than or equal to
Problem
Solve \(|x~-~2|~\ge~5\) and express the solution as a union.
Big Picture
What this problem is really about
A lower bound on absolute value asks for points at least a fixed distance from the center, so the solution occupies two outside regions. We’ll translate that condition into a negative-side case or a positive-side case, solve each inequality, and keep the equality bars at both boundaries. Checking the center and the boundary points distinguishes the outer union from the interval between them.
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