Solve and graph one-variable absolute-value equations and inequalities; interpret solutions in context.
Solve an absolute-value equation with a variable outside the absolute value
Problem
Solve the absolute-value equation \(|x~-~1|~=~x~+~3\) and check for extraneous solutions.
Big Picture
What this problem is really about
When the other side contains the variable, the usual fixed-target two-case shortcut does not apply directly. We’ll first require that side to be nonnegative, then use the piecewise definition of absolute value on either side of the inside expression’s zero. Each branch must be solved under its own condition, and every surviving candidate must satisfy both the branch restriction and the original equation.
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