Solve and graph one-variable absolute-value equations and inequalities; interpret solutions in context.
Solve an absolute-value inequality after isolating the absolute value
Problem
Solve the absolute-value inequality \(2|x~-~3|~+~1~\le~9\) after isolating the absolute-value expression.
Big Picture
What this problem is really about
The distance condition cannot be interpreted until the absolute-value expression is alone. We’ll undo the outside addition and multiplication in reverse order, watching the divisor’s sign, and then translate the isolated at-most distance into one compound inequality. Applying the same shift to all three parts finds the boundaries, while the equality bars and an endpoint check determine inclusion.
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