For any algebraic expressions $$u$$ and $$v$$, if $$|u| = |v|$$, then $$u = -v$$ or $$u = v$$. This property demonstrates that when two absolute value expressions are equal to each other, the expressions inside the absolute value bars must logically either be exactly equal to each other or be exact opposites of each other.

Claude

For any algebraic expression $$u$$ and any positive real number $$a$$, the condition $$|u| = a$$ means that the mathematical statement can be rewritten as two equivalent equations: $$u = -a$$ or $$u = a$$. This property stems from the fact that we are seeking all values whose distance from zero is $$a$$, and both $$a$$ and $$-a$$ satisfy this requirement. A single statement for the solutions can also be written as $$u = \pm a$$. Remember that an absolute value cannot be a negative number.

Absolute Value Equations

https://openstax.org/details/books/intermediate-algebra-2e

OpenStax Intermediate Algebra

Equations with Two Absolute Values

To solve an absolute value equation, follow a structured four-step procedure. First, isolate the absolute value expression on one side of the equation using standard algebraic properties. Second, write the equivalent equations by setting the algebraic expression inside the absolute value equal to both the positive and negative versions of the isolated constant. Third, solve each resulting equation individually. Finally, check each solution by substituting it back into the original absolute value equation to verify that it results in a true mathematical statement.

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