Try It: Solving $3|x - 4| - 4 = 8$ and $2|x - 5| + 3 = 9$

A quality control analyst uses the equation ${}2|x - 7| + 5 = 9$ to determine the upper and lower acceptable size limits for a newly manufactured part. According to the standard procedure for solving this type of equation, what foundational step must the analyst recall and perform before splitting the equation into two separate cases?

A quality control inspector is using the absolute value equation ${}3|L - 10| + 6 = 15$ to determine the acceptable length range ($L$) for a steel bolt. Arrange the following procedural steps in the correct order to solve for $L$ by first isolating the absolute value expression.

An electrical technician uses the equation ${}3|V - 12| + 6 = 18$ to determine the acceptable voltage variance ($V$) for a specific component. According to the standard algebraic procedure for solving such equations, the technician must first __________ the absolute value expression $|V - 12|$ before it can be rewritten as two separate linear equations.

A quality control technician uses the equation ${}5|p - 100| + 50 = 250$ to calculate the pressure tolerance ($p$) for a valve. To solve for $p$, the technician must follow a specific sequence of steps to isolate the absolute value expression. Match each equation state with the correct next action required by the standard algebraic procedure.

When an accountant uses an absolute value equation with extra terms to calculate acceptable budget variances, they should immediately split the equation into two separate cases before isolating the absolute value expression.