Procedure for Solving Absolute Value Inequalities with $<$ or $\leq$

Application of Absolute Value Inequalities to Tolerance

Example: Solving $|5x - 6| \leq 4$

Try It: Solving $|2x - 1| \leq 5$ and $|4x - 5| \leq 3$

In a manufacturing facility, the quality control team uses an absolute value inequality to represent the acceptable tolerance for a machine part's dimensions. If the acceptable deviation $d$ must be less than or equal to a specific positive tolerance value $t$, which is written as $|d| \leq t$, how is this algebraic condition correctly rewritten as a compound inequality to show the full range of acceptable values?

Precision Machining Tolerance Calibration

In technical fields such as precision manufacturing and quality control, tolerances are often defined using absolute value inequalities. Match each absolute value inequality with its correct algebraic translation or geometric description, where $a$ is a positive real number representing the allowable tolerance.

A quality control technician in a pharmaceutical plant is monitoring the weight deviation $x$ of a capsule, which must satisfy the condition $|x| \leq 0.02$ mg. To express this as a range, the technician translates the absolute value inequality into a compound inequality. Complete the following: The deviation $x$ must be between ${}-0.02$ mg and ____ mg.

When setting up safety constraints for a chemical mixture, an engineer models the acceptable pressure deviation $u$ with the absolute value inequality $|u| \leq a$ (where $a$ is a positive real number). This mathematical rule translates to the compound inequalities $u \leq -a$ or $u \geq a$.