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 $u \geq a$.

Translating Tolerance Thresholds in Software

As a Quality Control Technician in a manufacturing facility, you are drafting a Standard Operating Procedure (SOP) to help new hires verify that machined engine bolts meet strict safety tolerances. The acceptable deviation $u$ of a bolt's diameter must satisfy the tolerance constraint $|u| \leq a$, where $a$ represents the maximum allowable positive deviation.

To ensure new hires can mathematically determine the range of acceptable diameters, you need to show them how to translate and solve this constraint. Arrange the steps of this procedure in the correct logical and mathematical order, from the initial setting of the tolerance inequality to identifying the final acceptable range on a number line.