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

As a quality assurance technician in a manufacturing plant, you are tasked with identifying defective parts. A part is considered defective and must be rejected if its thickness variance, $x$, in millimeters, satisfies the inequality $|2x - 3| \geq 5$. Recalling the systematic procedure for solving 'greater than' absolute value inequalities, arrange the exact steps you would follow to determine the range of defective thicknesses.

In your role as a retail inventory manager, you use the formula $|2x - 3| \geq 5$ to identify items with a high variance that require a manual audit. When applying the systematic five-step procedure to solve this inequality, which of the following is the correct compound inequality used in Step 2 to remove the absolute value bars?

As a facility manager, you monitor a pressure system where an alarm is triggered if the pressure variance, $x$, satisfies $|2x - 3| \geq 5$. To ensure your team follows the safety protocol, match each stage of the five-step solution process with its correct mathematical representation.

A materials engineer testing the load-bearing capacity of a new alloy uses the inequality $|2x - 3| \geq 5$ to identify samples that fail a stress test. According to the five-step systematic procedure for this specific inequality, the range of values for $x$ that would cause a sample to fail is expressed in interval notation as $(-\infty, -1] \cup [4, \infty)$.

Determining Operational Boundaries

Verifying Machine Safety Alert Thresholds

As a quality control inspector at a manufacturing facility, you verify the thickness variation of specialized brackets. The thickness variation $x$ (in millimeters) is flagged for secondary inspection if it satisfies the absolute value inequality $|2x - 3| \geq 5$. To determine the range of flagged brackets, you apply the five-step systematic procedure for 'greater than' absolute value inequalities. In Step 2, you must rewrite this absolute value expression as two separate linear inequalities that form a compound inequality. Fill in the blank with the logical conjunction (such as 'and' or 'or') that correctly connects these two inequalities: ${}2x - 3 \leq -5$ ____ ${}2x - 3 \geq 5$.