In a manufacturing environment, a technician is calibrating a sensor. The sensor is considered active only when the input value $x$ satisfies both ${}2(3x+1) \leq 20$ and ${}4(x-1) < 2$. Based on the mathematical solution for this compound inequality, which interval notation correctly represents the values of $x$ for which the sensor is active?

In a manufacturing facility, a quality control engineer is establishing the allowable pressure adjustment range $x$ for a specialized valve. The adjustment must simultaneously satisfy two safety protocols: ${}2(3x+1) \leq 20$ and ${}4(x-1) < 2$. Arrange the following steps in the correct order to solve this compound inequality and identify the final overlapping safety interval in interval notation.

In a manufacturing facility, a technician is calibrating a sensor that must operate within the limits defined by the compound inequality ${}2(3x+1) \leq 20$ and ${}4(x-1) < 2$. Match each part of the solution process with its corresponding mathematical result.

True or False: When solving the compound inequality ${}2(3x+1) \leq 20$ and ${}4(x-1) < 2$, the solution to the first inequality is $x \leq 3$ and the solution to the second inequality is $x < \frac{3}{2}$.

Quality Control Safety Intervals

An HVAC technician is configuring the safety control limits for a smart building's environmental chamber. To ensure system stability, the temperature calibration variable, $x$, must simultaneously satisfy two safety constraints: the primary sensor constraint ${}2(3x+1) \leq 20$ and the backup sensor constraint ${}4(x-1) < 2$.

By solving both inequalities and finding their intersection, the technician determines the safe operational range for $x$.

In interval notation, the overlapping safe operational range for $x$ is ____ (use -infinity for $-\infty$, represent the fraction in simplified form as 3/2, and do not include any spaces. For example, your format should look like (-infinity,3/2)).

Water Filtration System Calibration