1Cademy - Try It: Solving and Graphing $$2(3x+1) \leq 20$$ and $$4(x-1)

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Procedure for Solving Compound Inequalities with "and"

Example

Try It: Solving and Graphing $2(3x+1) \leq 20$ and $4(x-1) < 2$

To practice solving a compound inequality, solve $2(3x+1) \leq 20$ and $4(x-1) < 2$ . Solving the first inequality gives $6x + 2 \leq 20$ , which simplifies to $6x \leq 18$ or $x \leq 3$ . Solving the second inequality yields $4x - 4 < 2$ , which simplifies to $4x < 6$ or $x < \frac{3}{2}$ . Graphing both results shows that the intersection of $x \leq 3$ and $x < \frac{3}{2}$ is the interval where $x < \frac{3}{2}$ . In interval notation, this overlapping solution is written as $(-\infty, \frac{3}{2})$ .

Updated 2026-06-05

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OpenStax Intermediate Algebra

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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 ope
Water Filtration System Calibration

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