Question

A time study was conducted to determine the length of time required to perform a particular task in a manufacturing process. The times required by approximately two-thirds of the workers in the study satisfied the inequality \(\left|\frac{t-15.6}{1.9}\right|<1\) where \(t\) is time in minutes. Determine the interval on the real number line in which these times lie.

Short answer

The times lie in the interval (13.7, 17.5) minutes.

Step-by-step solution

Understanding of the given inequality

The statement \(\left| \frac{t-15.6}{1.9} \right| < 1\) presents a condition for the real number \(t\) and is related to the concept of absolute value. One key property to know is that the inequality \(\left| x \right| < a\), where \(a > 0\), is equivalent to \(-a < x < a\). So this inequality can be rewritten as \(-1 < \frac{t-15.6}{1.9} < 1\).

Solving for \(t\) in the inequality

To solve for \(t\), we multiply each part of the inequality by 1.9, we get \(-1.9 < t - 15.6 < 1.9\). Adding 15.6 across the inequality, we will obtain \(13.7 < t < 17.5\).

Interpretation of the result

This result implies that the time, \(t\), taken by approximately two-thirds of the workers to complete the task lies between 13.7 minutes and 17.5 minutes.

Key concepts

Inequality Solving

Inequality solving is an essential part of understanding mathematical relationships and constraints. When faced with an absolute value inequality like \(\left|\frac{t-15.6}{1.9}\right|<1\), it represents a range of values for which the expression within the absolute value is less than 1. To solve such inequalities, a fundamental step is to eliminate the absolute value by setting up a compound inequality. This involves interpreting the inequality as two separate conditions: where the expression inside is less than 1, and where it's greater than -1, giving us \( -1 < \frac{t-15.6}{1.9} < 1 \).

From there, isolation of the variable of interest – in this case, the time \(t\) – involves algebraic manipulation, such as multiplying through by the denominator and adding or subtracting terms. For the inequality above, we multiply each part by 1.9 and add 15.6, to isolate \(t\) and find the solution interval \(13.7 < t < 17.5\). It's critical that when performing operations on all parts of a compound inequality, we maintain the inequality direction, unless we're multiplying or dividing by a negative number, in which case we must reverse it. This forms the crux of inequality solving, a technique that can be applied across various mathematical contexts.

Real Number Line Interval

Understanding intervals on the real number line is crucial for visualizing solutions to inequalities and for describing ranges of permissible values. An interval is a set of numbers between two endpoints, which can be a mixture of inclusive and exclusive bounds. For example, in the context of the inequality \(13.7 < t < 17.5\), the time interval is (13.7, 17.5), an open interval since it does not include the exact endpoints.

To depict this graphically, one would draw a real number line and use parentheses to denote that 13.7 and 17.5 are not part of the solution. The numbers inside this interval satisfy the original absolute value inequality. Knowing how to translate inequality solutions into interval notation or a number line diagram is a key skill in math, as it enables clear communication and understanding of the range of solutions. Moreover, this understanding is valuable when assessing quantities in real-world scenarios, such as time in process studies or measurements in scientific experiments.

Manufacturing Process Time Study

In the realm of industrial engineering, a manufacturing process time study is a comprehensive analysis aimed at determining the time required for a worker to complete a specific task. The goal is to establish standard times, identify efficiency improvements, and ensure the effective management of workforces and processes. The inequality \(\left|\frac{t-15.6}{1.9}\right|<1\) from the time study suggests that the times recorded for approximately two-thirds of the workers fall within a particular interval.

This interval, \(13.7 < t < 17.5\) minutes, aids in pinpointing the acceptable duration range for the task at hand. With this information, managers can set realistic production targets and expectations, design incentives, and improve scheduling. Furthermore, identifying outliers in the time study can help in recognizing areas where training or process refinement is needed. Thus, time studies are pivotal in ensuring that manufacturing processes are both efficient and cost-effective, thereby optimizing labor utilization and maintaining product quality.