Solve the compound inequality below.
$m-2<-8 \text { or } \frac{m}{8}>1$
The problem asks us to solve the compound inequality \(m - 2 < -8 \text{ or } \frac{m}{8} > 1\). A compound inequality connected by "or" indicates that any value of \(m\) satisfying either of the individual inequalities is a solution. Our goal is to solve both inequalities individually and then form their union.
First, we address the inequality \(m - 2 < -8\). To isolate \(m\), we add 2 to both sides of the inequality:
\[m - 2 + 2 < -8 + 2\]
\[m < -6\]
This gives us the first part of our solution set.
Next, we address the inequality \(\frac{m}{8} > 1\). To isolate \(m\), we multiply both sides of the inequality by 8:
\[\frac{m}{8} \times 8 > 1 \times 8\]
\[m > 8\]
This gives us the second part of our solution set.
Since the two inequalities are connected by the word "or," the final solution is the union of the two individual solution sets. Combining our results, we find that the inequality holds when \(m < -6\) or \(m > 8\). Comparing this result to the given options, we find it corresponds to the first option.
In real life, compound inequalities are useful for defining ranges of acceptable conditions. For example, if a machine in a factory is designed to operate safely only when its temperature \(T\) is below \(10^\circ C\) or above \(80^\circ C\), we can express this condition as the compound inequality \(T < 10 \text{ or } T > 80\). Understanding how to solve these inequalities helps in monitoring systems, safety protocols, and setting threshold alerts for sensors in various engineering and environmental applications.