- Solve the first inequality \(m - 2 < -8\) to find \(m < -6\).
- Solve the second inequality \(\frac{m}{8} > 1\) to find \(m > 8\).
- Combine these solutions using the "or" operator, resulting in the union of the two sets.
- Select the matching option, which is \(\boxed{m < -6 \text{ or } m > 8}\).
Explanation
- Analyzing the Problem
We need to solve the compound inequality \[m - 2 < -8 \quad \text{or} \quad \frac{m}{8} > 1.\] A compound inequality joined by "or" means that any value of \(m\) that satisfies at least one of the inequalities is a part of the solution set. We will solve each inequality separately and then combine their solution sets.
- Solving the First Inequality
To solve the first inequality, \(m - 2 < -8\), we add \(2\) to both sides of the expression to isolate the variable \(m\): \[m - 2 + 2 < -8 + 2\] \[m < -6.\] This gives us the first part of our solution set.
- Solving the Second Inequality
To solve the second inequality, \(\frac{m}{8} > 1\), we multiply both sides of the inequality by \(8\). Since \(8\) is a positive number, the direction of the inequality remains unchanged: \[\frac{m}{8} \times 8 > 1 \times 8\] \[m > 8.\] This gives us the second part of our solution set.
- Conclusion and Selection of the Correct Option
Since the original problem is a compound inequality with the conjunction "or," we combine the two solution sets: \(m < -6\) and \(m > 8\). The final solution set is the union of these two conditions: \(m < -6 \text{ or } m > 8\). Comparing this to the given options, we identify that the first option is the correct one. Therefore, the final answer is \(m < -6 \text{ or } m > 8\).
Examples
In finance, compound inequalities are used to define ranges for acceptable risk or performance. For example, a mutual fund might be considered "stable" if its daily change is either less than \(\$-6\) per share or greater than \(\$8\) per share, representing volatility zones. By solving this as a compound inequality, an investor can quickly determine which market conditions meet these specific criteria.