- Define the conditions: \(h < 7\) for working less than 7 hours and \(h > 9\) for working more than 9 hours.
- Combine these conditions using the logical "or" operator to represent the union of the two intervals.
- Identify the compound inequality as \(h < 7\) or \(h > 9\).
- Final answer: \(\boxed{h < 7 \text{ or } h > 9}\).
Explanation
- Problem Analysis and Variable Definition
Antonio's working hours, denoted by \(h\), need to meet specific conditions to avoid heavy traffic. We are given two separate conditions:
- Antonio can drive home if he works less than 7 hours, represented as \(h < 7\).
- Antonio can drive home if he works more than 9 hours, represented as \(h > 9\).
We need to combine these two conditions into a single compound inequality using the logical operator "or".
- Formulating the Compound Inequality
In mathematics, the "or" operator denotes that a value satisfies at least one of the given conditions. Therefore, if we want to represent the set of all hours \(h\) that satisfy either the first condition or the second condition, we connect the two individual inequalities with the word "or". The expression becomes:
\[h < 7 \text{ or } h > 9\]
This compound statement describes all possible values of \(h\) that allow Antonio to avoid heavy traffic.
- Comparison and Selection
Comparing our result to the given options:
- \(7 < h < 9\) (This represents hours worked between 7 and 9, which is the opposite of our requirement.)
- \(7 > h > 9\) (This is logically impossible, as a number cannot be both greater than 9 and less than 7 simultaneously.)
- \(h < 7\) or \(h > 9\) (This matches our derived condition.)
- \(h > 7\) or \(h < 9\) (This covers all real numbers \(h\), which is not the intended condition.)
Thus, the correct choice is \(h < 7\) or \(h > 9\).
Examples
Compound inequalities are frequently used in logistics and resource management to define "safe" or "optimal" operating ranges. For example, a refrigerator might have an optimal temperature range for food safety (e.g., \(T < 3^\circ\text{C}\) or \(T > 6^\circ\text{C}\) for certain experimental storage conditions, though usually it is a single interval). By using compound inequalities, automated systems can trigger alarms or adjustments whenever the variable falls outside the acceptable range, ensuring efficiency and safety in real-world environments.