- The inequality \(64 < x < 68\) signifies that \(x\) must be both greater than 64 and less than 68.
- The term "and" represents the intersection of these two conditions, which is required for the value to be within the specified range.
- The equivalent expression is the first option provided.
- Final Answer: \(\boxed{x > 64 \text{ and } x < 68}\)
Explanation
- Analyzing the Inequality Condition
The problem asks to find an equivalent representation of the inequality \(64 < x < 68\), where \(x\) represents an actor's height. This compound inequality represents a specific range of values for \(x\). The range is defined by two conditions that must be met simultaneously: the height \(x\) must be greater than 64 inches and the height \(x\) must be less than 68 inches.
- Evaluating Logical Connectors
The compound inequality \(64 < x < 68\) is a concise way of writing the intersection of two individual inequalities: \(x > 64\) and \(x < 68\). In mathematics, when we describe a value as being between two numbers, we are essentially saying that the value must satisfy both boundaries at once. Therefore, we use the logical connector "and" to join these two requirements. If we were to use the connector "or", it would imply that satisfying either one of the conditions (being taller than 64 inches or shorter than 68 inches) would be sufficient, which would cover a much broader set of values and not correctly represent the specified range.
- Identifying the Correct Option
Comparing our derived expression \(x > 64 \text{ and } x < 68\) with the given options, it matches the first choice exactly. The other options involve either different logical operators ("or") or different range directions (less than 64), which would not correctly represent the requirement of being between 64 and 68 inches. Thus, the alternative way to express the inequality is \(x > 64\) and \(x < 68\).
Examples
In everyday life, inequalities like this are used whenever there is a specific target range. For example, if you are looking for a car tire that fits a rim size between 15 and 17 inches, you might express this as \(15 < s < 17\), which is equivalent to saying the size \(s\) must be greater than 15 inches AND less than 17 inches. Similar logic applies to temperature settings for home thermostats, dosage ranges for medications, or size requirements for shipping packages. Understanding these logical connectors helps ensure that all constraints in a given situation are met accurately.