- Define the condition for a small vase: \(c \leq 4.5\).
- Define the condition for a large vase: \(c \geq 12\).
- Use the logical 'or' to combine these conditions for either size: \(c \leq 4.5 \text{ or } c \geq 12\).
- The final answer is \(\boxed{c \leq 4.5 \text{ or } c \geq 12}\).
Explanation
- Analyzing the Conditions for Vase Sizes
The problem asks us to determine the compound inequality for the amount of clay, \(c\), used for a single vase which can be either a small or a large size. We are given the following information:
- For a small vase, the amount of clay \(c\) is "no more than 4.5 ounces". Mathematically, "no more than" translates to the less-than-or-equal-to symbol (\(\leq\)). Thus, the condition for a small vase is \(c \leq 4.5\).
- For a large vase, the amount of clay \(c\) is "at least 12 ounces". Mathematically, "at least" translates to the greater-than-or-equal-to symbol (\(\geq\)). Thus, the condition for a large vase is \(c \geq 12\).
- Constructing the Compound Inequality
We need the compound inequality that represents the clay usage for a vase that could be either small or large. In mathematics, the word "either... or..." is represented by the logical connector "or". Since a vase cannot be both small and large simultaneously (as their clay ranges do not overlap), we represent the set of all possible clay amounts as the union of the two individual conditions. Therefore, we combine the two inequalities:
\[c \leq 4.5 \text{ or } c \geq 12\]
- Selecting the Correct Option
Comparing our result \(c \leq 4.5 \text{ or } c \geq 12\) with the given choices:
- \(4.5 < c < 12\)
- \(4.5 \leq c \leq 12\)
- \(c < 4.5\) or \(c > 12\)
- \(c \leq 4.5\) or \(c \geq 12\)
Our result matches the fourth option. Thus, the compound inequality that correctly represents the amount of clay \(c\) is \(\boxed{c \leq 4.5 \text{ or } c \geq 12}\).
Examples
Compound inequalities are very useful in real-life quality control and manufacturing. For example, a quality inspector at a factory might define an acceptable range for the weight of a product. If a product is expected to be either in a "light" category (e.g., weighing 100 grams or less) or a "heavy" category (e.g., weighing 200 grams or more), an inspector can use the inequality \(w \leq 100 \text{ or } w \geq 200\) to identify products that meet the specified standards. This helps in categorizing items efficiently based on their physical properties.