A quality assurance engineer randomly checks manufactured parts to ensure that they are within 0.002 mm of the desired specifications. If the desired length of the part is 3 mm, then the compound inequality $-0.002 \leq x-3 \leq 0.002$ represents the acceptable lengths.
Choose the absolute value inequality that represents the situation.
$|x-3| \leq 0.002$
$|x-3| \geq 0.002$
$|x-0.002| \leq 3$
$|x-0.002| \geq 3$
Answer (2)
The correct absolute value inequality representing the acceptable lengths of the manufactured parts is ∣ x − 3∣ ≤ 0.002 . This reflects that the actual length x can vary within 0.002 mm of the desired length of 3 mm. Thus, the answer is ∣ x − 3∣ ≤ 0.002 .
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The compound inequality − 0.002 ≤ x − 3 ≤ 0.002 represents acceptable lengths.
Convert the compound inequality to an absolute value inequality.
The absolute value inequality is ∣ x − 3∣ ≤ 0.002 .
The final answer is ∣ x − 3∣ ≤ 0.002 .
Explanation
Understanding the Problem We are given a compound inequality − 0.002 ≤ x − 3 ≤ 0.002 that represents the acceptable lengths of manufactured parts. We need to choose the absolute value inequality that represents the same situation.
Converting to Absolute Value Inequality The compound inequality − 0.002 ≤ x − 3 ≤ 0.002 can be directly translated into an absolute value inequality. The expression x − 3 represents the difference between the actual length x and the desired length 3. The inequality states that this difference must be within 0.002 mm.
Choosing the Correct Option The absolute value inequality that represents this situation is ∣ x − 3∣ ≤ 0.002 . This inequality means that the distance between x and 3 is less than or equal to 0.002.
Final Answer Therefore, the correct absolute value inequality is ∣ x − 3∣ ≤ 0.002 .
Examples
In manufacturing, ensuring parts meet specific size requirements is crucial. The absolute value inequality ∣ x − 3∣ ≤ 0.002 helps quality control engineers define the acceptable range of a part's length around the desired length of 3 mm. For example, if a machine produces parts with lengths ranging from 2.998 mm to 3.002 mm, this inequality confirms that all these parts are within the acceptable tolerance. This concept is also applicable in other fields, such as electronics, where components must have precise electrical characteristics, or in construction, where building materials must meet specific dimensional standards.
