When the arrows of an inequality go in opposite directions, or the absolute value is greater than, what type of a compound inequality are we working with?
A. Simple
B. Or
C. Complex
D. And
Answer (2)
'And' inequalities require both conditions to be true.
'Or' inequalities require at least one condition to be true.
Opposite direction arrows in inequalities indicate an 'Or' condition.
Absolute value greater than a number indicates an 'Or' condition, so the answer is O r .
Explanation
Understanding the Problem Let's analyze the problem. We need to identify the type of compound inequality described by the given conditions: arrows pointing in opposite directions or absolute value being greater than a certain value.
Definitions of Compound Inequalities Recall the definitions of 'And' and 'Or' compound inequalities.
An 'And' compound inequality consists of two inequalities joined by the word 'and'. The solution set includes only the numbers that satisfy both inequalities. For example, 2"> x > 2 and x < 5 .
An 'Or' compound inequality consists of two inequalities joined by the word 'or'. The solution set includes the numbers that satisfy either inequality. For example, x < − 1 or 3"> x > 3 .
Opposite Directions Imply 'Or' When the arrows of an inequality go in opposite directions, it indicates an 'Or' compound inequality. For example, consider 3"> ∣ x ∣ > 3 . This is equivalent to x < − 3 or 3"> x > 3 , which is an 'Or' compound inequality.
Absolute Value Greater Than Imply 'Or' When the absolute value is greater than a number, it also indicates an 'Or' compound inequality. For example, a"> ∣ x ∣ > a is equivalent to x < − a or a"> x > a , which is an 'Or' compound inequality.
Conclusion Therefore, the type of compound inequality we are working with is 'Or'.
Examples
Understanding compound inequalities is crucial in various fields. For instance, in engineering, when designing a bridge, engineers must ensure that the materials can withstand stress within a certain range. This can be expressed as a compound inequality, where the stress must be greater than a minimum threshold (to ensure structural integrity) and less than a maximum threshold (to prevent material failure). Similarly, in economics, analyzing market prices often involves compound inequalities to define acceptable price ranges for consumers and producers. These ranges ensure that prices are neither too high (unaffordable for consumers) nor too low (unprofitable for producers).
When the arrows of an inequality go in opposite directions or when dealing with absolute values greater than a number, we work with an 'Or' compound inequality. This means the solution includes values that satisfy either condition. Thus, the answer is B. Or .
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