Write the following as an inequality.
2 is greater than x, and -6 is less than x
Use x only once in your inequality.
Answer (2)
Convert "2 is greater than x" to x < 2 .
Convert "-6 is less than x" to − 6 < x .
Combine the inequalities to form a compound inequality.
The compound inequality is − 6 < x < 2 .
Explanation
Understanding the Problem We need to translate the sentences "2 is greater than x" and "-6 is less than x" into a single compound inequality using x only once.
Converting the First Sentence The sentence "2 is greater than x" can be written as the inequality x"> 2 > x , which is equivalent to x < 2 .
Converting the Second Sentence The sentence "-6 is less than x" can be written as the inequality − 6 < x , which is equivalent to -6"> x > − 6 .
Combining the Inequalities Now, we combine the two inequalities x < 2 and -6"> x > − 6 into a single compound inequality. This means that x must be greater than -6 AND less than 2. We can write this as − 6 < x < 2 .
Final Answer Therefore, the compound inequality is − 6 < x < 2 .
Examples
Compound inequalities are useful in many real-world situations. For example, suppose a company wants to hire workers who are at least 18 years old but not older than 30 years old. If we let x represent the age of the workers, then we can write this condition as a compound inequality: 18 ≤ x ≤ 30 . This means that the age of the worker must be greater than or equal to 18 AND less than or equal to 30. Another example is when defining acceptable ranges for measurements in manufacturing or engineering. For instance, a bolt's diameter must be within a certain tolerance, expressed as a < x < b , where x is the diameter and a and b are the lower and upper bounds, respectively.
The compound inequality that represents the statements "2 is greater than x" and "-6 is less than x" is − 6 < x < 2 . This indicates that x is greater than -6 but less than 2. Thus, x can take any value within this range.
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