What would be the interval notation of the following compound inequality: $-4
A. $(-4,5]$
B. $[-4,5]$
C. $(-4,5)$
D. $[-4,5)$
Answer (1)
The compound inequality is − 4 < x ≤ 5 .
Express -4"> x > − 4 as ( − 4 , ∞ ) .
Express x ≤ 5 as ( − ∞ , 5 ] .
Combine the intervals to get ( − 4 , 5 ] .
The interval notation is ( − 4 , 5 ] .
Explanation
Understanding the problem We are given the compound inequality − 4 < x ≤ 5 and asked to express it using interval notation. Interval notation is a way to represent a set of numbers using parentheses and brackets to indicate whether the endpoints are included or excluded.
Expressing the first part of the inequality in interval notation The inequality − 4 < x means that x is greater than − 4 . In interval notation, this is represented as ( − 4 , ∞ ) . The parenthesis indicates that − 4 is not included in the interval.
Expressing the second part of the inequality in interval notation The inequality x ≤ 5 means that x is less than or equal to 5 . In interval notation, this is represented as ( − ∞ , 5 ] . The square bracket indicates that 5 is included in the interval.
Combining the intervals The compound inequality − 4 < x ≤ 5 means that x must satisfy both conditions: x is greater than − 4 AND x is less than or equal to 5 . Combining these, we get the interval that starts just above − 4 (not including − 4 ) and goes up to and includes 5 . This is written as ( − 4 , 5 ] .
Final Answer Therefore, the interval notation for the compound inequality − 4 < x ≤ 5 is ( − 4 , 5 ] .
Examples
Imagine you're describing the acceptable temperature range for a sensitive chemical reaction. The reaction only works if the temperature is strictly above -4 degrees Celsius but can be up to and including 5 degrees Celsius. Expressing this range as (-4, 5] degrees Celsius clearly communicates the operational limits. This notation is also used in various fields like statistics, computer science, and economics to define ranges of values or parameters.
