Write the inequality $3
Answer (2)
Convert the inequality 3 < x to interval notation: ( 3 , ∞ ) .
Convert the inequality x ≤ 7 to interval notation: ( − ∞ , 7 ] .
Combine the intervals to represent 3 < x ≤ 7 : ( 3 , 7 ] .
The final answer in interval notation is ( 3 , 7 ] .
Explanation
Understanding the Problem We are given the inequality 3 < x ≤ 7 and asked to express it in interval notation and sketch it on a number line.
Understanding Interval Notation Interval notation is a way to write sets of real numbers. Parentheses indicate that the endpoint is not included, while brackets indicate that the endpoint is included.
Expressing the Left Side in Interval Notation The inequality 3 < x means that x is greater than 3, but not equal to 3. In interval notation, this is written as ( 3 , ∞ ) .
Expressing the Right Side in Interval Notation The inequality x ≤ 7 means that x is less than or equal to 7. In interval notation, this is written as ( − ∞ , 7 ] .
Combining the Intervals Combining the two inequalities, 3 < x ≤ 7 , we have that x is greater than 3 and less than or equal to 7. Therefore, the interval notation is ( 3 , 7 ] . This means that x can take any value between 3 and 7, including 7 but not including 3.
Sketching on a Number Line To sketch this on a number line, we draw a number line and mark the points 3 and 7. At 3, we draw an open circle to indicate that 3 is not included in the interval. At 7, we draw a closed circle (or a filled-in circle) to indicate that 7 is included in the interval. Then, we shade the region between 3 and 7 to represent all the values of x that satisfy the inequality.
Final Answer Therefore, the inequality 3 < x ≤ 7 in interval notation is ( 3 , 7 ] .
Examples
Interval notation is useful in many areas of mathematics, including calculus and analysis. For example, when discussing the domain or range of a function, interval notation provides a concise way to specify the set of possible input or output values. In real life, interval notation can be used to describe a range of acceptable values for a measurement, such as temperature or height. For instance, if a plant thrives in temperatures between 60 and 80 degrees Fahrenheit, this could be expressed as the interval ( 60 , 80 ) .
The inequality 3 < x ≤ 7 translates to the interval notation ( 3 , 7 ] . This indicates that x can take any value between 3 (not included) and 7 (included). Thus, the final answer is ( 3 , 7 ] .
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