Write the inequality $3
Answer (2)
The inequality 3 < x ≤ 7 is expressed in interval notation as ( 3 , 7 ] . This notation indicates that x is greater than 3 and less than or equal to 7. The final answer is ( 3 , 7 ] .
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Convert the inequality 3 < x to interval notation: ( 3 , ∞ ) .
Convert the inequality x ≤ 7 to interval notation: ( − ∞ , 7 ] .
Combine the two inequalities to get the interval notation for 3 < x ≤ 7 : ( 3 , 7 ] .
The final answer 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 subsets of the real number line. 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 Interval Notations Combining the two inequalities, we have 3 < x ≤ 7 . This means that x is greater than 3 and less than or equal to 7. In interval notation, this is written as ( 3 , 7 ] . The parenthesis indicates that 3 is not included, and the bracket indicates that 7 is included.
Sketching the Inequality on a Number Line To sketch the inequality on a number line, we draw a number line and mark the points 3 and 7. At 3, we draw an open circle (or a parenthesis) to indicate that 3 is not included in the solution. At 7, we draw a closed circle (or a bracket) to indicate that 7 is included in the solution. We then shade the region between 3 and 7 to represent all values of x that satisfy the inequality.
Final Answer Therefore, the inequality 3 < x ≤ 7 in interval notation is ( 3 , 7 ] .
Examples
Interval notation and number line representation are used in various fields, such as physics and engineering, to define the range of possible values for a variable. For example, when measuring the temperature of a chemical reaction, we might find that the temperature must be greater than 20 degrees Celsius but cannot exceed 50 degrees Celsius. This can be represented in interval notation as (20, 50], indicating that the temperature can be any value between 20 (exclusive) and 50 (inclusive) degrees Celsius. This helps in setting safety limits and understanding the operational range of the reaction.
