Graph the set {x |-1 < x <= 5}.
Answer (2)
Graph the set on a number line with open circles at -1 and 5, shading the region between them.
Use parentheses in interval notation because the endpoints are not included.
Write the set in interval notation as ( − 1 , 5 ) .
The set of all x such that − 1 < x < 5 in interval notation is ( − 1 , 5 ) .
Explanation
Understanding the Problem The problem asks us to graph the set of all numbers x such that − 1 < x < 5 on a number line and then express this set using interval notation. This means we need to visualize all numbers strictly between -1 and 5 and then write this set in a standard notation.
Graphing the Set To graph the set on the number line, we draw a line and mark the points -1 and 5. Since the inequality is strict (i.e., x is strictly greater than -1 and strictly less than 5), we use open circles at -1 and 5 to indicate that these points are not included in the set. Then, we shade the region between -1 and 5 to represent all the numbers between them.
Writing in Interval Notation To write the set in interval notation, we use parentheses because -1 and 5 are not included in the set. The interval notation is ( − 1 , 5 ) . This notation represents all real numbers between -1 and 5, not including -1 and 5 themselves.
Examples
Imagine you're measuring the temperature of a room, and you find that the temperature, x , is always between 20 and 25 degrees Celsius, but never exactly 20 or 25. You could represent this range of temperatures on a number line, shading the area between 20 and 25, with open circles at each endpoint. In interval notation, you'd write this as ( 20 , 25 ) . This notation is a concise way to describe a range of values, which is useful in many real-world situations, such as specifying acceptable ranges for measurements, ages, or quantities.
To graph the set { x ∣ − 1 < x ≤ 5 } , mark an open circle at -1 and a closed circle at 5 on a number line, shading the region between. This set in interval notation is expressed as ( − 1 , 5 ] .
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