Graph the set \{x \mid x \geq 1\} on the number line. Then, write the set using interval notation.
Answer (2)
Graph the set on the number line by marking x = 1 with a closed circle and shading the region to the right.
The set includes all numbers from 1 (inclusive) to infinity.
Express the set using interval notation.
The interval notation is [ 1 , ∞ ) .
Explanation
Understanding the Problem We are asked to graph the set of all numbers x such that x is greater than or equal to 1 on the number line and then to express this set using interval notation.
Graphing the Set To graph the set on the number line, we draw a number line and mark the point x = 1 . Since the set includes x = 1 , we use a closed circle (or a bracket) at x = 1 . Then, we shade the region to the right of x = 1 to represent all x greater than 1.
Writing in Interval Notation To write the set in interval notation, we note that the set includes all numbers from 1 (inclusive) to infinity. Therefore, the interval notation is [ 1 , ∞ ) .
Examples
Understanding sets and intervals is crucial in many real-world applications. For example, if a store offers a discount only to customers who spend at least 50 , t h ese t o f e l i g ib l es p e n d in g am o u n t sc anb ere p rese n t e d a s {x | x \geq 50} . I nin t er v a l n o t a t i o n , t hi s w o u l d b e [50, \infty) . S imi l a r l y , i f a t e m p er a t u rer an g e f or a c h e mi c a l re a c t i o nm u s t b e main t ain e d b e tw ee n 20 an d 30 d e g rees C e l s i u s ( in c l u s i v e ) , t h ese t o f a cce pt ab l e t e m p er a t u res i s {x | 20 \leq x \leq 30} , w hi c hinin t er v a l n o t a t i o ni s [20, 30]$. These notations help to clearly define conditions and ranges in various practical scenarios.
To graph the set {x \mid x \geq 1} on the number line, draw a closed circle at x = 1 and shade to the right. In interval notation, this set is expressed as [ 1 , ∞ ) . This indicates all values of x that are greater than or equal to 1, extending to infinity.
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