Interval Notation

Interval NotationOpen Intervals: Less than or greater than but not equal to ()If there are two real numbers a and b, then a < x < b means that the numbers represented by x arebetween a and b, where a is smaller than b, but the values of x cannot equal a or b.On the number line, there are open circles at a and b and the line is shaded between them.Interval notation uses parentheses ( ) to show the open circles; the numbers in the parentheses representthe ends of the shaded area.a < x < b ⇒ Interval notation: (a, b) O Oabx < a means that the numbers represented by x are less than but not equal to a. This interval includesnumbers from negative infinity ( − ∞)up to but not including a.On the number line, there is an open circle at a and the number line is shaded to the left.Interval notation uses parentheses before − ∞ and after the number a to show the open circle.x < a ⇒ Interval notation: ( − ∞,a)Oax > b means that the numbers represented by x are greater than but not equal to b. This intervalincludes numbers from b (but not including b) through positive infinity ( ∞ ).On the number line, there is an open circle at b and the number line is shaded to the right.Interval notation uses parentheses before the number b to show the open circle and to the right of ∞ .x > b ⇒ Interval notation: (b, ∞) ObEx:x < 5 ⇒ Interval notation: ( − ∞,5 )O5x > –3 ⇒ Interval notation: (–3, ∞) O–3–7 < x < 1 ⇒ Interval notation: (–7, 1) O O–7 1

Closed Intervals: Less than or equal to, or greater than or equal to (≤, ≥)If there are two real numbers a and b, then a ≤ x ≤ b means that the numbers represented by x arebetween a and b, where a is smaller than b, and the values of x can equal a or b.On the number line, there are closed circles at a and b and the line is shaded between them.Interval notation uses brackets [ ] to show the closed circles; the numbers in the parentheses representthe ends of the shaded area.a ≤ x ≤ b ⇒ Interval notation: [a, b] abx ≤ a means that the numbers represented by x are less than or equal to a. This interval includesnumbers from negative infinity ( − ∞)up to and including a.On the number line, there is a closed circle at a and the number line is shaded to the left.Interval notation uses a parenthesis before − ∞ and a bracket after the number a to show the closedcircle.x ≤ a ⇒ Interval notation: ( − ∞,a]ax ≥ b means that the numbers represented by x are greater than or equal to b. This interval includesnumbers from b (and including b) through positive infinity ( ∞ ).On the number line, there is a closed circle at b and the number line is shaded to the right.Interval notation uses a bracket before the number b to show the closed circle and a parenthesis to theright of ∞ .x ≥ b ⇒ Interval notation: [b, ∞) bEx:x ≥ 10 ⇒ Interval notation: [ 10 ,∞)10–5 ≤ x ≤ –3 ⇒ Interval notation: [–5, –3] –5 –3x ≤ 4 ⇒ Interval notation: ( − ∞,4]4

Half-open, half-closed intervals: a combination of parentheses and bracketsa < x ≤ b ⇒ (a, b] a ≤ x < b ⇒ [a, b)◦ ◦a b a bEx:2 ≤ x < 7 ⇒ Interval notation: [2, 7) O2 72 < x ≤ 7 ⇒ Interval notation: (2, 7] O 2 7

Summary of Interval NotationAn interval on the real number line is a set of points on the line less than or less than or equal to somevalue, greater than or greater than or equal to some value, or between two numbers, either including ornot including the endpoints. We can indicate intervals by writing an inequality, graphing them on thereal number line, or by using interval notation.Values of x Graph of Interval Interval Notationall x on the line( −∞,∞)Openx < 3less than 3, notincluding 3 3x < 3less than 3 orequal to 3 3x > 3greater than 3,not including 3x > 3greater than 3 orequal to 33 < x < 5between 3 and 5,not including 3and 5 3 53< x < 5between 3 and 5,including 3 butnot including 5 3 53 < x < 5between 3 and 5including 5 butnot including 3 3 53 < x < 5between 3 and 5including both 3and 5 3 533(- ∞ , 3)Open(- ∞ , 3]Half Open(3, ∞ )Open[3, ∞ )Half Open(3,5)Open[3,5)Half Open(3,5]Half Open[3,5]Closed