1.1 Interval Notation And Set Notation - Big Ideas Learning

2y ago
388 Views
21 Downloads
613.24 KB
6 Pages
Last View : 16d ago
Last Download : 2m ago
Upload by : Luis Waller
Transcription

1.1TEXAS ESSENTIALKNOWLEDGE AND SKILLSPreparing for 2A.6.K, 2A.7.IInterval Notation and Set NotationEssential QuestionWhen is it convenient to use set-buildernotation to represent a set of numbers?A collection of objects is called a set. You can use braces { } to represent a set bylisting its members or by using set-builder notation to define the set in terms of theproperties of its members. For instance, the set of the numbers 1, 2, and 3 can bedenoted asList the members of the set in braces.{1, 2, 3}and the set of all odd whole numbers can be denoted as{x x is a whole number and x is odd}Set-builder notationwhich is read “The set of all real numbers x such that x is a whole number andx is odd.”If all of the members of a set A are also members of a set B, then set A is a subset ofset B.For instance, if set A {a, b} and set B {a, b, c, d}, then set A is a subset of set B.Writing Subsets in Set NotationWork with a partner. Write all the nonempty subsets of each set.ANALYZINGMATHEMATICALRELATIONSHIPSTo be proficient in math,you need to connectand communicatemathematical ideas.a. {4, 5}b. {c, d}c. {2, 4, 6}d. {e, f, g, h}Writing Subsets in Set NotationWork with a partner. Write each given subset of the real numbers in set-buildernotation. Describe each set-subset relationship among these sets.a. the integersb. the whole numbersc. the natural numbersd. the rational numberse. the irrational numbersf. the positive integersWriting Subsets in Set NotationWork with a partner. Write each indicated set of numbers using either braces to listits members or set-builder notation. Explain your choice of notation.a. the whole numbers 50 through 54b. the real numbers 0 through 4c. the prime whole numbersd. the integers 100 through 100Communicate Your Answer4. When is it convenient to use set-builder notation to represent a set of numbers?5. What are some relationships between subsets of the real numbers?Section 1.1Interval Notation and Set Notations3

1.1LessonWhat You Will LearnRepresent intervals using interval notation.Represent intervals using set-builder notation.Core VocabulVocabularylarryset, p. 4subset, p. 4endpoints, p. 4bounded interval, p. 4unbounded interval, p. 5set-builder notation, p. 6Using Interval NotationIn mathematics, a collection of objects is called a set. You can use braces { } torepresent a set by listing its members or elements. For instance, the set{1, 2, 3}A set with three memberscontains the three numbers 1, 2, and 3. Many sets are also described in words, such asthe set of real numbers.If all the members of a set A are also members of a set B, then set A is a subset of setB. The set of natural numbers {1, 2, 3, 4, . . .} is a subset of the set of real numbers.The diagram shows several important subsets of the real numbers.Real Numbers (ℝ)UNDERSTANDINGMATHEMATICALTERMSRational Numbers (ℚ)IrrationalNumbersIntegers (ℤ)Whole Numbers (𝕎)The symbols representsubsets of the realnumbers.Natural Numbers (ℕ)ℝ: Real numbersℚ: Rational numbersℤ: Integers𝕎: Whole numbersℕ: Natural numbersMany subsets of the real numbers can be represented as intervals on the realnumber line.Core ConceptBounded Intervals on the Real Number LineLet a and b be two real numbers such that a b. Then a and b are the endpointsof four different bounded intervals on the real number line, as shown below.A bracket or closed circle indicates that the endpoint is included in the intervaland a parenthesis or open circle indicates that the endpoint is not included inthe interval.InequalityInterval Notationa x b[a, b]a x ba x ba x bGraphxabababab(a, b)x[a, b)x(a, b]xThe length of any bounded interval, [a, b], (a, b), [a, b), or (a, b], is the distancebetween its endpoints: b a. Any bounded interval has a finite length. An interval thatdoes not have a finite length is called unbounded or infinite.4Chapter 1Linear Functions

Core ConceptUnbounded Intervals on the Real Number LineLet a and b be real numbers. Each interval on the real number line shown belowis called an unbounded interval.InequalityInterval Notationx a[a, )x a(a, )x bGraphxaxa( , b]xbx b( , b)xb( , )xThe symbols (infinity) and (negative infinity) are used to represent theunboundedness of intervals such as [7, ) and ( , 7]. Because these symbolsdo not represent real numbers, they are always enclosed by a parenthesis.Writing Interval NotationWrite each interval in interval notation.a. 2 x 3b. x 1c.x 5 4 3 2 1012345 5 4 3 2 1012345d.xSOLUTIONa. The graph of 2 x 3 is the bounded interval [ 2, 3].b. The graph of x 1 is the unbounded interval ( 1, ).c. The graph represents all the real numbers between 3 and 4, including theendpoint 3. This is the bounded interval [ 3, 4).d. The graph represents all the real numbers less than or equal to 3. This is theunbounded interval ( , 3].Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comWrite the interval in interval notation.1. 7 x 42. x 53.x 5 4 3 2 10Section 1.112345Interval Notation and Set Notation5

Using Set-Builder NotationAnother way to represent intervals is to write them in set-builder notation.Core ConceptSet-Builder NotationSet-builder notation uses symbols to define a set in terms of the properties ofthe members of the set.Set-builder notationWords{x x b}the set of all real numbers x such that x is less than bSet-builder NotationGraph{x x a or x b}xab{x x a}xaUsing Set-Builder NotationSketch the graph of each set of numbers.a. {x 2 x 5}b. {x x 0 or x 4}SOLUTIONa. The real numbers in the setsatisfy both x 2 and x 5.b. The real numbers in the setsatisfy either x 0 or x 4.x 1UNDERSTANDINGMATHEMATICALTERMSThe symbol denotesmembership in a set.The expression x ℤmeans that x is a member(or element) of the setof integers.012345x 2 16012345Writing Set-Builder NotationWrite the set of numbers in set-builder notation.a. the set of all integers greater than 5b. ( , 1) or ( 1, )SOLUTIONa. x is greater than 5 and x isan integer.b. x can be any real numberexcept 1.{x x 5 and x ℤ}Monitoring Progress{x x 1}Help in English and Spanish at BigIdeasMath.comSketch the graph of the set of numbers.4. {x 6 x 2}5. {x x 0 or x 10}Write the set of numbers in set-builder notation.6. ( , 1] or (1, )6Chapter 1Linear Functions7. the set of all integers except 4

1.1ExercisesDynamicDynamic SolutionsSolutions availableavailable atat BigIdeasMath.comBigIdeasMath.comVocabulary and Core Concept Check1. COMPLETE THE SENTENCE Two real numbers a and b are the of four differentintervals on the real number line.2. WHICH ONE DOESN’T BELONG? The graph of which set of numbers does not belong with theother three? Explain.x 3 and x 5( 3, 5]{x 3 x 5}the set of all integers greater than 3 and less than or equal to 5Monitoring Progress and Modeling with MathematicsIn Exercises 3 6, use braces to list the elements ofthe set.3. the set of whole numbers less than 1018. {x 10 x 15}19. {x x 5 or x 10}20. {x x 4}4. the set of odd whole numbers less than 24In Exercises 21 28, write the set of numbers inset-builder notation. (See Example 3.)5. the set of integers greater than 506. the set of integers less than 8In Exercises 7 16, write the interval in intervalnotation. (See Example 1.)8. 5 x 207. 3 x 99. x 1321. [ 5, 16)22. (22, 98]23. ( , 4] or [4, )24. ( , 5] or [14, )25. the set of all integers less than 2026. the set of all real numbers greater than 19 and10. x 5811.x 6 4 20246x010203027. the set of all real numbers except 100812. 20 10less than 324013.28. the set of all whole numbers except 5029. ERROR ANALYSIS Describe and correct the errorin rewriting the interval ( , 8] in set-buildernotation.x 2 101 10 505234514. x10 15 20 2515. the real numbers from 10 through 1016. the real numbers between 110 and 220{x x 8}30. ERROR ANALYSIS Describe and correct the error inrewriting the interval [ 7, 24) in set-builder notation.In Exercises 17 20, sketch the graph of the set ofnumbers. (See Example 2.) 17. {x 3 x 12}Section 1.1{x x 7 or x 24}Interval Notation and Set Notation7

31. MODELING WITH MATHEMATICS The elevation35. NUMBER SENSE Write each set using braces to listrelative to sea level in the United States rangesfrom 282 feet in Death Valley, California to20,320 feet on Mount McKinley, Alaska. Writethe range of elevations in interval notation and inset-builder notation.the elements, in interval notation, and in set-buildernotation. If not possible, explain why.a. the set of even whole numbersb. the set of real numbers less than 4c. the set of real numbers 10 or more units from 5032. MODELING WITH MATHEMATICS The the main floorof an auditorium ranges from 6 feet below the stage to8 feet above the stage. The floor of the balcony rangesfrom 26 to 37 feet above the stage. Write the range ofthe floor levels relative to the stage in interval notationand in set-builder notation.33. MAKING AN ARGUMENT Your friend says that it is36. THOUGHT PROVOKING Explain how you can addthe same number to each member of the set of wholenumbers to produce another important subset of thereal numbers.37. MATHEMATICAL CONNECTIONS You are markingimpossible to write the seta rectangular paintball zone that must be 34 meterswide and have a perimeter of at least 140 meters butnot more than 260 meters. Find the interval for thelength x of the rectangular paintball zone.{x x 30 and x 60, and x 𝕎}in interval notation. Is your friend correct? Explain.34. HOW DO YOU SEE IT? The graphs show the legaldriving speeds (in miles per hour) on two al speeds(miles per hour)Legal speeds(miles per hour)38. MATHEMATICAL CONNECTIONS You have 20 gallonsa. Write the legal driving speeds shown in Graph Ain interval notation, in set-builder notation, andin words.b. Write the legal driving speeds shown in Graph Bin interval notation, in set-builder notation, andin words.c. One of the roadways is a state highway andthe other is a residential street. Which graphrepresents each roadway?of roof coating to apply to the roof of a mobile homethat is 16 feet wide. Twenty gallons covers 760 to1000 square feet. Find the interval for the length x thatyou will cover before you need to buy moreroof coating.16 ftxMaintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsComplete the table of values for the function f. Then graph the function. (Skills Review Handbook)39. f(x) 4xx40. f(x) 2x 2 2 101241. f(x) 3x 2 1012 2 101242. f(x) 2x 2 3 101Chapter 12xf (x)f (x)8 2f (x)f (x)xxLinear Functions 2

properties of its members. For instance, the set of the numbers 1, 2, and 3 can be denoted as {1, 2, 3} List the members of the set in braces. and the set of all odd whole numbers can be denoted as {x x is a whole number and x is odd} Set-builder notation which is read “The set of all real numbers x such

Related Documents:

It's Practice with Scientific Notation! Review of Scientific Notation Scientific notation provides a place to hold the zeroes that come after a whole number or before a fraction. The number 100,000,000 for example, takes up a lot of room and takes time to write out, while 10 8 is much more efficient.File Size: 290KBPage Count: 8People also search forscientific notation worksheet answersscientific notation worksheet keyscientific notation worksheet pdf answersscientific notation worksheet with answersscientific notation worksheetscientific notation worksheet with answer key

Inequality Notation: Interval Notation: Remember the connections between Inequality Notation and Interval Notation: Examples: Teacher Slide 8 / 103 3 Which of the following is interval notation for: A [-3,

Recall the definition for scientific notation 1. Change these LARGE scientific notation numbers to standard notation and vice versa. Make up a number for the blank cells. Scientific Notation Standard Notation Scientific Notation Standard Notation a. 6.345 10 e. 5,320 b. 8.04 10 % f. 420,000 c. 4.26 10 & g. 9,040,000,000 d. h. 2. Now try .

Notice that when writing in interval notation, we always write our intervals in increasing order. That is, we always have the smaller numbers on the left. Example 1 Write the following in interval notation a. 3d x 1 b. 0 x 2 c. x! 3 d. xd2 Solution: a. This is a bounded interval. It may prove helpful to File Size: 274KBPage Count: 7

Scientific Notation (SN)- A shorthanded way of writing really large or really small numbers. In SN a number is written as the product of two factors. !Ex: 280,000,000 can be written in scientific notation as 2.8!!!10. First Factor Regular Notation ! Scientific Notation Regular Notation How to Change Scientific Notation 420,000.

Convert interval notation to inequality notation or vice versa. Find the endpoints and state whether the interval is bounded, its type, and graph the interval. (a) (b) 12 (c) SOLUTION (a) The interval corresponds to and is bounded and half-open (see Figure P.4a). The endpoints are and 3. (b

¾ convert solution sets written in interval notation to inequality notation and vice versa ¾ graph solution sets on the real number line given either interval notation or inequality notation ¾ identify absolute

known as the scientific notation. If A is a number between 1 and 10 or 1 and n is an integer, then A x10n is a number written in scientific notation (1 A 10) Examples 1).write 80 000 in scientific notation 80 000 8 x 10 000 8 x 104 2).Write 354 in scientific notation 354 3.54 x 100 3.54 102 3).Write 63.33 in scientific notation 63. .