Solve inequality write solution set interval notation

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Solve inequality write solution set interval notation

Solution to linear inequality: introduction and formatting (p. 1 of 3) Chapters: Introduction and Formatting, Elementary Examples, Advanced Examples The solution to linear inequality is almost exactly the same as solving linear equations. If they gave me x + 3 = 0, I would have known how to solve: I would have taken 3 on both sides. I can do the same here: Then the solution is: The formatting of the above answer is called inequality notation, because the solution is written as inequality. This is probably the easiest of the decision notations, but there are three others with which you may need to know. Copyright ? Elizabeth Stapel 2002-2011 All Rights Reserved Set Notation writes the decision as a set of points. The above solution would be written as {x | xis real number x < -3}, which is pronounced as a set of all x values, such that x is a real number and x is less than minus three. A simpler form of this notation would be something like {x | x < -3}, which is pronounced as all x such that x is less than minus three. Interval notation writes the solution as an interval (that is, as a section or length along a string of numbers). The judgment in x < ?3 would be written as a range from negative infinity to minus three, or simply minus infinity to minus three. Interval notation is easier to write than to pronounce ambiguity as to whether endpoints are included in the interval or not. (For example, marking x < ?3 would be a range that would be pronounced as minus infinity over (not just up to) minus three or minus infinity up to minus three, inclusive, which means that ?3 will be added. Right brackets in x < -3 indicate that ?3 was not included; right bracket x < ?In case 3, this indicates that this is the case.) The last notation is more illustration. You can be directed to a graph solution. This means that you draw a line of numbers and select the part included in the solution. First, mark the edge of the solution range, in this case -3. Since ?3 is not included in the solution (that is, less than, remember that it is not less than or equal to), mark this point with an open point or an open bracket pointing in the direction of the remaining decision interval: ... or: Then you could color the corresponding side: ... or: Why the shade to the left? Because they want all values that are less than-3, and those values are to the left of the boundary point. If they wanted higher than points, you would be shaded to the right. In total, we've seen four ways, with a couple of options, to mark the solution to the above inequality: notation format pronunciation inequality x < ?3 x is less than minus three set i) {x | x is a real number, x < ?3} ... or: (ii) {x | x < ?3} (i) a set of all xs, such as x is a real number xis less than minus three ii) all x such that x is less than minus three interval interval from minus infinity to minus three graph in one of the following graphs: Here is another example, along with different answer formats: If they were me x - 4 = 0, then I would have solved adding four to each side. I can do the same here: Then the solution is: x > 4 As before, this solution can be given in any of four ways: notation format pronunciation inequality x > 4 x is greater than or equal to four set i) {x | x is a real number, x > 4} ... or: (ii) {x | x> 4} (i) a set of all x, such that x is a real number, and x is greater than or equal to four (ii) all x such that x is greater than or equal to four ranges in the range of four to infinity, including four graphs, in any of the following graphs: In terms of solution graphs, square brackets contain parentheses notation, and closed (filled) point notation passes with an open point. While your current textbook may require that you only know one or two of the above formats for your answers, this topic of inequality usually arises in other contexts of other books in other courses. Since you may need later to be able to understand other formats, make sure now that you know them all. However, I will only use the rest of this lesson to noting inequality; I like it the most. Top | 1 | 2 | 3 | Go to Index Next >> Cite for this article as: Stapel, Elizabeth. Solving linear inequality: introduction and formatting. Purplemath. Available from . Open Algebraic InequalityPresses associated with the characters , <, , and >., such as x2, are written x is greater than or equal to 2. This inequality has a lot of solutions x. Some decisions are 2, 3, 3.5, 5, 20 and 20,001. Since it is not possible to list all solutions, a system is required that allows you to communicate clearly with this endless set. The two common ways to express solutions to inequality are their graph of numbers in the line of numbersSolutions to algebraic inequality, expressed by shaved decision in a string of numbers. and the use of interval notation Textual system for expressing solutions to algebraic inequality.. To graphically express a solution, draw a line and shade of numbers across all the values that are solutions to inequality. Interval notation is textual and uses a specific notation as follows: Set up interval notation by setting the resolution set on the number line. Interval notation numbers must be written in the same order as the number string, and smaller numbers in the collection appear first. This example contains inclusive inequality Inequality, which includes a threshold point that is referenced by a portion of the characters and or equal to, and a closed point in a number string., which means that the decision includes a lower limit of 2. Mark this with a dot on a string of numbers and a square bracket in an interval notation. The symbol () reads as infinity The symbol () indicates that the interval is limitless on the right. and indicates that the set is linked to the right side of the number row. Interval notation requires a parenthese to add infinity. A square bracket indicates that the limit is included in the solution. Parentheses indicate that the limit is not included. Infinity is the upper limit to real numbers, but in itself is not a real number: it cannot be included in the set of solutions. Now compare the range notes in the previous example with the harsh or unconvincing inequality: The strict marking of inequality in building relationships using the symbol < less than and > greater than that means that decisions can be very close to the threshold point, in this case 2, but do not actually include it. Select this idea with an open dot in a number row and a circular parenthese displayed in an interval notation. Example 1: Graph and specify the notation equivalent of the interval: x<3. Solution: Use open point 3 and color all real numbers strictly less than 3. Use negative infinityThat (-) indicates that the interval is limitless to the left. (-) to indicate that the solution set is bound to the left side of the number row. Answer: Interval notation: (-, 3) Example 2: Graph and equivalent of interval notation: x5. Solution: Use a closed point and color all numbers less than 5. Answer: Interval notation: (-, 5] It is important to see that 5x is the same as x5. In both cases, the x-values must be less than or equal to 5. To avoid confusion, it is good practice to rewrite all inequalities with the left-hand variable. Also, use inf as an abbreviated infinity form when using text. For example ( -, 5] can be expressed in textiles as (-inf, 5]. Compound inequality Of action inequality in a single statement, combined by a word and/or word or, in fact, there are two or more inequalities in a single statement, which is combined by a word and/or a word or. Compound inequality with logical or requires that any condition be met. Therefore, the set of solutions to inequality of this type of compound consists of all the elements of each set of solutions to inequality. When we connect to these sets of individual solutions, it is called a union When you combine individual sets of solutions that are referenced by logical use of a word or use and that are labeled with the symbol ., denoted by . For example, solutions to the compound inequality x<3 or x6 can be arranged as follows: Sometimes we are faced with cases of compound inequality, where the individual sets of solutions overlap. In the event that compound inequality is a word or, we combine all the elements of both sets to create a single set that contains all the elements of each. Example 3: Graph and indicate the notation equivalent of the interval: x-1 or x<3. Solution: Combine all the solutions to both inequalities. solutions to inequalities are above the number line as a means of identifying the union that is set out in the number line below. Answer: Interval notation: (-, 3) Any real number less than 3 in the shaded region in the row of numbers will satisfy at least one of the two missing differences. Example 4: Graph and indicate the notation equivalent of the interval: x<3 or x-1. Solution: Both sets of decisions are arranged above the union, which is set out below. Answer: Interval notation: R = (-, ) When you combine both sets of solutions and form a union, you can see that all the real numbers correspond to the original inequality of the compound. To sum up, inequalities such as -1 are less than or equal to x, and x is less than three. This is compound inequality because it can be broken down as follows: Logical and requires that both conditions be fair. Both inequalities are satisfied with all the elements of the intersectionState consisting of the total values of the sets of individual decisions, indicated by the logical use of the word and the use indicated by the symbol of each set of decisions ., denoted by example . 5: Graph and indicate the equivalent of the interval notation: x<3 and x-1. Solution: Set the intersection or overlap between two sets of solutions. Solutions to each inequality are sketched above the string of numbers as a means of determining the intersection, which is arranged in the number row below. Here x = 3 is not the solution, because it solves only one of the inequalities. Answer: Interval notation: [-1, 3) Or we can interpret -1x<3 as all possible x-values between or bounded by -1 and 3 numbers in a string. For example, one such solution is x=1. Note that 1 contains -1 to 3 numbers per line or -1 < 1 < 3. We also see that other possible solutions are -1, -0.99, 0, 0.0056, 1.8 and 2.99. Since there are an infinite number of real numbers from -1 to 3, we must express the decision graphically and/or with a notation of the interval, in this case [-1, 3]. Example 6: Graph and indicate the equivalent of an interval notation: -32<x<2. Solution: Color any real numbers that are limited or severely limited by -32=-112 and 2. Answer: Interval notation: (-32, 2) Example 7: Graph and interval notation equivalent: -5<x15. Solution: Colour all actual numbers from -5 to 15 and indicate that upper limit 15 is included in the solution set using a closed point. Answer: Interval notation: (-5, 15] In the previous two examples we did not disintegrate inequality; instead we decided to think about all the real numbers between the two certain boundaries. In conclusion, we use interval notation in this text. However, other resources that you may encounter use an alternative method to describe collections using a familiar mathematical notation for collections called set-builder notationA system.. We used set notation to list items such as integers in parentheses set items and colon tags indicate that integers last forever. In this case, the norime apibdinti reali skaici intervalus, pvz., tikrieji skaiciai yra didesni arba lygs 2. Kadangi rinkinys yra per didelis, kad bt galima isvardyti, set-builder notacija leidzia mums j apibdinti naudojant pazstam matematin notacij. Toliau pateikiamas set-builder notation pavyzdys: cia xR aprasomas numerio tipas, kai simbolis () skaitomas kaip elementas. Tai reiskia, kad kintamasis x reiskia real skaici. Vertikali juosta (|) rasoma tokia. Galiausiai, pareiskimas x2 yra slyga, kuri apibdina rinkin naudojant matematin notacij. Siuo metu ms algebros tyrime daroma prielaida, kad visi kintamieji yra reals skaiciai. Dl sios priezasties galite praleisti R ir rasyti {x|x2}, kuris skaitomas vis reali skaici x rinkinys, toks, kad x yra didesnis arba lygus 2. Aprasyti jungini nelygyb, pvz., x <3 or= x6,= write=></3> <3 or= x6},= which= is= read= "the= set= of= all= real= numbers= x= such= that= x= is= less= than= 3= or= x= is= greater= than= or= equal= to= 6." = write= bounded= intervals,= such= as=></3> <3, as=></3,> <3}, which= is= read= "the= set= of= all= real= numbers= x= such= that= x= is= greater= than= or= equal= to= -1= and= less= than= 3." = key= takeaways= inequalities= usually= have= infinitely= many= solutions,= so= rather= than= presenting= an= impossibly= large= list,= we= present= such= solutions= sets= either= graphically= on= a= number= line= or= textually= using= interval= notation.= inclusive= inequalities= with= the= "or= equal= to"= component= are= indicated= with= a= closed= dot= on= the= number= line= and= with= a= square= bracket= using= interval= notation.= strict= inequalities= without= the= "or= equal= to"= component= are= indicated= with= an= open= dot= on= the= number= line= and= a= parenthesis= using= interval= notation.= compound= inequalities= that= make= use= of= the= logical= "or"= are= solved= by= solutions= of= either= inequality.= the= solution= set= is= the= union= of= each= individual= solution= set.= compound= inequalities= that= make= use= of= the= logical= "and"= require= that= all= inequalities= are= solved= by= a= single= solution.= the= solution= set= is= the= intersection= of= each= individual= solution= set.= compound= inequalities= of= the= form=></3} ,> <> <m can= be= decomposed= into= two= inequalities= using= the= logical= "and." = however,= it= is= just= as= valid= to= consider= the= argument= a= to= be= bounded= between= the= values= n= and= m.= part= a:= simple= inequalities= graph= all= solutions= on= a= number= line= and= provide= the= corresponding= interval= notation.= 1.= x10= 2.= x=>-5 3. x>0 4. x0 5. x-3 6. x-1 7. -4 <x 8.= 1x= 9.=></x> <-12 10.= x-32= 11.= x-134= 12.=></-12> <34 part= b:= compound= inequalities= graph= all= solutions= on= a= number= line= and= give= the= corresponding= interval= notation.= 13.=></34> <> <5 14.= -5x-1= <x20 16.=></x20> <15 17.=></15> <x40 18.=></x40> <-10 19.=></-10> <x 50= 20.=></x> <> <0 21.=></0> <> <18 22.= -34x12= 23.=></18> <112 24.=></112> <> <-12 25.=></-12> <-3 or= x=>3 26. x<-2 or= x4= 27.= x0= or= x=>10 28. x-20 or x-10 29. x<-23 or= x=>13 30. x-43 or x>-13 31. x>-5 or x <5 32.=></5> <12 or= x=>-6 33. x<3 or= x3= 34.= x0= or= x=>0 35. x <-7 or=></-7> <2 36.= x-3= or= x=>0 37. x5 or x>0 38. x<15 or= x10= 39.= x=> <3 40.= x0= and=></3> <5 41.= x-5= and= x-1= 42.=></5> <-4 and and=></-4 and > -2 and x</15> </2> </3> </12> </-23> </-2> </-3> </m> </m> 43. x3 ir x>3 44. x5 ir x5 45. x0 ir x0 46. x<2 and= x-1= 47.= x=>0 ir x-1 48. x <5 and=></5> <2 part= c:= interval= notation= determine= the= inequality= given= the= answers= expressed= in= interval= notation.= 49.= (-,= 7]= 50.= (-4,= )= 51.= [-12,= )= 52.= (-,= -3)= 53.= (-8,= 10]= 54.= (-20,= 0]= 55.= (-14,= -2)= 56.= [23,= 43]= 57.= (-34,= 12)= 58.= (-,= -8)= 59.= (8,= )= 60.= (-,= 4)[8,= )= 61.= (-,= -2][0,= )= 62.= (-,= -5](5,= )= 63.= (-,= 0)(2,= )= 64.= (-,= -15)(-5,= )= write= an= equivalent= inequality.= 65.= all= real= numbers= less= than= 27.= 66.= all= real= numbers= less= than= or= equal= to= zero.= 67.= all= real= numbers= greater= than= 5.= 68.= all= real= numbers= greater= than= or= equal= to= -8.= 69.= all= real= numbers= strictly= between= -6= and= 6.= 70.= all= real= numbers= strictly= between= -80= and= 0.= part= d := discussion= board= topics= 71.= compare= interval= notation= with= set-builder= notation.= share= an= example= of= a= set= described= using= both= systems.= 72.= explain= why= we= do= not= use= a= bracket= in= interval= notation= when= infinity= is= an= endpoint.= 73.= research= and= discuss= the= different= compound= inequalities,= particularly= unions= and= intersections.= 74.= research= and= discuss= the= history= of= infinity.= 75.= research= and= discuss= the= contributions= of= georg= cantor.= 76.= what= is= a= venn= diagram?= explain= and= post= an= example.= 1:= (-,= 10]= 3:= (0,= )= 5:= (-,= -3]= 7:= (-4,= )= 9:= (-,= -12)= 11:= [-134,= )= 13:= (-2,= 5)= 15:= (-5,= 20]= 17:= (10,= 40]= 19:= (0,= 50]= 21:= (-58,= 18)= 23:= [-1,= 112)= 25:= (-,= -3)(3,= )= 27:= (- ,= 0](10,= )= 29:= (-,= -23)(13,= )= 31:= r= 33:= r= 35:= (-,= 2)= 37:= (0,= )= 39:= (-2,= 3)= 41:= [-5,= -1]= 43:= = 45:= {0}= 47:= (0,= )= 49:= x7= 51:= x-12= 53:=></2> <x10 55:=></x10> <> <-2 57:=></-2> <> <12 59:= x=>8 61: x-2 arba x0 63: x<0 or= x=>2 65: x<27 67:= x=>5 69: -6 <> <6></6> <> <6 ></6 > </27> </0> </12> </2>

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