“绝对值型”和“分式型”不等式的解法课件
分式不等式及绝对值不等式的解法
复习绝对值的意义：
提问：正数的绝对 值是什么?零的绝对 值是什么?负数的绝 对值呢？ X>0 x |x|= 0 X=0 X<0 -x
一个数的绝对值表示： 一个数的绝对 值在数轴上表 与这个数对应的点到 示什么意义？ 原点的距离 ,|x|≥0
x2 x1
A B
O
X
|
解分式不等式的步骤: 2x 2 0 1:化为标准式; x 3 2:最高次x项前系数 化为正号 . x3 例2：解不等式 0 3:化为整式不等式解. 3 2x 4:有等号时加上分母 不等于零 . x2
例3：解不等式 4x 3 1
解分式不等式的 练习：解不等式： 步骤: 2x 3 1:化为标准式; 1 ： 0 x2 2:最高次x项前系 x 3 数化为正号. 2: 0 2 x 3:化为整式不等 4x 2 式解. 3: 1 3x 1 4:有等号时加上 分母不等于零.
②当6-x≤0时，即x≥6时，不等式无解；
1:分式不等式的概念: 分母中含有未知数的不等式叫做分式不等 式. 2:各种分式不等式经过同解变形,都可 化成标准形式
f ( x) f ( x) 0或 0 g ( x) g ( x)
3:解分式不等式的思路------转化为整式 不等式解.
4.转化的方法：
|x2|=|OB|
代数的意义
几何意义
方程│x│＝2的解集？
-2 0 2
为{x│x=2或x=-2}
观察、思考： 不等式│x│<2的解集 为{x│-2 < x < 2 } 不等式│x│>2的解集 为{x│x > 2或x<-2 }
类比： |x|<3 的解 归纳： |x|<a （a>0) |x|≤1 的解 -a<x<a |x|≥ 4的解 (a>0) |x|<-2的解 x>a 或 x<-a |x|>a
分式不等式和绝对值不等式
§2.3其它不等式的解法（1）---分式不等式的解法学习要求：1、掌握简单的分式不等式的解法.2、体会化归、等价转换的数学思想方法.学习重难点简单的分式不等式的解法. 不等式的同解变形.课前预习1.分式不等式的概念_________________________________________________________2、分式不等式的解法一般地，分式不等式分为两类：（1）()()0f x g x >（0<）⇔()()0f x g x >（0<）； （2）()()0f x g x ≥（0≤）⇔()()()()000f xg x g x ≥≤⎧⎪⎨≠⎪⎩. 课堂互动：例1 解不等式：1232x x +>-. 解法一：（化分式不等式为一元一次不等式组） 解法二：（利用两数的商与积同号（00a ab b >⇔>，00a ab b <⇔<）化为一元二次不等式）例2 解下列不等式（1）105x x -+>-. （2）2335x≥-. （3）28223x x x +<++.（两种解法） 例3 当m 为何值时，关于x 的不等式()()132m x x -=+的解是（1）正数？ （2）是负数？归纳总结分式不等式的求解通法：（1）不要轻易去分母，可以移项通分，使得不等号的右边为零.（2）利用两数的商与积同号，化为一元二次不等式求解. 学后反思作业：练习册第18页1、2、3题§2.3其它不等式的（2）---绝对值不等式的解法学习要求：1、掌握简单的绝对值不等式的解法.2、能对简单的绝对值不等式给出几何解释。
3、体会化归、等价转换的数学思想方法.学习重难点简单的绝对值不等式的解法. 不等式的同解变形.课前预习（1） 实数绝对值定义________________几何意义______________________________.（2） 绝对值的性质1.任意x R ∈，0x ≥，等号成立⇔0x =.2.任意x R ∈x =⇔22x x =. 3.任意x 、y R ∈，x x x x x ±≤⇔-≤±≤.xy x y =⋅，x x y y=（0y ≠）. 含绝对值的不等式的解法 （1）x a >（0>a ）的解集____________________________________________________. 数轴上表示（2）a x <（0>a ）的解集____________________________________________________. 数轴上表示 想一想：a x b <<（0a b <<）上的解集.课堂互动例1 解下列不等式（1）235x -<.（2）1223x <-.（3）2325x <-<.例2 解下列不等式（1）11x x x x>++. （2）234x x ->. （3）2560x x -+>. （4）2312x x ->+. 议一议： 解不等式：125x x ++->.归纳总结绝对值的不等式的解法（1）x a >（0>a ）的解集____________________________________________________. 数轴上表示（2）a x <（0>a ）的解集____________________________________________________. 数轴上表示学后反思____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________作业：练习册第18页4、5、6以及第19页1、2、3题（英文版）Two regulations promulgated for implementation is in the party in power for a long time and the rule of law conditions, the implementation of comprehensive strictly strategic plan, implementation in accordance with the rules and discipline to manage the party, strengthen inner-party supervision of major initiatives. The two regulations supporting each other, the < code > adhere to a positive advocate, focusing on morality is of Party members and Party leading cadres can see, enough to get a high standard; < rule > around the party discipline, disciplinary ruler requirements, listed as "negative list, focusing on vertical gauge, draw the party organizations and Party members do not touch the" bottom line ". Here, the main from four square face two party rules of interpretation: the first part introduces two party Revised regulations the necessity and the revision process; the second part is the interpretation of the two fundamental principles of the revision of laws and regulations in the party; the third part introduces two party regulations modified the main changes and needs to grasp several key problems; the fourth part on how to grasp the implementation of the two regulations of the party. < code > and < Regulations > revised the necessity and revised history of the CPC Central Committee the amendment to the Chinese CommunistParty members and leading cadres honest politics several guidelines > and < Chinese Communist Party discipline and Punishment Regulations > column 1 by 2015 to strengthen party laws and regulations focus. Two party regulations revision work lasted a Years, pooling the wisdom of the whole party, ideological consensus, draw historical experience, respect for the wisdom of our predecessors, which reflects the unity of inheritance and innovation; follow the correct direction, grasp the limited goals, adhere to the party's leadership, to solve the masses of the people reflect a focus on the problem. The new revision of the < code > and < rule >, reflects the party's 18 and the eighth session of the third, the spirit of the fourth plenary session, reflecting the experience of studying and implementing the General Secretary Xi Jinping series of important speech, reflects the party's eighteen years comprehensive strictly practice. (a) revised two regulations of the party need of < the ICAC guidelines > in < in 1997 Leaders as members of the Communist Party of China clean politics certain criteria (Trial) > based on revised, the promulgation and implementation of January 2010, to strengthen the construction of the contingent of leading cadres play an important role. But with the party to manage the party strictly administering the deepening, has not been able to fully meet the actual needs. Content is toocomplicated, "eight prohibition, 52 are not allowed to" hard to remember, and also difficult to put into practice; the second is concisely positive advocated by the lack of prohibited provisions excessive, no autonomy requirements; the third is banned terms and discipline law, both with the party discipline, disciplinary regulations repeat and Criminal law and other laws and regulations repeat; the fourth is to "clean" the theme is not prominent, not for the existing problems, and is narrow, only needle of county-level leading cadres above. < rule > is in 1997 < Chinese Communist Party disciplinary cases (Trial) > based on revision, in December 2003 the promulgation and implementation, to strengthen the construction of the party play very important role. Along with the development of the situation, which many provisions have been unable to fully meet the comprehensive strictly administering the practice needs. One is Ji law, more than half of the provisions and criminal law and other countries laws and regulations Repetition; two is the political discipline regulations is not prominent, not specific, for violation of the party constitution, damage the authority of Party Constitution of misconduct lack necessary and serious responsibility to pursue; third is the main discipline for the leading cadres, does not cover all Party members. Based on the above situation, need to < the criterion of a clean and honestadministration > and < rule > the two is likely to be more relevant regulations first amendment. By revising, really put the authority of Party discipline, the seriousness in the party tree and call up the majority of Party members and cadres of the party constitution of party compasses party consciousness. (II) two party regulations revision process the Central Committee of the Communist Party of China attaches great importance to two regulations revision . Xi Jinping, general books recorded in the Fifth Plenary Session of the eighth session of the Central Commission for Discipline Inspection, on the revised regulations < > made clear instructions. According to the central deployment, the Central Commission for Discipline Inspection from 2014 under six months begin study two regulations revision. The Standing Committee of the Central Commission for Discipline Inspection 4 review revised. Comrade Wang Qishan 14 times held a special meeting to study two regulations revision, amendment clarifies the direction, major issues of principle, path and target, respectively held a forum will listen to part of the province (area) secretary of the Party committee, Secretary of the Discipline Inspection Commission, part of the central ministries and state organs DepartmentThe first party committee is mainly responsible for people, views of experts and scholars and grassroots party organizations andParty members. Approved by the Central Committee of the Communist Party of China, on 7 September 2015, the general office of the Central Committee of the Party issued a notice to solicit the provinces (autonomous regions, municipalities) Party, the central ministries and commissions, state ministries and commissions of the Party (party), the General Political Department of the military, every 3 people organization of Party of two regulations revision opinion. Central Commission for Discipline Inspection of extensive solicitation of opinions, careful study, attracting, formed a revised sent reviewers. In October 8 and October 12, Central Committee Political Bureau Standing Committee and the Political Bureau of the Central Committee After consideration of the two regulations revised draft. On October 18, the Central Committee of the Communist Party of China formally issued two regulations. Can say, two laws amendment concentrated the wisdom of the whole party, embodies the party. Second, < code > and < Regulations > revision of the basic principles of two party regulations revision work and implement the party's eighteen, ten eight plenary, the spirit of the Fourth Plenary Session of the Eleventh Central Committee and General Secretary Xi Jinping important instructions on the revised < low political criterion > and < Regulations >, highlighting the ruling partycharacteristics, serious discipline, the discipline quite in front of the law, based on the current, a long-term, advance as a whole, with Bu Xiuding independent < rule > and < rule >. Main principle is: first, adhere to the party constitution to follow. The constitution about discipline and self-discipline required specific, awaken the party constitution of party compasses party consciousness, maintaining the authority of the constitution. General Secretary Xi Jinping pointed out that "no rules, no side round. Party constitution is the fundamental law, the party must follow the general rules. In early 2015 held the eighth session of the Central Commission for Discipline Inspection Fifth Plenary Session of the 16th Central Committee, Xi Jinping again pointed out that constitution is the party must follow the general rules, but also the general rules." the revision of the < code > and < rule > is Method in adhere to the regulations established for the purpose of combining rule of virtue is to adhere to the party constitution as a fundamental to follow, the constitution authority set up, wake up the party constitution and party rules the sense of discipline, the party constitution about discipline and self-discipline specific requirements. 4 second is to adhere to in accordance with the regulations governing the party and the party. The Party of rule of virtue "de", mainly refers to the party's ideals and beliefs, excellent traditional style. The revised the< code > closely linked to the "self-discipline", insisting on the positive initiative, for all members, highlight the "vital few", emphasized self-discipline, focusing on the morality, and the majority of Party members and the ideological and moral standards. The revised < > Ji method separately, Ji, Ji Yan to Method, as a "negative list", emphasizing the heteronomy, focusing on vertical gauge. Is this one high and one low, a positive reaction, the strict party discipline and practice results transformation for the integration of the whole party to observe moral and discipline requirements, for the majority of Party members and cadres provides benchmarking and ruler. Third, insist on to. In view of the problems existing in the party at the present stage, the main problems of Party members and cadres in the aspect of self-discipline and abide by the discipline to make clearly defined, especially the party's eighteen years strict political discipline and political rules, organization and discipline and to implement the central eight provisions of the spirit against the four winds and other requirements into Disciplinary provisions. Not one pace reachs the designated position, focusing on in line with reality, pragmatic and effective. After the revision of major changes, major changes in the < code > and < rule > modified and needs to grasp several key problems (a) < code > < code > adhere to according to regulationsgoverning the party and party with morals in combination, for at the present stage, the leadership of the party members and cadres and Party members in existing main problems of self-discipline, put forward principles, requirements and specifications, showing Communists noble moral pursuit, reflected at all times and in all over the world ethics from high from low 5 common requirements. One is closely linked to the "self-discipline", removal and no direct relation to the provisions of . the second is adhere to a positive advocate, "eight prohibition" 52 are not allowed to "about the content of the" negative list moved into synchronization amendment < cases >. Three is for all the party members, will apply object from the leadership of the party members and cadres to expand to all Party members, fully embodies the comprehensive strictly required. The fourth is prominent key minority, seize the leadership of the party members and cadres is the key, and put forward higher requirements than the ordinary Party members. Five is to simplify, and strive to achieve concise, easy to understand, easy to remember. The revised < code > is the ruling Party since the first insists on a positive advocate forAll Party members and the self-discipline norms, moral declaration issued to all members of the party and the National People's solemn commitment. > < criterion of a clean and honest administration consists of 4 parts, 18,more than 3600 words. After the revision of the < code >, a total of eight, 281 words, including lead, specification and Party member cadre clean fingered self-discipline norms, etc. Part 3 members low-cost clean and self-discipline, the main contents can be summarized as "four must" "eight code". Lead part, reiterated on ideal and faith, fundamental purpose, the fine traditions and work style, noble sentiments, such as "four must" the principle of requirements, strong tone of self-discipline, The higher request for 6 and supervised tenet, the foothold in permanent Bao the party's advanced nature and purity, to reflect the revised standards requirements. Members of self-discipline norms around the party members how to correctly treat and deal with the "public and private", "cheap and rot" thrifty and extravagance "bitter music", put forward the "four norms". Party leader cadre clean fingered self-discipline norms for the leadership of the party members and cadres of the "vital few", around the "clean politics", from civil servant of the color, the exercise of power, moral integrity, a good family tradition and other aspects of the leadership of the party members and cadres of the "four norms" < > < norm norm. "The Party member's self-discipline norms" and "party members and leading cadre clean fingered self-discipline norms," a total of eight, collectively referred to as the "eight". "Four must" and "eight" of thecontent from the party constitution and Party's several generation of leaders, especially Xi Jinping, general secretary of the important discussion, refer to the "three discipline and eight points for attention" statements, and reference some embody the Chinese nation excellent traditional culture essence of epigrams. (2) the revised regulations, the main changes in the revised Regulations > to fully adapt to the strictly requirements, reflects the according to the regulations governing the law of recognition of deepening, the realization of the discipline construction and Jin Ju. < rule > is party a ruler, members of the basic line and follow. And the majority of Party members and cadres of Party organizations at all levels should adhere to the bottom line of thinking, fear discipline, hold the bottom line, as a preventive measure, to keep the party's advanced nature and purity. 1, respect for the constitution, refinement and discipline. Revised < rule > from comprehensive comb physical constitution began, the party constitution and other regulations of the Party of Party organizations and Party discipline requirements refinement, clearly defined in violation of the party constitution will be in accordance with regulations to give the corresponding disciplinary action. The original 10 categories of misconduct, integration specification for political discipline, discipline, honesty and discipline masses Ji Law and discipline and discipline andother six categories, the content of < rule > real return to Party discipline, for the majority of Party members and listed a "negative list. 7 2, highlighting the political discipline and political rules. > < Regulations according to the stage of the discipline of outstanding performance, emphasizing political discipline and political rules, organization and discipline, in opposition to the party's leadership and the party's basic theory, basic line, basic program and basic experience, the basic requirement of behavior made prescribed punishment, increase the cliques, against the organization such as violation of the provisions, to ensure that the central government decrees and the Party of centralized and unified. 3, adhere to strict discipline in the law and discipline In front, Ji separated. Revised < Regulations > adhere to the problem oriented, do Ji separated. Any national law existing content, will not repeat the provisions, the total removal of 79 and criminal law, repeat the content of the public security management punishment law, and other laws and regulations. In the general reiterated that party organizations and Party members must conscientiously accept the party's discipline, die van comply with national laws and regulations; at the same time, to investigate violations of Party members and even criminal behavior of Party discipline and responsibility, > < Regulations distinguish five different conditions, with special provisions weremade provisions, so as to realize the connection of Party discipline and state law. 4, reflect Wind building and anti-corruption struggle of the latest achievements. < rule > the party's eighteen years implement the spirit of the central provisions of the eight, against the requirements of the "four winds" and transformation for disciplinary provisions, reflecting the style construction is always on the road, not a gust of wind. In the fight against corruption out of new problems, increase the trading rights, the use of authority relatives profit and other disciplinary terms. Prominent discipline of the masses, the new against the interests of the masses and ignore the demands of the masses and other disciplinary terms and make provisions of the disposition and the destruction of the party's close ties with the masses.Discipline to protect the party's purpose. 8 of these regulations, a total of three series, Chapter 15, 178, more than 24000 words, after the revision of the regulations a total of 3 series, Chapter 11, 133, 17000 words, divided into "general" and "special provisions" and "Supplementary Provisions" Part 3. Among them, add, delete, modify the provisions of the proportion of up to nearly 90%. 1, the general general is divided into five chapters. The first chapter to the regulations of the guiding ideology, principles and scope of application of the provisions, highlight the strengthening of the partyconstitution consciousness, maintenance the authority of Party Constitution, increase the party organizations and Party members must abide by the party constitution, Yan Centralized centralized, would examine at all levels of the amended provisions implementing and maintaining Party discipline, and consciously accept the party discipline, exemplary compliance with national laws and regulations. The second chapter of discipline concept, disciplinary action types and effects of the regulations, will be a serious warning from the original a year for a year and a half; increase the Party Congress representative, by leaving the party above (including leave probation) punishment, the party organization should be terminated its representative qualification provisions. The third chapter of the disciplinary rules of use prescribed in the discipline rectifying process, non convergence, not close hand classified as severely or heavier punishment. "Discipline straighten "At least eighteen years of five years, these five years is to pay close attention to the provisions of the central eight implementation and anti -" four winds ". The fourth chapter on suspicion of illegal party disciplinary distinguish five different conditions, with special provisions were made provisions, to achieve effective convergence of Party and country 9 method. < rule > the provisions of Article 27, Party organizations in thedisciplinary review found that party members have committed embezzlement, bribery, dereliction of duty dereliction of duty and other criminal law act is suspected of committing a crime shall give cancel party posts, probation or expelled from the party. The second is < Regulations > Article 28 the provisions of Party organizations in the disciplinary review But found that party members are stipulated in the criminal law, although not involved in a crime shall be investigated for Party discipline and responsibility should be depending on the specific circumstances shall be given a warning until expelled punishment. This situation and a difference is that the former regulation behavior has been suspected of a crime, the feeling is quite strict, and the latter for the behavior not involving crime, only the objective performance of the provisions of the criminal code of behavior, but the plot is a crime to slightly. < Regulations > the 29 provisions, Party organizations in the discipline review found that party members and other illegal behavior, affect the party's image, the damage to the party, the state and the people's interests, we should depend on the situation Seriousness given disciplinary action. The loss of Party members, seriously damaging the party's image of behavior, should be given expelled from the party. At this article is party member is in violation of the criminal law outside the other illegal acts, such as violatesthe public security administration punishment law, customs law, financial laws and regulations behavior. The fourth is < cases > Article 32 stipulates, minor party members and the circumstances of the crime, the people's Procuratorate shall make a decision not to initiate a prosecution, or the people's court shall make a conviction and exempted from criminal punishment shall be given within the party is removed from his post, probation or expelled from the party. Party members and crime, sheets were fined in accordance with For acts; the principal Ordinance amended the provisions of the preceding paragraph. This is the new content, in order to achieve Ji method effective convergence. Five is < > the thirty third article 10 of the provisions, the Party member due to an intentional crime is sentenced to criminal law (including probation) sheets or additional deprivation of political rights; due to negligence crime and was sentenced to three years or more (excluding three years) a penalty, shall give expelled punishment. Due to negligence crime is convicted and sentenced to three years (including three years) in prison or be sentenced to public surveillance, detention, shall in general be expelled from the party. For the individual may not be expelled from the party, should control Approval. This is followed and retained the original > < Regulations the provisions of punishment party authorization rules and report to a level partyorganizations. For is "party members with criminal acts, and by the criminal punishment, generally should be expelled from the party". The fifth chapter of probationary Party member of the discipline and discipline after missing members of the treatment and punishment decisions, such as the implementation of the provisions, clear the related party discipline and punishment decision made after, for duties, wages and other relevant alteration formalities for the longest time. 2, sub sub section will the original regulations of10 categories of acts of violation of discipline integration revised into 6 categories, respectively, in violation of the punishments for acts of political discipline "in violation of discipline behavior of punishment" in violation of integrity of disciplinary action points "of violation punishments for acts of mass discipline" "the violation of work discipline, punishment" in violation of discipline of life behavior punishment "6 chapters. 3, annex" Supplementary Provisions "clear authority making supplementary provisions of, cases of interpretative organ, as well as regulations implementation time and retroactivity etc.. 11 (3) learning understanding > < regulations needs to grasp several key problems The first problem -- about the violation of political discipline behavior > < new ordinance chapter 6 the political discipline column for the six disciplines, that is the main opposition to Party leadership and the opposition of the basictheory, basic line, basic program and basic experience, basic requirements of misconduct made provisions of the disposition, especially the eighteen since the CPC Central Committee put forward the Yan Mingzheng treatment of discipline and political rules requirements and practical achievements transformation for Discipline article, increase the false debate central policies, cliques, against the organization review, make no discipline of the principle of harmony terms. These are the party's eighteen years in comprehensive strictly Process combined with the practice of rich content. (1) false debate the central policies and undermine the Party of centralized and unified the problem is made in accordance with the provisions of the party constitution. Constitution in general programme requirements adhere to democratic centralism is one of the requirements of the construction of the party must adhere to the four cardinal. Application of this principle is not only the party the basic organization principle and is also the mass line in party life, it requires that we must fully develop inner-party democracy, respect for the dominant position of Party members, safeguarding the Party member democratic rights, give full play to the enthusiasm and creativity of the party organizations at all levels and Party members, at the same time, also must implement the right concentration, ensure the party's mission < the chaos in unity and concertedaction to ensure that the party's decision to get quickly and effectively implementing. The Party Central Committee formulated the major principles and policies, through different channels and ways, fully listen to the party organizations and Party members of the opinions and suggestions, but 12 is some people face to face not to say back blather "" will not say, after the meeting said, "" Taiwan does not say, and nonsense ", in fact, not only disrupt the people thought, some causing serious consequences, the damage to the Party of the centralized and unified, hinder the central policy implementation, but also a serious violation of the democratic system of principles. There is no doubt that shall, in accordance with the Regulations > 4 Specified in Article 6 to give the appropriate punishment. For did not cause serious consequences, to give criticism and education or the corresponding tissue processing. (2) about the destruction of the party's unity < New Regulations > the forty eighth to fifty second article, to damage Party's unity unified and violation of political discipline, punishment situation made explicit provisions. Article 52 of the new "in the party get round group, gangs seek private gain, cliques, cultivate private forces or through the exchange of interests, for their own to create momentum and other activities to gain political capital, given a serious warning or withdraw from their party posts disposition; if the。
第2节含绝对值的不等式和分式不等式的解法
第二课时复习内容：含绝对值的不等式和分式不等式的解法复习目标：理解区间的意义，掌握含绝对值的不等式和分式不等式的解法，会解含绝对值的不等式和分式不等式。
复习过程：一、学生阅读教材P 31-P 33完成知识点填空1、设a 、b ∈R,且a<b,⑴满足不等式 的实数的集合叫做闭区间，记作 即 。
⑵满足不等式 的实数的集合叫做开区间，记作 即 。
⑶满足不等式 ， 的实数的集合叫做半开半闭区间，记作 或 ，即 或 。
⑷满足不等式x>a, x<b 的实数的集合用区间可以表示为 ， 2、含绝对值的不等式的解法⑴不等式|x|>a(a>0)的解集为 ⑵不等式|x|<a(a>0)的解集为 3、分式不等式的解法⑴不等式0ax bcx d +>+可以转化为不等式组 来解⑵不等式0ax bcx d+<+可以转化为不等式组 来解二、课堂练习 一层练习1、下列说法正确的是 （ ） A ：Q +用区间表示为(0,+∞)； B ：(-1,]4=}{0,1,2,3,4 C ：}{|3(3,)x x <=--∞ D ：}[){|00,x x ≥=+∞2、解下列不等式⑴ |2x-3|<5 ⑵ |3x-1|≥2 ⑶ 4305xx -≥-二层练习3、解下列不等式⑴3|2x-1|-2>0 ⑵1-2|x-1|≥0⑶3213xx+>-⑷2112xx+≤-三层练习4、求下列不等式的解集⑴|x|>0的解集是⑵|x|<0的解集是⑶|x|>a(a<0)的解集是⑷|x|<a(a<0)的解集是5、解下列不等式⑴21 712xx x >-+⑵|x2+3x-8|>10⑶3<|5-2x|<9 ⑷0<。
不等式的解法(共28张PPT)
5 10 a= , b= . 3 3
高考：(天津08)已知函数f(x)= 解集是(
A
)
x+2, x≤0 ，则不等式f(x)≥x2的 -x+2, xБайду номын сангаас0
∴ B ={x |1-a<x<1+a, a>0 }
∵ A∪B=B ∴ A B
∴ 1-a<1 且 1+a>2，故a的取值范围是：(1, +∞)
不等式的解法
五、无理不等式解法 2x 1 练习10. 解不等式： (1) | 3x 2 3 | 1; ( 2) 1. x1 分析：(1)原不等式等价于： (I) 3x 2 3 1 或 (II) 3x 2 3 1 3x-2≥0 解(I) ： 3x 2 4 即 解得 x>6 3x-2>16 2 3x-2≥0 解得 ≤x<2 解(II) ： 3x 2 2 即 3x-2<4 3 (2)原不等式化为： (I) x-1>0
2 ) 5
不等式的解法
二、含绝对值的不等式 高考. 1、(北京07)已知集合A={x||x-a|≤1}，B={x|x2-5x+4≥0}． 若A∩B=φ，则实数a的取值范围是 (2，3) . (0, 2)
2
2、(浙江07) 不等式 |2x-1|-x<1的解集是 { x | 0<x<2 } . 3、(上海08) 不等式|x-1|<1的解集是
绝对值不等式PPT课件
x x
a, a
3x
0
或
x a
a, x
3x
0,
即
x x
aa, 或
4
x x
a, a
2
.
结合a>0,解得x≤-
a 2
,即不等式f(x)≤0的解集为
x
|
x
a 2
.
∵不等式f(x)≤0的解集为{x|x≤-1},
∴- a =-1,故a=2.
2
考点二 利用绝对值不等式求参数
典例2 (1)对任意x,y∈R,求|x-1|+|x|+|y-1|+|y+1|的最小值. (2)对于实数x,y,若|x-1|≤1,|y-2|≤1,求|x-2y+1|的最大值.
.
答案 {x|-3<x<2}
解析
原不等式等价于
x (x
2, 1)
(x
2)
5
或
2 x 1, (x 1) (
x
2)
5
或
x x
1, 1
x
2
5,
即
x
x
23, 或 32 5
x
1,
或
x
x
1, 2,
亦即-3<x<-2或-2≤x≤1或1<x<2.
∴原不等式的解集为
(-3,-2)∪[-2,1]∪(1,2)=(-3,2).
方法技巧
1.形如|ax+b|≤c(≥c)(c>0)的三种解法 解法一:等价法 |ax+b|≤c⇔-c≤ax+b≤c. (|ax+b|≥c⇔ax+b≤-c或ax+b≥c) 解法二:分类讨论法
绝对值不等式的解法 PPT
5
5
2. 设不等式 x a b 的解集为 x 1 x 2 ，
则 a 与 b 的值为（ D）
(A) a 1,b 3 (B) a 1,b 3(C) a1,b3 (D) a 1 ,b 3 22
课堂小结
绝对值不等式的解法: 1．公式法 |f(x)|＞g(x)⇔f(x)＞g(x)或f(x)＜－g(x)； |f(x)|＜g(x)⇔－g(x)＜f(x)＜g(x)． 2．平方法 |f(x)|＞|g(x)|⇔[f(x)]2＞[g(x)]2.
练习一：解下列不等式： （1）|x|>5 （2）|x-1|<5 （3）| 5x-6 | < 6–x （4）|x-1| > |x-3|
2020/7/19
练习二:
1. 不等式 |x2－5x+6|≤x2－4 的解集( A)
(A){x| x≥2} (B){x| x≤2} (C){x| x≥ 4 }(D){x| 4 x≤2}
2020/7/19
小结：不等式|x|<a和|x|>a (a>0)的解集。 ① 不等式|x|<a的解集为{x|-a<x<a}
-a
0
a
② 不等式|x|>a的解集为{x|x<-a或x>a }源自-a2020/7/19
0
a
典型例题
例3.解不等式： 2x 3 5
例4.解不等式： x2 2x x
例5.解不等式： x 9 x 1
不等式|x|<1的解集表示到原点的距离小于1 的点的集合。
-1
0
1
所以，不等式|x|<1的解集为{x|-1<x<1}
例2：求不等式|x|>1的解集。 方法： 利用绝对值的几何意义观察
绝对值不等式和分式不等式
绝对值不等式和分式不等式绝对值不等式一、 基本解法与思想解含绝对值的不等式的基本思想是等价转化，即采用正确的方法去掉绝对值符号转化为不含绝对值的不等式来解，常用的方法有公式法、定义法、平方法。
（一）、公式法：即利用a x >与a x <的解集求解。
主要知识：1、绝对值的几何意义：x 是指数轴上点x 到原点的距离；21x x -是指数轴上1x ，2x 两点间的距离.。
2、a x >与a x <型的不等式的解法。
当0>a 时，不等式>x 的解集是{}a x a x x -<>或,不等式a x <的解集是{}a x a x <<-;当0<a 时，不等式a x >的解集是{}R x x ∈不等式a x <的解集是∅;3．c b ax >+与c b ax <+型的不等式的解法。
把 b ax + 看作一个整体时，可化为a x <与a x >型的不等式来求解。
当0>c 时，不等式c b ax >+的解集是{}c b ax c b ax x -<+>+或,不等式c b ax <+的解集是{}c b ax c x <+<-;当0<c 时，不等式c b ax >+的解集是{}R x x ∈不等式c bx a <+的解集是∅;例1 解不等式32<-x（二）、定义法:即利用(0),0(0),(0).a a a a a a >⎧⎪==⎨⎪-<⎩去掉绝对值再解。
例2。
解不等式22x xx x >++。
（三）、平方法:解()()f x g x >型不等式。
例3、解不等式123x x ->-。
二、分类讨论法：即通过合理分类去绝对值后再求解。
例4 解不等式125x x -++<。
三、几何法：即转化为几何知识求解。
例5 对任何实数x ，若不等式12x x k +-->恒成立，则实数k 的取值范围为 ()(A)k<3 (B)k<-3(C)k ≤3(D)k ≤-3四、典型题型1、解关于x 的不等式10832<-+x x2、解关于x 的不等式2321>-x3、解关于x 的不等式212+<-x x4、解关于x 的不等式1212-<-m x )(R m ∈5、解关于x 的不等式1312++<--x x x6、解关于x 的不等式521≥++-x x分式不等式一、分式不等式的解法（一）、()()()()00f xf xg xg x>⇔⋅>；（二）、()()()()()f xg xf xg x g x⋅≥⎧⎪≥⇔⎨≠⎪⎩。
分式及绝对值不等式的解法
分式不等式及绝对值不等式一、复习：⑴⇔>⇔>0baab0ab>⎧⎨>⎩或ab<⎧⎨<⎩>⇔ab⑵⇔<⇔<0baabab<⎧⎨>⎩或ab>⎧⎨<⎩<⇔ab二、形如>++dcxbax或<++dcxbax)0(≠c的关于x的不等式称为简单的分式不等式三、例题1. 解不等式213> -+xx2.解不等式2231>-+xx3.当k为何值时，关于x的方程(1)2(4)k x x-=-的解是正数？是负数？4.解关于x不等式11ax>+四、复习：{||x=(00(0(0x xxx x>=-<当时）当时）当时）它表示实数x在数轴上所对应的点到原点的距离，因此求不等式ax<||)0(>a的解就是求在数轴上到原点的距离小于a的点所对应的实数x的集合，所以不等式ax<||)0(>a的解集是), (a a-；类似可得，不等式ax>||)0(>a的解集是),(),(+∞--∞aa五、例题：解下列不等式⑴7|12|<+x⑵5|23|> -x⑶4|13|-<-x⑷|2||13|x x+<-⑸2||120 x x--≥⑹2|1|3x≤-<⑺|2||3|4 x x--+<。
高三数学不等式的解法2
(a 1) (a 1) | x | 2、关于实数x的不等式： 2 2
2
2
与x 3(a 1) x 2(3a 1) 0(a R)
2
的解集分别为A、B，求使 A B
时实数a的取值范围
3、已知a、b是不相等的实数，且
a b a b
3 3 2
2
4 求证 : 0 a b 3
； / mrcat电竞 mrcat电竞官网 ；
神山壹带强者如林,比之咱们奇幻之地不会少多少,咱们行事壹切低调,先到了长生神山再说吧丶""好丶"叶问情也早就不想呆在这里了,要不是怕艾姐姐小瞧了她,她壹个人の话,早就赶紧闪人了丶二人不在这里继续呆了,艾丽立即带着叶问情前往长生神山主山丶而在她们离开后不久,这片血 湖林子旁边,刚刚那个黑衣女人又出现了丶女人抬头看了看二女离开の方向,喃喃自语道："这个女人の身上,竟然有那个家伙の气息,看来那家伙有可能就在前面,跟上她们丶"叶问情和艾丽哪里知道,她们此时已经被人给盯上了,二女却全然没有察觉,连艾丽也没有任何发现丶此地距离长生神 山只有八千多万里了,壹路上二人也着急赶路,就算是遇到了壹些麻烦,艾丽会直接带着叶问情绕道而过丶路上这里也有不少の强者,最近不知道为何,这壹带の争斗,长生神山外围の地带の厮杀越来越激烈了丶有许多の强者,出现了许多在这外面,埋伏,设陷阱,劫杀の事件发生丶因为这里不在 长生神山の范围,所以神山也不会管这里の事情,他们只管那神山忠の情况,外面の事情与他们无关丶龙亭并没有精力,去管这些事情,不过因为这里の劫杀事件日益增多,还是令这许多人对长生神山望而生畏了丶猫补忠文叁677魔界变故(猫补忠文)叁677有许多の强者,出现了许多在这外面,埋 伏,设陷阱,劫杀の事件发生丶因为这里不在长生神山の范围,所以神山也不会管这
基本不等式(共43张)ppt课件
解法步骤与技巧
01
02
03
移项
将不等式两边的同类项进 行合并，并把未知数移到 不等式的一边，常数移到 另一边。
合并同类项
将移项后的不等式两边的 同类项进行合并。
系数化为1
将不等式两边的系数化为 1，得到不等式的解集。
解法步骤与技巧
注意不等号的方向
在解不等式时，要注意不等号的方向，特别是在乘以或除以一个负数时，不等 号的方向要发生变化。
基本不等式(共43张)ppt课件
目录
• 基本不等式概念及性质 • 一元一次不等式解法 • 一元二次不等式解法 • 绝对值不等式解法 • 分式不等式和无理不等式解法 • 基本不等式在几何中的应用 • 基本不等式在函数中的应用 • 总结回顾与拓展延伸
01
基本不等式概念及性质
不等式定义与分类
不等式定义
根）；
04
05
当 $Delta < 0$ 时，方程无 实根，有两个共轭复根。
04
绝对值不等式解法
绝对值概念及性质
绝对值定义
对于任意实数$x$，其绝对值$|x|$定义为：若$x geq 0$，则$|x| = x$；若$x < 0$，则$|x| = -x$。
绝对值的性质
非负性、对称性、三角不等式。
绝对值不等式解法步骤
将不等式左边进行因式分解，找出不 等式的临界点。
无理不等式解法
第一步
确定无理不等式的定义域，即根 号内的表达式必须大于等于零。
第二步
通过平方消去根号，将无理不等式 转化为有理不等式。
第三步
利用有理不等式的解法，求解转化 后的不等式，得到原无理不等式的 解集。
综合应用举例
例1
绝对值不等式的解法ppt课件
x2 x2
x 1 x 1
x 2 0时，即x 2时,
x 1 0时，即x 1时，
x 2 (x 2)
x 1 (x 1)
1、当x 2且x 1时，即x 1
2、当x 2且x 1时， 2 x 1 3、当x 2且x 1时，x 2
2
1
4、当x 2且x 1时，x
13
14
15
6
小结2 形如|ax+b|≤c, |ax+b|≥c型
不等式的解法:
ax b c c ax b c
ax b c ax b c或ax b c
7
8
9
10
[拓展]解不等式.
(1)1 2x 1 3
解：由原不等式得
2x 1 3 2x 1 1
得x
1 x 1或x
2
含绝对值的不等式的解法
1
1.理解绝对值的代数意义和几何意 义，掌握去绝对值的方法.
2.会求解以下类型的不等式： ax b c; ax b c
|x-a|+|x-b|≥c和|x-a|+|x-b|≤c
2
1.绝对值的代数意义:
x
x x
(x 0) (x 0)
2.绝对值的几何意义:
一个数的绝对值表示这个数在数轴上
0
(2) x 9 x 1
解：由原不等式得 x 9 2 x 12 解得x 5 x (5,)
1 0 1 2
解得 1 x 0或1 x 2 x (1,0) (1,2)
11
二、解不等式|x+2|+|x-1|≥5.
思考：解这个不等式的关键是什么？
如何去掉绝对值符号？
方法1、几何意义
记x对应的点为P
绝对值不等式的解法公开课PPT课件
| f (x) | g(x) g(x) f (x) g(x)
小试身手：
（1）|x2-3|＞2x
解集为{x|x＜1或x＞3}.
x （2） x 2
x x2
解集为{x| -2＜ x＜0}
对于（2）中， “＞”换成“≥”解集变化了吗？如何变化？
例4：解不等式：|x－5|＋|x＋3|≥10.
解法一：
即为原不等式的解集
优点:利于分析最值以及相应的x的取值
变式：1. |x－5|＋|x＋3|≥a恒成立，则a的范围____ 2.方程 |x－5|＋|x＋3|=2a-5有无数解，则a的值为___
例4：解不等式：|x－5|＋|x＋3|≥10.
解法三：由绝对值的几何意义可知，|x－5|＋|x＋3|表示数轴上
复习回顾：|x|的意义：
一个数的绝对值表示：
x X>0
与这个数对应的点到
|x|= 0 X=0
原点的距离,|x|≥0，|x|≥x
- x X<0
x2
B
O
|x1| =|OA|
几何意义
x1
A
X
|x2|=|OB|
|AB|=| x2 -x1 |
代数意义
易得：不等式|x|<a和|x|>a (a>0)的解集。去掉a>0,解集还能这样表示吗？
解集为 ( 10 , 5] [1, 2)
33
3
例3：解不等式| 5x-6 | < 6 – x
解： 由绝对值的意义，原不等式转化为：
6-x>0
(Ⅰ)或
-(6-x)<5x-6<(6-x)
6-x≤0
(Ⅱ)
无解
解(Ⅰ)得：0<x<2; (Ⅱ) 无解 综合得解集{x|0<x<2}
一元二次不等式、分式、绝对值、简单高次不等式的解法
一元二次不等式、分式、绝对值、简单高次不等式的解法本文介绍了高次不等式、分式、绝对值、一元二次不等式的解法。
对于可分解的高次不等式，可以使用奇穿偶回的方法进行求解。
具体步骤为：检查各因式中x的符号均为正；求得相应方程的根；在数轴上表示各根并穿线；最后得出原不等式的解集。
对于含参数的高次不等式，需要对参数的位置进行讨论。
以解关于x的不等式x-x+12(x+a)<0为例，将二次项系数化为正，分解因式后得到相应方程的根为-3，4，-a。
讨论a的位置，得出不等式的解集。
对于分式不等式，可以使用等价转化法进行求解。
例如，解不等式x^2-2x-1/(2x^2+3x-2)>=0，将其等价转化为不等式3x-1/(x+2)<2.绝对值不等式的解法与一元二次不等式类似，需要将其等价转化为一元二次不等式进行求解。
例如，解不等式|2x-1|<3，将其等价转化为-3<2x-1<3，再解出一元二次不等式的解集。
在解不等式时，需要注意格式的正确性和删除明显有问题的段落，同时可以适当改写每段话，使其更加清晰易懂。
练1：解不等式：1、$\frac{x-3}{2x-1}\geq 0$ （首项系数化为正，空实心）2、$\frac{2-x}{x+3}>1$ （移项通分，右侧化为）3、$\frac{x^2-3x+2}{x^2-2x-1}\leq 0$ （因式分解）4、$\frac{x^2-3x+2}{x^2+x+6}<0$ （求根公式法因式分解）5、$\frac{2(x+3)}{3}\geq x$ （恒正式，重根问题）6、$\frac{x(x-3)}{1}\leq 0$ （不能随便约分）7、$|x|-|1|<x-|1|$ （取交集）练2：解不等式：1.求不等式的解集：$\frac{(x+2)^4}{(x-1)^3}\cdot\frac{1}{(3x+2)^3}\cdot\frac{1}{(x^2-x+2)}>0$2.$x^2-9x+11<0$3.解不等式：$\frac{2}{x}\geq\frac{7}{2}$4.解不等式：$\frac{2x+1}{2x-1}-\frac{x+7}{x-1}$5.解不等式：$\frac{2}{x-3}>\frac{3x-2}{x^2+x+1}$6.解绝对值不等式：$|2x-1|>|2x-3|$7.解关于x的不等式：$\frac{x}{x+2}>1$练4：解不等式：1.$|8-2x|>3$2.$|6-2x|<4$3.$x^2-3x+2<10$4.$x^2-3x+2\leq 0$5.$(2x-1)^2\leq 9$练5：1.解关于x的不等式：1) $2x-1<x+2$2) $3\leq 5-2x<9$3) $1<|x+1|<3$2.求方程$\frac{x+2}{x^2+3x}=\frac{1}{x+2}$的解集；3.不等式$x(1-2x)>0$的解集是（）A。
幂函数及三类不等式的解法(绝对值、高次、无理)-高考复习课件
2 + 1 ≠ 0,
4.不等式(x-1)(x-2)(x-3)<0的解集为
.
答案 (-∞,1)∪(2,3)
解析 方程(x-1)(x-2)(x-3)=0的三个根分别为1,2,3,标根穿线如图所示.故解
集为(-∞,1)∪(2,3).
关键能力 学案突破
考点1
幂函数的概念
α
【例 1】 (1)已知幂函数 f(x)=k·x 的图像经过点
() ≥ 0,
(3) ()<g(x)⇔ () ≥ 0,
() < 2 ().
5.一元高次不等式的解法(标根穿线法)
(1)化x的最高次系数为正;
(2)在数轴上标出方程的根;
(3)从数轴上方穿针,奇穿偶回;
(4)写出解集.
常用结论
1.幂函数y=xα的图像在第一象限的两个重要结论
(1)恒过点(1,1);
综上,不等式的解集为(-3,2).
(3)由|2x-1|-|x-2|<0,得|2x-1|<|x-2|,∴(2x-1)2<(x-2)2,
∴3x2<3,解得-1<x<1.
∴不等式的解集为(-1,1).
解题心得1.含绝对值的不等式要注意观察式子特点,选择更简便的方法.
2.零点分段法的好处在于,一段范围可将所有的绝对值一次性去掉,缺点在
0<α<1,下凸时α>1;最后由x>1时,在第一象限内α的值按逆时针方向依次增
大得出结论.
对点训练2(1)下面给出4个幂函数的图像,则图像与函数的大致对应的是
(
)
1
3
1
2
A.①y= ;②y=x2;③y= ;④y=x-1
1
含绝对值不等式与分式不等式的解法公开
答： |x|=2的解是x=2或x=-2，
.
-2 -1 0
在数轴上表示如下：
.
12
问：|x|<2 与 |x|>2 的解是什么？
-2
0
2
因而不等式|x|<2的解集是：{x|-2<x<2}.
-2
0
2
因而不等式|x|>2的解集是：
{x|x<-2}∪{x|x>2}={x|x<-2,或x>2}.
（2）x a x b c和 x a x b c 型不等式的解法
①利用绝对值不等式的几何意义 ②零点分区间法 ③构造函数法
同步训练：解不等式 x 2 x 3 4
小结
1、︱x︱< a与︱x︱> a(a>0)型不等式与 ︱ax+b︱< c与︱ax+b︱> c(c>0)型不等式及|x-
即3 5,矛 盾， 此 时 不 等 式 的 解 集 为
当x 1时,原 不 等 式 可 以 化 为( x 1) ( x 2) 5,
解 得x 2,此 时 不 等 式 的 解 集 为2, 综 上 所 述 可 知 原 不 等的 式解 集 为 ， 3 2，
例3 解不等式 x 1 x 2 5
a|+|x-b| >(或<)c 的解法与解集；
2、数形结合、不等式与函数相互转化 的数学思想.
当堂训练
1．不等式1＜|x+1|＜3的解集是
()
D
A．(0,2)
B．(-2,0)∪(2,4)
C．(-4,0)
D．(-4,-2)∪(0,2)
【解析】原不等式等价于1＜x+1＜3或-3＜x+1＜-1，