004分式不等式及简单的绝对值不等式的解法
专题：分式不等式和绝对值不等式的解法
专题：分式不等式和绝对值不等式的解法一、知识要点本讲义从以下两方面展开： 1. 分式不等式的解法分式不等式是一种常见的不等式，掌握其解法在高考中是非常重要的。
2. 绝对值不等式的解法绝对值不等式是一种常见的不等式，其解法主要要注意分类讨论，也是高考常考的一个内容。
➢ 知识点一：分式不等式的解法分式不等式的求解主要在于同解变形，将不等式化为整式不等式来进行求解。
一般地，对于分式不等式11()()()f x h xg x ≤，要将其通分化为()0()f x g x ≤的标准形式， 对于分式不等式()0()f x g x ≤，它与()()0()0f xg x g x ≤⎧⎨≠⎩同解。
这样，我们就可以将分式不等式化为整式不等式。
➢ 知识点二：绝对值不等式的解法与分式不等式类似的是，求解绝对值不等式也是要将不等式的绝对号去掉，进行同解变形。
一般的，()()f x g x >与()()()()f x g x f x g x ><-或同解；()()f x g x <与()()()g x f x g x -<<同解。
需要注意的是，如果不等式中有多个绝对值，那么就需要对每个绝对值号进行讨论。
二、典型例题1. 分式不等式的解法【例1】 （★☆☆☆）解不等式：22911721x x x x -+≥-+ 解：原不等式化为：2(43)(12)0(1)x x x -+≥-，它等价于： 4()(12)031x x x ⎧-+≤⎪⎨⎪≠⎩，得到：14[,1)(1,]23x ∈-⋃教学提示：此题是标准的求解分式不等式的题目。
分式不等式求解的关键在于把分式不等式进行等价变形成为整式形式。
在等价变形时要注意分母不为零。
一般地，对于分式不等式()0()f x g x ≤，它与()()0()0f x g x g x ≤⎧⎨≠⎩同解。
【例2 】解不等式：2121332x x x x ++≥-- 解：通分整理，原不等式化为：2(12)0(3)(32)x x x +>--，它等价于： (3)(32)0210x x x -->⎧⎨+≠⎩，得到：3x >或23x <且12x ≠- 教学提示：注意提醒学生，此题切忌直接把21x +约去，因为它的符号是未知的。
分式和绝对值不等式的解法
（一）分式不等式： 型如：0)()(>x x f ϕ或0)()(<x x f ϕ（其中）（、x x f ϕ)(为整式且0≠）（x ϕ）的不等式称为分式不等式。
（1）分式不等式的解法：解关于x 的不等式0231>-+x x 方法一：等价转化为： 方法二：等价转化为：⎩⎨⎧>->+02301x x 或⎩⎨⎧<-<+02301x x 0)23)(1(>-+x x 变式一：0231≥-+x x 等价转化为：⎩⎨⎧≠-≥-+0230)23)(1(x x x 比较不等式0231<-+x x 及0231≤-+x x 的解集。
（不等式的变形，强调等价转化，分母不为零） （2）归纳分式不等式与整式不等式的等价转化：（1）0)()(0)()(>⋅⇔>x x f x x f ϕϕ （3）0)()(0)()(<⋅⇔<x x f x x f ϕϕ （2）⎩⎨⎧≠≥⋅⇔≥0)(0)()(0)()(x x x f x x f ϕϕϕ （4）⎩⎨⎧≠≤⋅⇔≤0)(0)()(0)()(x x x f x x f ϕϕϕ （3）小结分式不等式的解法步骤：（1）移项通分，不等式右侧化为“0”，左侧为一分式（2）转化为等价的整式不等式（3）因式分解，解整式不等式（注意因式分解后，一次项前系数为正）练一练：解关于x 的不等式 051)1(>--x x 3532)2(≤-x例1、 解关于x 的不等式：232≥+-x x 解：0232≥-+-x x 03)3(22≥++--x x x 即，038≥+--x x 038≤++x x （保证因式分解后，保证一次项前的系数都为正） 等价变形为：⎩⎨⎧≠+≤++030)3)(8(x x x ∴原不等式的解集为[)3,8-- 例2、解关于x 不等式23282<+++x x x 方法一：322++x x 恒大于0，利用不等式的基本性质方法二：移项、通分，利用两式同号、异号的充要条件，划归为一元一次或一元二次不等式。
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
不等式的解法
解析：∵|x＋1|＞2，∴x＋1＞2或x＋1＜－2，x＞1或x
2．(2011· 乐山质检)不等式|2x－1|－|x－2|＜0的解集为
__________．
解析：|2x－1|－|x－2|＜0⇔|2x－1|＜|x－2| ⇔(2x－1)2＜(x－2)2⇔4x2－4x＋1＜x2－4x＋4 ⇔3x2＜3⇔－1＜x＜1.
答案：{m|m＜1}
若不等式解集为R，则m的范围为________． 解析：∵－1≤f(x)≤1，f(x)min＝－1，∴m＜－1.
答案：{m|m＜－1}
[归纳领悟] 对形如|x－a|＋|x－b|＜c或|x－a|－|x－b|＞c的不等 式，由于它们分别表示数轴上的点x到a，b点的距离之 和或距离之差，因而利用不等式的几何意义去解不等式， 更为直观、简捷.
(2)|x|>a⇔ x<－a或x>a (a>0)； (3)|f(x)|≤g(x)⇔ －g(x)≤f(x)≤g(x) ； (4)|f(x)|≥g(x)⇔ f(x)≤－g(x)或f(x)≥g(x) ．
[题组自测]
1．(2010· 广西桂林一模)不等式|x＋1|－2＞0的解集是
( A．(－∞，－1)∪(3，＋∞) C．(－∞，－3)∪(1，＋∞) ＜－3. 答案：C B．(－1,3) D.(－3,1) )
1 则当 a＝ 时，原不等式解集为{x|x＞－1，且 x≠2}． 2 1 1 当 0＜a＜ 时，原不等式解集为{x|x＞a或－1＜x＜2}； 2 1 1 当 a＞ 时，原不等式解集为{x|－1<x<a或 x>2}． 2
若将a≥0改为a＜0，其它条件不变，解不等式． 1 解：因 a＜0 时，原不等式等价于(x－a)(x＋1)(x－2)＜0.
简单不等式的解法
简单不等式的解法一、绝对值不等式的解法在解绝对值不等式时，我们需要分类讨论。
假设有一个不等式|a| < b，我们可以将其分解为两个部分，即a < b和-a < b，然后分别求解这两个不等式。
例如：|2x - 3| < 5，我们可以将它分为两个不等式：1) 2x - 3 < 5，解得 x < 4；2) -(2x - 3) < 5，解得 x > -1。
所以，该不等式的解集为-1 < x < 4。
二、分式不等式的解法当我们遇到分式不等式时，我们可以通过消去分母的方式将其化简成为一个多项式不等式。
例如：(x + 3) / (x - 2) ≥ 0，我们可以通过以下步骤解决：1) 确定分式的定义域，即x ≠ 2，因为分母不能为0。
2) 我们可以通过乘法的方式消去分母，得到(x + 3) ≥ 0。
3) 解不等式(x + 3) ≥ 0，得到x ≥ -3。
所以，该分式不等式的解集为x ≥ -3，且x ≠ 2。
三、一次不等式的解法一次不等式是指不等式中只涉及到一次幂的情况，也就是不含有平方项、立方项等高次项。
例如：3x + 5 > 2x - 1，我们可以通过以下步骤解决：1) 整理不等式，将x的系数移到一边，得到 x > -6。
2) 解不等式 x > -6，得到 x > -6。
所以，该一次不等式的解集为 x > -6。
四、二次不等式的解法二次不等式是指不等式中含有二次项的情况，比如 x^2 + 3x - 10 > 0。
解二次不等式的方法有两种：一种是通过绘制图像来求解，一种是通过求解二次函数的根来求解。
例如：x^2 + 3x - 10 > 0，我们可以通过以下步骤解决：1) 求解二次方程 x^2 + 3x - 10 = 0，得到 x = -5 和 x = 2。
2) 绘制出二次函数的图像，根据图像可以确定不等式的解集为 x < -5 或 x > 2。
分式不等式解法
分式不等式解法分式不等式是一个非常重要的数学概念，也是高中数学学习的重要内容之一。
本文将详细介绍分式不等式的解法，包括分式不等式的基本定义、求解分式不等式的步骤、常见的分式不等式类型及其解法等。
一、分式不等式的基本定义分式不等式是由分式式子构成的不等式，其形式如下：$\\frac{f(x)}{g(x)} < a$其中，$f(x)$ 和 $g(x)$ 分别为分式中的分子和分母，$a$ 为常数。
注意：这里假设 $g(x) \eq 0$，否则分式无定义。
二、求解分式不等式的步骤1. 将不等式两边乘上分母 $g(x)$。
2. 化简不等式左边的分式。
3. 将化简后的不等式左边与常数 $a$ 比较大小。
4. 根据不等式的性质确定不等式符号的方向，得到最终的解集。
三、常见的分式不等式类型及其解法1. 一次分式不等式一次分式不等式是指分子和分母都是一次函数的分式不等式，如：$\\frac{x-1}{2x+1}>1$解法：将不等式两边乘上分母 $2x+1$，得到：$x-1>2x+1$移项化简得到：$x<-2$解集为 $(-\\infty,-2)$。
2. 二次分式不等式二次分式不等式是指分子和（或）分母含有二次项的分式不等式，如：$\\frac{x^2+x-2}{x^2+2x+1}<0$解法：将不等式化简得到：$\\frac{(x+2)(x-1)}{(x+1)^2}<0$根据零点的性质，可以将数轴分为以下几段：$x<-2,-2<x<-1,x>1$在每个段内再分别确定原式的符号，并根据符号的性质确定最终的解集。
解集为 $(-\\infty,-2) \\cup (-1,1)$。
3. 无理分式不等式无理分式不等式是指分式中含有无理数的不等式，如：$\\frac{\\sqrt{2}x-1}{x^2+1}>0$解法：将不等式化简得到：$\\frac{\\sqrt{2}x-1}{x^2+1}\\cdot\\frac{\\sqrt{2}x+1}{\\sqrt{2}x+1}>0$$(\\sqrt{2}x-1)(\\sqrt{2}x+1)>0$根据乘积的性质，可以将数轴分为以下几段：$x<-\\frac{\\sqrt{2}}{2},-\\frac{\\sqrt{2}}{2}<x<\\frac{\\sqrt{2}}{2},x>\\frac{\\sqrt{2}}{2}$ 在每个段内再分别确定原式的符号，并根据符号的性质确定最终的解集。
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<。
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高三数学 序号004 高三 年级 班 教师 方雄飞 学生
分式不等式及简单的绝对值不等式的解法
学习目标
1、知识与技能：会求简单的分式不等式、简单的含绝对值不等式以及简单的高次不等式。
2、过程与方法： 通过知识点与实例巩固复习，体会数形结合及转化的思想在解题中的应用
3、情感态度与价值观：培养认真参与、积极交流的主体意识和乐于思考、踏实肯学的精神。
学习重点：绝对不等式与分式不等式的方法与步骤； 难点：注意数形结合和等价转化的思想在解题中的应用 教学过程 一、知识归纳
1、分式不等式的解法 思路是：“分式不等式−−→−转化整式不等式”； 主要方法有：分类讨论、转化为整式
即:
⇔>0)()(x g x f ⇔≥0)()
(x g x f 同理：
⇔<0)()(x g x f ⇔≤0)
()
(x g x f 2、简单的含绝对值不等式
思路是：“去掉绝对值符号”； 方法：平方法、定义法（讨论法）、几何意义法（等价变形） 即：a x a x a a x -<>⇔>>或)0(； a x a a a x <<-⇔><)0( 推广：⇔>≠>-)0,0(c a c b ax c b ax c b ax -<->-或 ⇔>≠<-)0,0(c a c b ax
3、简单的高次不等式的解法：标根法
思路：（1）、把不等式变形为一边是一次因式的积，另一边是0的形式； （2）、各因式中x 的系数全部变为1，约去偶次因式； （3）、把各个根从小到大依次排好标出，从右上方向左下方“穿针引线”； （4）、严格检查因式的根（特别是约去的偶次因式的根）是否在解集内； 二、例题讲解
题型1 分式不等式的解法
例1、解不等式：（1）、1213≥--x x （2）、05
46
52
2>--++x x x x ；
方法点拨1：解分式不等式的步骤：（1）先化为标准型，即
)0(0)
()
(<>或x g x f ； （2）转化为整式不等式； （3）解不等式时应注意“系数符号、不等号的方向以及考虑分母不为零” 练习1、解不等式：（1）、021≤-x （2）、021
2>--x
x
（3）21≤+x x （4）、22
06
x x
x x +<+-
题型2 绝对值不等式的解法
例2、解不等式：（1）392≥-x ； （2）125x x -++<
练习2、解不等式：（1）、12≤+x （2）、311<+<x
（3）、212+>+x x （4）、x x -<-665
（5）432≥++-x x （6）235x x --->
课后作业 1、不等式21
≥+x x
的解集为
2、0)86)(1(2
2
≥+--x x x 的解集为
3、不等式1|1|3x <+<的解集为
4、解下列不等式： （1）、02
32
>-+x x （2）、
2112x x ->-+
（3）、 21
1<≤-x
（4）、321-<x
（5）6431≤+<x （6）221>-+-x x
5、解不等式组：
（1）、⎪⎩⎪⎨⎧>-+>+-21
30862x x x x （2）、111<-+x x。