分式不等式和绝对值不等式(高中数学衔接内容)
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。
突破04 分式不等式与绝对值不等式的解法(重难点突破)(解析版)
突破04 分式不等式与绝对值不等式一、考情分析二、经验分享【重难点01 分式方程与分式不等式】1、分式方程：分母中含有未知数的方程叫做分式方程．(1)分式方程的解法①一般解法：去分母法，即方程两边同乘以最简公分母．②特殊解法：换元法．(2)验根：由于在去分母过程中，当未知数的取值范围扩大而有可能产生增根．因此，验根是解分式方程必不可少的步骤，一般把整式方程的根的值代人最简公分母，看结果是不是零，使最简公分母为零的根是原方程的增根，必须舍去．说明：解分式方程，一般先考虑换元法，再考虑去分母法．2、分式不等式的解法：分母恒为正时可去分母；分母不恒为正时不能去分母，应先移项使右边为0再通分并将分子分母分解因式，最后用标根法求解。
解分式不等式的主旨是化分式不等式为整式不等式，进行求解．3、可化为一元二次方程的分式方程1．去分母化分式方程为一元二次方程2．用换元法化分式方程为一元二次方程简单分式不等式的解法【重难点02 绝对值不等式】 1、实数绝对值的意义 ⎩⎨⎧<-≥=)0()0(||a a a a a2、a>0：①a x a a x a x <<-⇔<⇔<22||②a x a x a x -<⇔>⇔>22||或x>a3、解含有绝对值不等式关键是如何去绝对值符号．对于形如|()|()f x g x ≥和|()|()f x g x ≤的不等式，可利用绝对值的含义去绝对值符号得|()|()f x g x ≥⇔()()f x g x ≥或()()f x g x ≤；|()|()f x g x ≤⇔()()()g x f x g x -≤≤．三、题型分析(一) 分式方程与分式不等式的解法 例1、解方程21421224x x x x +-=+--． 【分析】：去分母，转化为整式方程． 【解析】：原方程可化为：14212(2)(2)2x x x x x +-=++--，方程两边各项都乘以24x -： 2(2)42(2)4x x x x -+-+=-即2364x x -=-， 整理得：2320x x -+=解得：1x =或2x =．检验：把1x =代入24x -，不等于0，所以1x =是原方程的解；把2x =代入24x -，等于0，所以2x =是增根．所以，原方程的解是1x =． 【点睛】：(1) 去分母解分式方程的步骤：①把各分式的分母因式分解； ②在方程两边同乘以各分式的最简公分母； ③去括号，把所有项都移到左边，合并同类项； ④解一元二次方程； ⑤验根．(2) 验根的基本方法是代入原方程进行检验，但代入原方程计算量较大．而分式方程可能产生的增根，就是使分式方程的分母为0的根．因此我们只要检验一元二次方程的根，是否使分式方程两边同乘的各分式的最简公分母为0．若为0，即为增根；若不为0，即为原方程的解．【变式训练1】解方程 2223()4011x x x x --=--【分析】：本题若直接去分母，会得到一个四次方程，解方程很困难．但注意到方程的结构特点，设21x y x =-，即得到一个关于y 的一元二次方程．最后在已知y 的值的情况下，用去分母的方法解方程21x y x =-． 【解析】：设21x y x =-，则原方程可化为：2340y y --= 解得4y =或1y =-．(1)当4y =时，241x x =-，去分母，得224(1)4402x x x x x =-⇒-+=⇒=； (2)当1y =-时，22215111012x x x x x x x -±=-⇒=-+⇒+-=⇒=-． 检验：把各根分别代入原方程的分母，各分母都不为0． 所以，2x =，152x -±=都是原方程的解． 例2．不等式302x x -<-的解是__________． 【答案】23x << 【解析】不等式302x x -<-等价于30{20x x ->-<或30{ 20x x -<->解得23x << 【变式训练2】不等式的解为____________．【答案】【解析】不等式化为，解一元二次不等式即可．不等式化为，解得，∴不等式的解集为，故答案为．【变式训练3】不等式的解为______．【解析】．点睛：解分式不等式的方法是：移项，通分化不等式为，再转化为整式不等式，然后利用二次不等式或高次不等式的结论求解．【变式训练4】不等式 501xx -≥-的解是__________． 【答案】15x <≤ 【解析】原不等式化为550,011x x x x -+-≥≤--，解得15x <≤． (二) 绝对值不等式的解法例3．（1）、不等式15x -≤的解集为__________． 【答案】[]4,6- 【解析】15,515x x -≤∴-≤-≤，解得46,x -≤≤∴原不等式的解集为[]4,6-，故答案为[]4,6-．（2）、已知的解集是，则实数，的值是（ ） A ．，B ．，C ．，D ．，【答案】D【解析】分析：先解不等式，再列方程组得实数a ，b 的值．由题得-b ＜x-a ＜b ，所以a-b ＜x ＜a+b ， 因为的解集是，所以a-b=-3且a+b=9，所以a=3，b=6．故答案为：D【点睛】(1)本题主要考查绝对值不等式的解法，意在考查学生对该基础知识的掌握能力． (2)绝对值不等式|ax+b|<c 等价于-c ＜ax+b ＜c ． |ax+b|＞c 等价于ax+b>c 或ax+b ＜-c ． 【变式训练1】关于的不等式在上恒成立，则的取值范围是__________．【答案】【解析】结合自变量的范围，若，可得：，不等式明显成立；若，由不等式可得，解得：，综上可得的取值范围是．例4．若关于x 的不等式20k x x -->恰好有4个整数解，则实数k 的取值范围是（ ）A ． 32,53⎛⎫⎪⎝⎭ B ． 32,53⎛⎤⎥⎝⎦ C ． 3,15⎛⎫⎪⎝⎭ D ． 3,15⎛⎤ ⎥⎝⎦【答案】B【 方法点睛】本题主要考查绝对值不等式的解法、排除法解选择题，属于难题． 用特例代替题设所给的一般性条件，得出特殊结论，然后对各个选项进行检验，从而做出正确的判断，这种方法叫做特殊法． 若结果为定值，则可采用此法． 特殊法是“小题小做”的重要策略，排除法解答选择题是高中数学一种常见的解题思路和方法，这种方法即可以提高做题速度和效率，又能提高准确性，这种方法主要适合下列题型:(1)求值问题（可将选项逐个验证）；（2）求范围问题（可在选项中取特殊值，逐一排除）；（3）图象问题（可以用函数性质及特殊点排除）；（4）解方程、求解析式、求通项、求前n项和公式问题等等．【变式训练1】．的解集为（）A．B．C．D．【答案】A【解析】：很明显，则不等式等价于：，解不等式组可得实数x的取值范围是：．本题选择A选项．四、迁移应用1．分式方程23122xx x+=--的解为：（ ） A 、1 B 、2 C 、13D 、0【答案】A【解析】根据分式方程的解法：去分母，得2-3x=x-2，移项后解得x=1，检验x=1是原分式方程的根． 答案为A2． 用换元法解方程22124312x x x x --=-时，设212x y x-=，则原方程可化为（ ） A ．130y y --= B ．430y y --= C ．130y y -+= D ．430y y-+= 【答案】B ．【分析】直接利用已知将原式用y 替换得出答案．【解析】∵设212x y x -=，∴22124312x x x x --=-，可转化为：43y y -=，即430y y--=．故选B ． 3．不等式32x x->的解集是（ ） A ． {}|1 3 x x x -或 B ． {}|10 3 x x x -<或C ． {}|10 3 x x x <-<<或D ． {}|100 3 x x x -<<<<或 【答案】B【解析】1x =时， 22->不成立，可排除,C D ，2x =-时，122->不成立，可排除A ，故选B ． 4．不等式3112x x -≥-的解集是（ ） A ． 3{|2}4x x ≤≤ B ． 3{|2}4x x ≤< C ． {2x x 或3}4x ≤ D ． {}2x x【答案】B 【解析】31102x x --≥-， 31202x x x --+≥-， 4302x x -≥-， ()()4320{2x x x --≤≠ ， 324x ≤<，选B ．5.解下列不等式：(1)2301x x -<+ (2)2301x x x +≥-+ 【分析】：(1) 类似于一元二次不等式的解法，运用“符号法则”将之化为两个一元一次不等式组处理；或者因为两个数(式)相除异号，那么这两个数(式)相乘也异号，可将分式不等式直接转化为整式不等式求解． (2) 注意到经过配方法，分母实际上是一个正数． 【解析】：(1) 解法(一)原不等式可化为：3323023031221010211x x x x x x x x x ⎧⎧-<-><>⎧⎧⎪⎪⇒⇒-<<⎨⎨⎨⎨+>+<⎩⎩⎪⎪>-<-⎩⎩或或解法(二)原不等式可化为：3(23)(1)012x x x -+<⇒-<<． (2) ∵ 22131()024x x x -+=-+>，原不等式可化为：303x x +≥⇒≥- 6．方已知关于x 的分式方程111k x kx x ++=+-的解为负数，则k 的取值范围是 ．【答案】k ＞12-且k≠0．7．关于x 的两个方程260x x --=与213x m x =+-有一个解相同，则m= ． 【答案】﹣8．【解析】解方程260x x --=得：x=﹣2或3； 把x=﹣2或3分别代入方程213x m x =+-，当x=﹣2时，得到21223m =-+--，解得m=﹣8． 故答案为：﹣8． 8．解方程：2717=---xx x ． 【答案】x=15．【解析】分式方程去分母转化为整式方程，求出整式方程的解得到x 的值，经检验即可得到分式方程的解． 去分母得：x+1=2x ﹣14，解得：x=15，经检验x=15是分式方程的解． 9．若关于x 的分式方程121k x -=+的解为负数，则k 的取值范围为 ． 【答案】k ＜3且k ≠1．【分析】分式方程去分母转化为整式方程，表示出整式方程的解，根据解为负数确定出k 的范围即可． 【解析】去分母得：k ﹣1=2x +2，解得：x =32k -，由分式方程的解为负数，得到32k -＜0，且x +1≠0，即32k -≠﹣1，解得：k ＜3且k ≠1，故答案为：k ＜3且k ≠1．10．分式方程2110051025x xx的解是 ．【答案】15x =．【解析】去分母得：5100x --=，解得：15x =，经检验15x =是分式方程的解．故答案为：15x =．。
第二讲：各类不等式的解法
初高中数学衔接教材第二讲各类不等式的解法教师版一、含绝对值不等式【引例】（1）2||<x （2）3||<x Guess ：a x <||（3）1||-<x【点评】公式：①a x a a a x <<-⇔><)0(||；②a x a x a a x ≥-≤⇔>≥或)0(||.【注】①“<”对着“两根之间”；“>”对着“两根之外”；②若a 的符号不定，要分类讨论.1、3|2-|2<x x 练习：1|32|2≥+x x二、含一个偶次根式的不等式【引例】（1）2112≥-x （2）2112-≥-x 【点评】解含一个偶次根式的不等式的步骤：S1：对不含偶次根式的式子分“0≥”和“0<”两类；S2：“0≥”两边平方、“0<”考虑式子“有意义”；S3：综上，两类答案取“并”.1、x x -≥-12 练习：11-≥+x x三、一元二次不等式1、解一元二次方程（1）0652=-+x x （2）0142=++x x （3）032-2=+x x【点评】（1）（2）解一元二次方程的步骤：S1：能否十字相乘（十字分解）；S2：若不能，用求根公式（3）不是所有的一元二次方程都能用求根公式：①0>∆，方程有两个不等的实根21,x x ；②……；③……2、画出函数652-+=x x y 的图象.【点评】二次函数画图象关注：①开口——二次项系数；②对称轴——配方法；③点（端点值）——与x ，y 轴的交点、已知点等；④∆3、解一元二次不等式：①0652>-+x x ；②0652≤-+x x ；③0652>+--x x【点评1】三个“二次关系”：【点评2】解一元二次不等式步骤：S1：二次项系数变为正（变正）；S2：求对应一元二次方程的根（求根）——方法：十字分解或求根公式；S3：画图象（定图）——熟悉后可省；S4：决定“两根之外”还是“两根之间”（判断）——只要开口向上，“>”对着“两根之外”；“<”对着“两根之间”. 注意：解一元二次不等式的目标：确定“根的大小”和“解集形式”.练习11、解不等式：①81032≤-x x ；②0142<++x x2、解不等式组：⎪⎩⎪⎨⎧≤-<++810301422x x x x . 答案：)23,32[--四、高次不等式1、01323≥++-x x x 练习：02-7523<+-x x x【点评】解高次不等式步骤：S1：猜根（试根）；S2：竖式除法（短除法）——因式分解；S3：穿针引线——口诀奇穿偶不穿（注意空心画圈，实心画点）；S4：下结论.五、分式不等式1、【引例】（1）11>x （2）01272<-+-x x （3）043>--x x（1）1472>--x x （2）11≥x【点评】解分式不等式的步骤：S1：移项；S2：通分；S3：化商为积；S4：变负为正；S5：穿针引线——口诀奇穿偶不穿（注意空心画圈，实心画点）；S6：下结论.（3）1232≥-+xx练习：1、05)4()3(2≥---xx x 2、1232≤--x x 3、六、含参不等式1、0)1(2>++-a x a x2、)0(01)1(2><++-a x a ax练习：1、0222≤--a ax x2、01)1(2<++-x a ax【点评】①高中考到“含参”题，有两个考向：①分类讨论；②给定等量条件，求出参数的值. ②解一元二次含参不等式的步骤：S1：因式分解（十字分解/求根公式）；S2：按根有无意义分类；S3：按开口分类；S4：按根比大小分类；S5：根据“开口”和“两根大小”确定解集形式.【反思收获】。
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。
初升高数学衔接课程(15节)
初升高数学衔接课程（例题+练习+习题+答案）1、一元二次不等式2、分式不等式3、绝对值不等式4、集合的含义与表示5、集合间的基本关系6、集合的基本运算7、映射与函数8、分式函数9、函数定义域10、函数值域11、函数单调性12、函数奇偶性13、函数解析式14、二次函数在闭区间上的最值15、集合与函数测试制作人：梁林庆时间：2015－7－1１、一元二次不等式１、１ 知识１、定义：含有一个未知数，并且未知数的最高次数是二次的不等式叫做一元二次不等式。
２、解一元二次不等式的步骤：（１）把二次项系数变为正，令一元二次不等式＝０，得到一元二次方程； （２）解一元二次方程得到两根（一根或无根）；（３）根据不等号判断取值范围。
（若>，两根之外，若<，两根之间）。
１、２ 例题例１、 解下列不等式１、02532>-+x x ２、01692>+-x x ３、0542>+-x x４、0122<++-x x ５、0442>-+-x x例２、 已知不等式012<-+bx ax 的解集是{}43|<<x x ，求实数a,b 的值。
例３、 解关于x 的不等式 0)12(22<+++-m m x m x例４、 解关于x 的不等式 0)1(2<--+a x a x１、解下列不等式（１）03422<++x x （２）08232≤+--x x （３）21618x x ≥-（４） ()()410x x +--<； （５）232x x -+>； （６）24410x x -+>．２、已知一元二次不等式210ax bx ++>的解集为113x x ⎧⎫-<<⎨⎬⎩⎭，求实数ab 的值。
３、若不等式210x mx ++>的解集为R ，求m 的取值范围。
解下列一元二次不等式1.03282>--x x2.031082≥-+x x3.041542<--x x4.02122>--x x5.021842>-+x x6.05842<--x x7.0121752≤-+x x 8.0611102>--x x 9.038162>--x x10.038162<-+x x 11.0127102≥--x x 12.02102>-+x x２、分式不等式２、１知识１、定义：分母中含有未知数的不等式叫做分式不等式。
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<。
高二不等式知识点总结
高二不等式知识点总结不等式是数学中一种重要的关系式，它描述了两个数或两个式子之间的大小关系。
在高二阶段学习数学时，不等式是必不可少的知识点之一。
本文将对高二阶段学习的不等式知识点进行总结和概述。
一、一元一次不等式1. 不等式的定义：不等式是含有不等号（<、>、≤、≥）的数学式子。
2. 不等式的解：解不等式可以通过移项和绘制数轴的方法。
解集通常用区间表示。
3. 不等式的性质：不等式在两边同时加上一个相等的数或者在两边同时乘以一个正数时，不等关系不变；在两边同时乘以一个负数时，不等关系会颠倒。
4. 一元一次不等式的解法：考虑到正负数以及系数的情况，可以分为以下几种情况进行讨论。
二、一元二次不等式1. 一元二次不等式的定义：一元二次不等式是含有平方项的不等式。
2. 一元二次不等式的解法：可通过化为标准形式，配方法或绘制图像等方式进行求解，解集常用区间来表示。
3. 一元二次不等式的性质：与一元一次不等式类似，需要注意平方项对不等式性质的影响。
三、绝对值不等式1. 绝对值不等式的定义：绝对值不等式是含有绝对值的不等式。
2. 绝对值不等式的解法：可通过绝对值的定义以及正负号的讨论来解决。
四、分式不等式1. 分式不等式的定义：分式不等式是含有分式的不等式。
2. 分式不等式的通解：利用分式不等式的定义，可通过化简、拆分分式等方式求得通解。
五、不等式组1. 不等式组的定义：含有多个不等式的组合形式。
2. 不等式组的解法：可通过图示法、代入法、消元法等不同的方法求解。
六、不等式的应用1. 不等式在数学问题中的应用：不等式常常被应用于解决实际问题，如优化问题、约束条件等。
2. 不等式在证明中的应用：不等式在数学证明中具有重要的作用，可通过不等式进行推导、化简等。
综上所述，高二阶段的不等式知识点主要包括一元一次不等式、一元二次不等式、绝对值不等式、分式不等式、不等式组等内容。
掌握这些知识点对高中数学的学习以及今后的学习和工作都具有重要的意义。
高一常见基本不等式知识点
高一常见基本不等式知识点不等式在高中数学中占据着非常重要的地位，它是代数与几何的重要桥梁之一。
本文将介绍高一阶段常见的基本不等式知识点，帮助同学们更好地理解和掌握这一内容。
1.直观理解首先，我们来简单回顾一下不等式的基本概念。
不等式是数学中描述两个数之间大小关系的一种符号表示方法。
比如，我们用"<"表示“小于”，用">"表示“大于”，用"≤"表示“小于等于”，用"≥"表示“大于等于”等。
在解不等式的过程中，我们通常需要找出满足不等式的数的取值范围。
2.一元一次不等式一元一次不等式是指只有一个未知数并且次数为一次的不等式。
例如，"2x+3>7"就是一个一元一次不等式。
解这类不等式的方法和解一元一次方程类似，主要是通过移项和化简来求解。
3.一元二次不等式一元二次不等式是指只有一个未知数并且次数为二次的不等式。
例如，"x^2-3x+2>0"就是一个一元二次不等式。
解这类不等式一般需要借助图像或者计算得到解集的范围。
4.绝对值不等式绝对值不等式是指不等式中含有绝对值符号的情况。
例如，"|x+1|<5"就是一个绝对值不等式。
解绝对值不等式的方法一般是将绝对值不等式拆分成两个简单的不等式来求解。
5.分式不等式分式不等式是指不等式中含有分式的情况。
例如，"1/(x+2)>3"就是一个分式不等式。
解分式不等式的关键是确定各个分式的定义域，并利用分式的性质进行求解。
6.不等式的性质在解不等式过程中，我们还需要掌握一些不等式的性质，以便更好地进行推导。
其中一些常见的不等式性质包括：- 两边加上（或减去）相同的数（或式子）时，不等号方向不变。
- 两边乘以（或除以）相同的正数时，不等号方向不变。
- 两边乘以（或除以）相同的负数时，不等号方向相反。
初高中数学衔接知识点
初高中数学衔接知识点从初中升入高中，数学学科的知识难度和深度都有了明显的提升。
为了帮助同学们更好地适应高中数学的学习，下面我们来梳理一下初高中数学衔接的重要知识点。
一、数与式1、绝对值初中阶段，我们对绝对值的理解主要是基于数轴上的距离。
例如，｜3| ＝ 3，｜－3| ＝ 3。
但在高中，绝对值的概念会被更深入地运用，例如在求解不等式|x 2| ＞ 5 时，需要分情况讨论 x 2 的正负，得到 x ＜－3 或 x ＞ 7。
2、二次根式初中我们学习了二次根式的基本运算，如化简、乘法法则和除法法则。
高中会在此基础上，结合函数、不等式等知识进行更复杂的运算和应用。
3、因式分解初中常见的因式分解方法有提公因式法、公式法（平方差公式、完全平方公式）。
高中数学中，因式分解的应用更加广泛，有时需要使用十字相乘法、分组分解法等更复杂的方法来分解因式，以解决方程和不等式的问题。
二、方程与不等式1、一元二次方程初中我们重点学习了一元二次方程的求解方法，如配方法、公式法和因式分解法。
高中则会更多地关注一元二次方程根与系数的关系（韦达定理），以及利用一元二次方程解决实际问题和函数问题。
2、不等式初中主要学习了一元一次不等式的解法。
高中会拓展到一元二次不等式、简单的分式不等式和绝对值不等式。
例如，求解不等式 x² 2x 3 ＜ 0，需要先求出方程 x² 2x 3 ＝ 0 的根，然后根据函数图象的开口方向和与 x 轴的交点来确定不等式的解集。
三、函数1、函数的概念初中对于函数的定义是基于变量之间的对应关系。
高中则会从集合的角度来重新定义函数，使函数的概念更加严谨和抽象。
2、一次函数与反比例函数初中我们对一次函数和反比例函数的性质有了一定的了解。
高中会在这些基础上，进一步研究它们的图象和性质，并与其他函数进行综合应用。
3、二次函数初中主要学习了二次函数的基本表达式、图象和简单的应用。
高中会深入探讨二次函数的最值问题、与一元二次方程和不等式的关系，以及二次函数在实际生活中的优化问题。
高中数学不等式知识点归纳
高中数学不等式知识点归纳
高中数学不等式知识点归纳主要包括以下几个方面：
1. 不等式的概念和性质：不等式是数学中比较基础的概念，它表示两个数之间的大小关系。
不等式的性质包括：对称性、传递性、加法法则、乘法法则等。
这些性质在解决不等式问题时非常重要。
2. 一元一次不等式：一元一次不等式是只含有一个未知数，且未知数的次数为1的不等式。
解决这类不等式问题，可以通过移项、合并同类项、化系数为1等方法，将其转化为一元一次方程，然后求解。
3. 一元二次不等式：一元二次不等式是含有一个未知数，且未知数的最高次数为2的不等式。
解决这类不等式问题，可以通过因式分解、配方、判别式等方法，将其转化为一元二次方程，然后求解。
4. 分式不等式：分式不等式是含有分式的不等式。
解决这类不等式问题，可以通过通分、分子分母同号或异号等方法，将其转化为整式不等式，然后求解。
5. 绝对值不等式：绝对值不等式是含有绝对值符号的不等式。
解决这类不等式问题，可以通过绝对值的定义，将其转化为分段函数，然后分别求解每一段的情况。
6. 不等式的应用：不等式在实际生活中有广泛的应用，如优化问题、最值问题、范围问题等。
在解决这些问题时，需要根据问题的实际情况，建立相应的不等式模型，然后求解。
以上是高中数学不等式知识点的主要归纳，希望对你有所帮助。
初高中衔接内容
初高中数学衔接教材现有初高中数学教材存在以下“脱节”：1、绝对值型方程和不等式，初中没有讲，高中没有专门的内容却在使用；2、立方和与差的公式在初中已经删去不讲，而高中还在使用；3、因式分解中，初中主要是限于二次项系数为1的二次三项式的分解，对系数不为1的涉及不多，而且对三次或高次多项式的分解几乎不作要求；高中教材中许多化简求值都要用到它，如解方程、不等式等；4、二次根式中对分子、分母有理化初中不作要求，而分子、分母有理化是高中数学中函数、不等式常用的解题技巧；5初中教材对二次函数的要求较低，学生处于了解水平。
而高中则是贯穿整个数学教材的始终的重要内容；配方、作简图、求值域（取值范围）、解二次不等式、判断单调区间、求最大最小值、研究闭区间上的函数最值等等是高中数学所必须掌握的基本题型和常用方法；6、二次函数、二次不等式与二次方程之间的联系，根与系数的关系（韦达定理）初中不作要求，此类题目仅限于简单的常规运算，和难度不大的应用题，而在高中数学中，它们的相互转化屡屡频繁，且教材没有专门讲授，因此也脱节；7、图像的对称、平移变换初中只作简单介绍，而在高中讲授函数时，则作为必备的基本知识要领；8、含有参数的函数、方程、不等式初中只是定量介绍了解，高中则作为重点，并无专题内容在教材中出现，是高考必须考的综合题型之一；9、几何中很多概念（如三角形的五心：重心、内心、外心、垂心、旁心）和定理（平行线等分线段定理、平行线分线段成比例定理、射影定理、相交弦定理）初中早就已经删除，大都没有去学习；10、圆中四点共圆的性质和判定初中没有学习。
高中则在使用。
另外，象配方法、换元法、待定系数法、双十字相乘法分解因式等等等等初中大大淡化，甚至老师根本没有去延伸发掘，不利于高中数学的学习。
新的课程改革，难免会导致很多知识的脱节和漏洞。
本书当然也没有详尽列举出来。
我们会不断的研究新课程及其体系。
将不遗余力地找到新的初高中数学教材体系中存在的不足，加以补充和完善。
高中4个基本不等式链
高中4个基本不等式链在高中数学学习中，基本不等式链是重要的概念之一。
它们是一系列基本不等式的组合，可以帮助我们解决各种数学问题。
本文将介绍高中阶段常见的4个基本不等式链，并探讨其应用。
1. 一次不等式链一次不等式链是最基本的不等式链形式。
它由一次不等式构成，即形如ax+b>0的不等式，其中a和b为实数，且a不等于零。
该类不等式链在解决实数范围内的一元一次不等式问题时非常有用。
例如，对于不等式3x + 4 > 0，可以通过求解一次不等式链来确定不等式的解集。
2. 二次不等式链二次不等式链由二次不等式构成，即形如ax^2 + bx + c > 0的不等式，其中a、b和c为实数，且a不等于零。
二次不等式链常用于解决实数范围内的一元二次不等式问题。
通过分析二次函数的图像并结合一次不等式的解法，可以确定不等式的解集。
例如，对于不等式x^2 -4 < 0，可以通过解二次不等式链来求解。
3. 绝对值不等式链绝对值不等式链由绝对值不等式构成，即形如|ax + b| > c的不等式，其中a、b和c为实数，且a不等于零。
绝对值不等式链常用于解决实数范围内的一元绝对值不等式问题。
通过分析绝对值函数的性质及符号的变化情况，可以确定不等式的解集。
例如，对于不等式|2x - 3| < 5，可以通过解绝对值不等式链来求解。
4. 分式不等式链分式不等式链由分式不等式构成，即形如f(x) > 0或f(x) < 0的不等式，其中f(x)为一个分式函数。
分式不等式链常用于解决实数范围内的分式不等式问题。
通过分析分式函数的性质及分母和分子的正负情况，可以确定不等式的解集。
例如，对于不等式(x - 2)/(x + 1) > 0，可以通过解分式不等式链来求解。
综上所述，高中阶段的数学学习中，我们经常遇到不等式问题。
掌握基本不等式链的概念和应用方法，可以帮助我们解决各种不等式问题。
一次不等式链、二次不等式链、绝对值不等式链和分式不等式链是常见的不等式链形式，通过对不等式的拆解和分析，我们能够准确求解不等式的解集。
数学初高中衔接绝对值不等式的解法与简单分式不等式解法
解：
3x-2
转化为
x 1 -2 0, 3x-2
整理，得
5x 5 0, 3x-2
即
(x 1)(3x 2) 0,
故不等式的解为 2 x 1 3
解法小结
ax b k (a ' x b ') 0
cx d
(cx d )
ax b k (a ' x b ') 0
cx d
(cx d )
(2) | x 2 | 1 33
(4) | 3 2x | 7
(1) 3 x 1 (2)x 1或x 1
44
3
(3)x 2或x 2（4）x 5或x 2 5
简单分式不等式的解法
引入：解不等式： x 1 0. 3x 2
分析：当且仅当分子 x 与1 分母 3x 2
同号时， 上述不等式成立.
式的解:
① 不等式|x|<c的解为-c<x<c
-c
0
c
② 不等式|x|>c的解为x<-c或x>c
-c
0
c
思考：若c 0呢？
一般地，可得规律: 题型2：形如|ax+b|<c和|ax+b|>c (c>0)的含绝对值
的不等式的解:
① 不等式|ax+b|<c的解为-c<ax+b<c
-c
0
c
② 不等式|ax+b|>c的解为ax+b<-c或ax+b>c
-c
0
c
基础练习： 解下列不等式： （1）|x|>5 （2）2|x|<5
（3）|2x|>5
高中阶段的不等式
高中阶段的不等式在高中阶段的数学学习中，我们经常会接触到不等式这个概念。
不等式是数学中一种比较大小关系的表示方式，它与等式不同，不要求两边的值相等，而是通过符号来表示大小关系。
下面我们来探讨一下高中阶段的不等式。
一、一元一次不等式一元一次不等式是高中阶段学习的第一种不等式形式。
它的形式通常为ax + b > 0或ax + b < 0，其中a和b为实数，x为未知数。
解一元一次不等式的关键是确定不等式中x的取值范围。
我们可以通过构建不等式的解集来解决这个问题。
比如，对于不等式2x + 1 > 0，我们可以将其转化为方程2x + 1 = 0，然后解得x = -1/2。
由于2x + 1是一个一次函数，它的图像是一条直线，我们可以根据直线的斜率和截距来确定函数的正负性。
当x < -1/2时，2x + 1 < 0；当x > -1/2时，2x + 1 > 0。
因此，不等式2x + 1 > 0的解集为x > -1/2。
二、一元二次不等式一元二次不等式是高中阶段学习的第二种不等式形式。
它的形式通常为ax^2 + bx + c > 0或ax^2 + bx + c < 0，其中a、b和c为实数，x为未知数。
解一元二次不等式的关键是确定不等式的解集。
我们可以通过求解一元二次方程来解决这个问题。
比如，对于不等式x^2 - 3x - 4 > 0，我们可以将其转化为方程x^2 - 3x - 4 = 0，然后求解得到x = -1和x = 4。
由于x^2 - 3x - 4是一个二次函数，它的图像是一个抛物线，我们可以根据抛物线的开口方向和顶点位置来确定函数的正负性。
当x < -1或x > 4时，x^2 - 3x - 4 > 0。
因此，不等式x^2 - 3x - 4 > 0的解集为x < -1或x > 4。
三、绝对值不等式绝对值不等式是高中阶段学习的第三种不等式形式。
高一数学高级不等式知识点
高一数学高级不等式知识点在高中数学中，不等式是一个非常重要的概念和工具。
不等式不仅存在于代数和几何中，还涉及到实际问题的建模和解决。
在高一数学中，学生们掌握了基本的不等式知识后，接下来将会学习高级的不等式知识。
本文将介绍高一数学高级不等式的一些重要知识点。
1. 绝对值不等式绝对值不等式是高级不等式中的一个重要概念。
它可以通过解决问题中的绝对值关系来确定变量的取值范围。
常见的绝对值不等式有：- |x| < a- |x| > a- |x| <= a- |x| >= a解决绝对值不等式时，可以利用绝对值函数性质、图像和特殊情况进行分析和推理。
同时，注意要正确地对于绝对值进行判断和分析。
2. 幂函数不等式幂函数不等式是高级不等式中比较常见的一类。
它可以通过幂函数的性质和图像来求解。
常见的幂函数不等式有：- x^a < b- x^a > b- x^a <= b- x^a >= b求解幂函数不等式时，可以利用幂函数的单调性、奇偶性、图像和特殊情况进行分析和推理。
3. 分式不等式分式不等式在数学中也是一类比较常见的不等式。
它可以通过分式的性质和图像来求解。
常见的分式不等式有：- (x + a)/(x + b) < 0- (x + a)/(x + b) > 0- (x + a)/(x + b) <= 0- (x + a)/(x + b) >= 0求解分式不等式时，可以利用分式的性质、图像和特殊情况进行分析和推理。
同时，注意要对分式的分母进行判断和分析。
4. 复合不等式复合不等式是由两个或多个不等式通过逻辑运算（如与、或、非等）组合而成的不等式。
在解决复合不等式时，需要考虑逻辑运算的优先级和运算规则。
常见的复合不等式形式有：- 不等式1并且不等式2- 不等式1或者不等式2- 不等式1与不等式2同时满足在解决复合不等式时，可以利用逻辑运算的概念、不等式的性质和图像来进行分析和推理。
#预备知识(二次函数及不等式,分式不等式,绝对值不等式)
1、一元二次方程：20(0)ax bx c a 求根公式：24.2b b acxa2=4b ac其中：.0有两个不相等的实数根；=0有两个相等的实数根；0无实数根.根与系数关系（韦达定理）12bx xa. 12cx xa.例题1：2x2＋5x－3＝0解：∵Δ＝49>0，∴方程2x2＋5x－3＝0有两个实数根，解得x1＝－3，x2＝1 2，例题2：9x2－6x＋1＝0解：∵Δ＝0，∴方程9x2－6x＋1＝0有两个相等的实数根，解得x1＝x2＝1 3 .2、一元二次不等式：1）判断根的个数2=4b ac ∆-，求根242bb acx a .2）根据开口方向画图.3）图像在x 轴上方的函数值大于0；图像在x 轴下方的函数值小于0.例题1：－x 2＋2x －23>0解：(1)两边都乘以－3，得3x 2－6x ＋2<0，因为3>0，且方程3x 2－6x ＋2＝0的解是x 1＝1－33，x 2＝1＋33，所以原不等式的解集是⎩⎨⎧⎭⎬⎫x |1－33<x <1＋33.例题2：x 2－6x ＋10<0解：原不等式可化为x 2－6x ＋10<0，∵Δ＝－4<0，∴方程x 2－6x ＋10＝0无实数根， 画出函数y ＝x 2－6x ＋10的图象如图④所示，由图象可得原不等式的解集为∅.3、二次函数恒成立 二次函数恒成立问题：可以通过开口与判别式Δ；或者开口与最值判断.例题1：不等式mx 2＋mx ＋1>0对一切x ∈R 恒成立，则实数m 的取值范围是________. 答案 [0,4)解：当m ＝0时，显然成立；当m ≠0时，由已知得⎩⎨⎧m >0，Δ＝m 2－4m <0，解得0<m <4.综上知，实数m 的取值范围是[0,4)．例题2：已知关于x 的不等式kx 2－6kx ＋k ＋8≥0对任意x ∈R 恒成立，则k 的取值范围是( )A ．[0,1]B ．(0,1]C ．(－∞，0)∪(1，＋∞)D ．(－∞，0]∪[1，＋∞) 答案 A解：当k ＝0时，不等式kx 2－6kx ＋k ＋8≥0可化为8≥0，其恒成立；当k ≠0时，要满足关于x 的不等式kx 2－6kx ＋k ＋8≥0对任意x ∈R 恒成立，只需⎩⎨⎧k >0，Δ＝36k 2－4k k ＋8≤0，解得0<k ≤1.综上，k 的取值范围是[0,1]．4、分式不等式1）()0(0)()()0(0)()f x f x g x g x2）()0(0)()()0(0)()()f x f xg x g x g x 且注：（注意不等号右边要化为0.）例题1：不等式3x －12－x< 0的解集是 . 解： X<13或x>2例题2：不等式3x －12－x≥1的解集是( ) A.⎩⎪⎨⎪⎧⎭⎪⎬⎪⎫x |34≤x ≤2 B.⎩⎪⎨⎪⎧⎭⎪⎬⎪⎫x |34≤x <2 C.⎩⎪⎨⎪⎧⎭⎪⎬⎪⎫x |x >2或x ≤34 D.⎩⎪⎨⎪⎧⎭⎪⎬⎪⎫x |x ≥34答案 B解：3x －12－x ≥1⇒4x －32－x ≥0⇒34≤x <2.故选B.5、简单的绝对值不等式1）()()()()()f x g x g x f x g x ()()()()()()f x g x f x g x f x g x >⇔><-或.2）22()()[()][()]f x g x f x g x .例题1：11x -≤解：0≤x ≤2例题2：+21x x >-解：x ≥ - 12。
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新高一衔接讲义
分式不等式和绝对值不等式
知识详解
一、分式不等式概念
1．分式不等式的概念：分母中含有未知数的不等式称为分式不等式。
2. 各种分式不等式经过变形都可化为标准形式a
b >0(a
b
≥0)或a
b
<0(a
b
≤0)，
其中a,b分别为整式，且b≠0。
二、分式不等式解法
解分式不等式的思路:化为标准形式，变形为整式不等式求解。
【例1】解不等式
（1）x−3
x+7<0（2）x+1
3x−3
>0（3）2+7x
x−9
≥0
（4）3x+4
x−1≤0（5）2x
x−1
≤1（6）x−4
3−x
≥1
【变式1.1】不等式x－1
2x＋1≤0的解集为___________
【变式1.2】与不等式x−3
2−x
≥0同解的不等式是（）
A．(x－3)(2－x)≥0B．0＜x－2≤1C．2−x
x−3
≥0D．(x－3)(2－x)≤0
【变式1.3】（1）2x−15
x+2≤0（2）3x−2
2x
≥1
【例2】解下列不等式
（1）0)23)(1(2<+--x x x （2）
0)
3)(2(1>---x x x
【变式2.1】（1）2
1222-≤++-x x （2）x−5x 2+9x+18<0
【提高习题】
1．不等式1+x >
11−x 的解集为___________
2．不等式ax x−1<1的解集为x <1或x >2，则a 的值为___________
3．解不等式3x−7x 2+2x−3≥2
三、含有绝对值的不等式
1．绝对值的定义及性质
绝对值：在数轴上表示数a 的点到原点的距离叫做a 的绝对值。
绝对值的性质：|a |={a (a >0)
0 (a =0)−a (a <0)
绝对值的几何意义：
2．含绝对值不等式的解法
探究不等式|x |<1的解集。
方法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。
所以，不等式|x|<1的解集为−1<x <1
方法2：对原不等式两边平方得x 2<1，即 x 2－1<0，
即 (x+1)(x －1)<0，−1<x <1
思考：|x |<3.6的解集________________；|x |<√2的解集______________.
|x |>34
的解集__________________；|x |>2√3的解集______________.
【例3】解下列不等式
（1）|3x −9|<4 （2）|4x −5|>1
（3）|x +1|−|4−x|<0 （4）|2x −8|−|x −4|>0
【变式3.1】（1）|√2x −3|<1 （2）|34x +5|<4
（3）|x −1|<|9−x| （4）|x −3|−|2−x|<0
※注意：解含绝对值不等式的三种常用思路
（1）利用绝对值的几何意义观察
（2）利用绝对值的定义去掉绝对值符号，需要分类讨论
（3）两边同时平方去掉绝对值符号
小结：形如不等式|x|<a 和|x|>a (a >0)的解集为
（1）不等式|x|<a 的解集为 −a <x <a ； （2）不等式|x|>a 的解集为x <−a 或x >a 。
【例4】（1）解不等式|x−8|−|x−4|>2（2）解不等式|x +1| + |3−x|>2 + x.【变式4.1】解不等式|x−3|−|x+1|<1习题：解不等式| x −1 | + | 2x−4 |>3 + x 总结（1）解含绝对值的不等式的关键是要去掉绝对值的符号，其基本思想是把含绝对值的不等式转为不含绝对值的不等式。
（2）零点分段法解含有多个绝对值的不等式。
提高练习
1．不等式 3<|3−2x|≤5的解集为__________
2．不等式|x2－3x|>4的解集是________．
3．若不等式|3x－b|＜4的解集中的整数有且仅有1,2,3，则b的取值范围为________
4．解不等式（1） |2x2−x|<1（2）|x2−3x|<4。