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    2024年数学高考大一轮复习第十三章13.3 绝对值不等式 试卷

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    2024年数学高考大一轮复习第十三章 §13.3 绝对值不等式

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    这是一份2024年数学高考大一轮复习第十三章 §13.3 绝对值不等式,共5页。
    §13.3 绝对值不等式考试要求 1.理解绝对值的几何意义,并了解下列不等式成立的几何意义及取等号的条件:|ab||a||b|(abR)|ac||ab||bc|(abcR).2.会利用绝对值的几何意义求解以下类型的不等式:|axb|c|axb|c|xa||xb|c|xa||xb|c.知识梳理1.绝对值不等式的解法(1)含绝对值的不等式|x|<a|x|>a的解集不等式a>0a0a<0|x|<a |x|>a(,-a)(a,+)(0)(0,+)R (2)|axb|c(c>0)|axb|c(c>0)型不等式的解法|axb|c________________________.|axb|c________________________.(3)|xa||xb|c(c>0)|xa||xb|c(c>0)型不等式的解法利用绝对值不等式的几何意义求解,体现了数形结合的思想.利用零点分段法求解,体现了分类讨论的思想.通过构造函数,利用函数的图象求解,体现了函数与方程的思想.2.含有绝对值的不等式的性质(1)如果ab是实数,则________|a±b|________.(2)如果abc是实数,那么______________,当且仅当________________时,等号成立.思考辨析判断下列结论是否正确(请在括号中打“√”“×”)(1)|x|>c的解集为R,则c0.(  )(2)不等式|x1||x2|<2的解集为.(  )(3)|ab||a||b|当且仅当a>b>0时等号成立.(  )(4)|ab||a||b|当且仅当ab0时等号成立.(  )教材改编题1.不等式3|52x|<9的解集为(  )A[2,1)[4,7)   B(2,1](4,7]C(2,-1][4,7)   D(2,1][4,7)2.不等式|x1||x5|<2的解集为________3.设abR|ab|>2,则关于实数x的不等式|xa||xb|>2的解集是________题型一 绝对值不等式的解法1 (2021·全国乙卷)已知函数f(x)|xa||x3|.(1)a1时,求不等式f(x)6的解集;(2)f(x)>a,求a的取值范围.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思维升华 解绝对值不等式的基本方法(1)利用绝对值的定义,通过分类讨论转化为解不含绝对值符号的普通不等式.(2)当不等式两端均为正数时,可通过两边平方的方法,转化为不含绝对值符号的普通不等式.(3)利用绝对值的几何意义,数形结合求解.跟踪训练1 已知函数f(x).(1)画出函数yf(x)的图象;(2)解不等式1.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________题型二 利用绝对值不等式的性质求最值2 已知函数f(x)|2x1||x4|.(1)解不等式f(x)6(2)若不等式f(x)|x4|<a28a有解,求实数a的取值范围.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思维升华 求含绝对值函数的最值时,常用的方法有三种(1)利用绝对值的几何意义.(2)利用绝对值的三角不等式,即|a||b||a±b|||a||b||.(3)利用零点分段法,转化为分段函数求最值.跟踪训练2 已知函数f(x)mR.(1)m3,求不等式f(x)>1的解集;________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2)若对xR,不等式f(x)24都成立,求实数m的取值范围.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 题型三 绝对值不等式的综合应用3 设函数f(x)|2x1||x1|.(1)画出yf(x)的图象;(2)x[0,+)时,f(x)axb,求ab的最小值.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思维升华 (1)解决与绝对值有关的综合问题的关键是去掉绝对值,化为分段函数.(2)数形结合是解决与绝对值有关的综合问题的常用方法.跟踪训练3 (2023·成都联考)已知函数f(x)|x2|a|x1|.(1)a1时,求不等式f(x)<x的解集;________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2)a2时,若关于x的不等式f(x)>m1恰有2个整数解,求实数m的取值范围.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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