§13.3　絕對值不等式考試要求　1.理解絕對值的幾何意義，并了解下列不等式成立的幾何意義及取等號的條件：|a＋b|≤|a|＋|b|(a，b∈R)；|a－c|≤|a－b|＋|b－c|(a，b，c∈R).2.會利用絕對值的幾何意義求解以下類型的不等式：|ax＋b|≤c，|ax＋b|≥c；|x－a|＋|x－b|≥c，|x－a|＋|x－b|≤c.知識梳理1．絕對值不等式的解法(1)含絕對值的不等式|x|<a與|x|>a的解集不等式a>0a＝0a<0|x|<a ??|x|>a(－∞，－a)∪(a，＋∞)(－∞，0)∪(0，＋∞)R (2)|ax＋b|≤c(c>0)和|ax＋b|≥c(c>0)型不等式的解法①|ax＋b|≤c?________________________.②|ax＋b|≥c?________________________.(3)|x－a|＋|x－b|≥c(c>0)和|x－a|＋|x－b|≤c(c>0)型不等式的解法①利用絕對值不等式的幾何意義求解，體現(xiàn)了數(shù)形結(jié)合的思想．②利用“零點(diǎn)分段法”求解，體現(xiàn)了分類討論的思想．③通過構(gòu)造函數(shù)，利用函數(shù)的圖象求解，體現(xiàn)了函數(shù)與方程的思想．2．含有絕對值的不等式的性質(zhì)(1)如果a，b是實(shí)數(shù)，則________≤|a±b|≤________.(2)如果a，b，c是實(shí)數(shù)，那么______________，當(dāng)且僅當(dāng)________________時(shí)，等號成立．思考辨析判斷下列結(jié)論是否正確(請?jiān)诶ㄌ栔写?/span>“√”或“×”)(1)若|x|>c的解集為R，則c≤0.(　　)(2)不等式|x－1|＋|x＋2|<2的解集為?.(　　)(3)對|a＋b|≥|a|－|b|當(dāng)且僅當(dāng)a>b>0時(shí)等號成立．(　　)(4)對|a－b|≤|a|＋|b|當(dāng)且僅當(dāng)ab≤0時(shí)等號成立．(　　)教材改編題1．不等式3≤|5－2x|<9的解集為(　　)A．[－2,1)∪[4,7)   B．(－2,1]∪(4,7]C．(－2，－1]∪[4,7)   D．(－2,1]∪[4,7)2．不等式|x－1|－|x－5|<2的解集為________．3．設(shè)a，b∈R，|a－b|>2，則關(guān)于實(shí)數(shù)x的不等式|x－a|＋|x－b|>2的解集是________．題型一　絕對值不等式的解法例1 (2021·全國乙卷)已知函數(shù)f(x)＝|x－a|＋|x＋3|.(1)當(dāng)a＝1時(shí)，求不等式f(x)≥6的解集；(2)若f(x)>－a，求a的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華 解絕對值不等式的基本方法(1)利用絕對值的定義，通過分類討論轉(zhuǎn)化為解不含絕對值符號的普通不等式．(2)當(dāng)不等式兩端均為正數(shù)時(shí)，可通過兩邊平方的方法，轉(zhuǎn)化為不含絕對值符號的普通不等式．(3)利用絕對值的幾何意義，數(shù)形結(jié)合求解．跟蹤訓(xùn)練1 已知函數(shù)f(x)＝－.(1)畫出函數(shù)y＝f(x)的圖象；(2)解不等式≥1.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________題型二　利用絕對值不等式的性質(zhì)求最值例2 已知函數(shù)f(x)＝|2x＋1|＋|x－4|.(1)解不等式f(x)≤6；(2)若不等式f(x)＋|x－4|<a2－8a有解，求實(shí)數(shù)a的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華 求含絕對值函數(shù)的最值時(shí)，常用的方法有三種(1)利用絕對值的幾何意義．(2)利用絕對值的三角不等式，即|a|＋|b|≥|a±b|≥||a|－|b||.(3)利用零點(diǎn)分段法，轉(zhuǎn)化為分段函數(shù)求最值．跟蹤訓(xùn)練2 已知函數(shù)f(x)＝－，m∈R.(1)若m＝3，求不等式f(x)>1的解集；________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2)若對?x∈R，不等式f(x)＋2≥4都成立，求實(shí)數(shù)m的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 題型三　絕對值不等式的綜合應(yīng)用例3 設(shè)函數(shù)f(x)＝|2x＋1|＋|x－1|.(1)畫出y＝f(x)的圖象；(2)當(dāng)x∈[0，＋∞)時(shí)，f(x)≤ax＋b，求a＋b的最小值．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________思維升華 (1)解決與絕對值有關(guān)的綜合問題的關(guān)鍵是去掉絕對值，化為分段函數(shù)．(2)數(shù)形結(jié)合是解決與絕對值有關(guān)的綜合問題的常用方法．跟蹤訓(xùn)練3 (2023·成都聯(lián)考)已知函數(shù)f(x)＝|x－2|－a|x＋1|.(1)當(dāng)a＝1時(shí)，求不等式f(x)<x的解集；________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2)當(dāng)a＝2時(shí)，若關(guān)于x的不等式f(x)>m＋1恰有2個(gè)整數(shù)解，求實(shí)數(shù)m的取值范圍．________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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