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1、不等式選講（高考試題匯編）一、知識點(diǎn)整合：1 含有絕對值的不等式的解法(1)|f(x)|a(a0)f(x)a或f(x)a；(2)|f(x)|0)af(x)a.(3)對形如|xa|xb|c，|xa|xb|c的不等式，可利用絕對值的幾何意義求解2 含有絕對值的不等式的性質(zhì)|a|b|ab|a|b|.3 柯西不等式(1)設(shè)a，b，c，d均為實(shí)數(shù)，則(a2b2)(c2d2)(acbd)2，當(dāng)且僅當(dāng)adbc時等號成立(2)若ai，bi(iN*)為實(shí)數(shù)，則(a)(b)(aibi)2，當(dāng)且僅當(dāng)(當(dāng)某bj0時，認(rèn)為aj0，j1,2，n)時等號成立(3)柯西不等式的向量形式：設(shè)，為平面上的兩個向量，則|，當(dāng)且僅當(dāng)

2、這兩個向量共線時等號成立4 不等式的證明方法證明不等式常用的方法有比較法、綜合法、分析法、反證法、放縮法、數(shù)學(xué)歸納法等二、典型題型題型一含絕對值的不等式的解法例1(2013課標(biāo)全國)已知函數(shù)f(x)|2x1|2xa|，g(x)x3.(1)當(dāng)a2時，求不等式f(x)1，且當(dāng)x時，f(x)g(x)，求a的取值范圍審題破題(1)可以通過分段討論去絕對值；(2)在x時去絕對值，利用函數(shù)最值求a的范圍解(1)當(dāng)a2時，不等式f(x)g(x)化為|2x1|2x2|x30.設(shè)函數(shù)y|2x1|2x2|x3，則y其圖象如圖所示，由圖象可知，當(dāng)且僅當(dāng)x(0,2)時，y0，所以原不等式的解集是x|0x1，則0的解集

3、；(2)若關(guān)于x的不等式f(x)2的解集是R，求m的取值范圍題型二不等式的證明例2(2012福建)已知函數(shù)f(x)m|x2|，mR，且f(x2)0的解集為1,1(1)求m的值；(2)若a，b，cR，且m，求證：a2b3c9.審題破題(1)從解不等式f(x2)0出發(fā)，將解集和1,1對照求m；(2)利用柯西不等式證明(1)解因為f(x2)m|x|，f(x2)0等價于|x|m.由|x|m有解，得m0，且其解集為x|mxm又f(x2)0的解集為1,1，故m1.(2)證明由(1)知1，又a，b，cR，由柯西不等式得a2b3c(a2b3c)29.反思?xì)w納不等式證明的基本方法是比較法、綜合法、分析法、反證法

4、、放縮法和數(shù)學(xué)歸納法，其中以比較法和綜合法最為基礎(chǔ)，使用綜合法證明不等式的關(guān)鍵就是通過適當(dāng)?shù)淖儞Q后使用重要不等式，證明過程注意從重要不等式的形式入手達(dá)到證明的目的變式訓(xùn)練2已知f(x)|x1|x1|，不等式f(x)4的解集為M.(1)求M；(2)當(dāng)a，bM時，證明：2|ab|0時，x，得a2.(2)記h(x)f(x)2f，則h(x)所以|h(x)|1，因此k1.反思?xì)w納不等式f(a)g(x)恒成立時，要看是對哪一個變量恒成立，如果對于aR恒成立，則f(a)的最小值大于等于g(x)，再解關(guān)于x的不等式求x的取值范圍；如果對于xR不等式恒成立，則g(x)的最大值小于等于f(a)，再解關(guān)于a的不等式

5、求a的取值范圍變式訓(xùn)練3已知函數(shù)f(x)log2(|x1|x5|a)(1)當(dāng)a2時，求函數(shù)f(x)的最小值；(2)當(dāng)函數(shù)f(x)的定義域為R時，求實(shí)數(shù)a的取值范圍變式訓(xùn)練4設(shè)f(x)|x|2|xa|(a0)(1)當(dāng)a1時，解不等式f(x)8；(2)若f(x)6恒成立，求正實(shí)數(shù)a的取值范圍三、專題限時規(guī)范訓(xùn)練一、填空題1 不等式|x3|x2|3的解集為_2 設(shè)x0，y0，M，N，則M、N的大小關(guān)系為_3 對于實(shí)數(shù)x，y，若|x1|1，|y2|1，則|x2y1|的最大值為_4 若關(guān)于x的不等式|a|x1|x2|存在實(shí)數(shù)解，則實(shí)數(shù)a的取值范圍是_二、解答題5 設(shè)不等式|2x1|a2|1對于一切非零實(shí)

6、數(shù)x均成立，求實(shí)數(shù)a的取值范圍7 (2012江蘇)已知實(shí)數(shù)x，y滿足：|xy|，|2xy|，求證：|y|.8 已知函數(shù)f(x)|xa|.(1)若不等式f(x)3的解集為x|1x5，求實(shí)數(shù)a的值；(2)在(1)的條件下，若f(x)f(x5)m對一切實(shí)數(shù)x恒成立，求實(shí)數(shù)m的取值范圍9 已知函數(shù)f(x)|2x1|2x3|.(1)求不等式f(x)6的解集；(2)若關(guān)于x的不等式f(x)|a1|的解集非空，求實(shí)數(shù)a的取值范圍10設(shè)a，b，c為正實(shí)數(shù)，求證：abc2.一、填空題 （2013年高考湖北卷（理）設(shè),且滿足:,則_.二、解答題 （2013年普通高等學(xué)校招生統(tǒng)一考試新課標(biāo)卷數(shù)學(xué)（理）（純WORD版

7、含答案）選修45;不等式選講設(shè)均為正數(shù),且,證明:(); (). （2013年普通高等學(xué)校招生統(tǒng)一考試遼寧數(shù)學(xué)（理）試題（WORD版）選修4-5:不等式選講已知函數(shù),其中.(I)當(dāng)時,求不等式的解集; (II)已知關(guān)于的不等式的解集為,求的值. （2013年普通高等學(xué)校招生統(tǒng)一考試福建數(shù)學(xué)（理）試題）不等式選講:設(shè)不等式的解集為,且,.(1)求的值;(2)求函數(shù)的最小值.（2013年高考湖南卷（理）在平面直角坐標(biāo)系xOy中,將從點(diǎn)M出發(fā)沿縱、橫方向到達(dá)點(diǎn)N的任一路徑成為M到N的一條“L路徑”.如圖6所示的路徑都是M到N的“L路徑”.某地有三個新建的居民區(qū),分別位于平面xOy內(nèi)三點(diǎn)處.現(xiàn)計劃在x

8、軸上方區(qū)域(包含x軸)內(nèi)的某一點(diǎn)P處修建一個文化中心.(I)寫出點(diǎn)P到居民區(qū)A的“L路徑”長度最小值的表達(dá)式(不要求證明);(II)若以原點(diǎn)O為圓心,半徑為1的圓的內(nèi)部是保護(hù)區(qū),“L路徑”不能進(jìn)入保護(hù)區(qū),請確定點(diǎn)P的位置,使其到三個居民區(qū)的“L路徑”長度值和最小. 四，高考試題匯編1 (2013重慶)若關(guān)于實(shí)數(shù)x的不等式|x5|x3|0,求證:4 (2012山東)若不等式|kx4|2的解集為x|1x3，則實(shí)數(shù)k_.5.（2012、江蘇）已知實(shí)數(shù)x，y滿足：求證：6 (2011湖南)設(shè)x，yR，且xy0，則的最小值為_1.(2011山東)不等式的解集為（A）-5.7 （B）-4,6 （C） （D

9、）2.(2011年高考天津卷理科13)已知集合，則集合=_.3.對于實(shí)數(shù)x，y，若，則的最大值為 .4.(2011年高考廣東卷理科9)不等式的解集是_.4(2011年高考陜西卷理科15)（不等式選做題）若關(guān)于x的不等式存在實(shí)數(shù)解，則實(shí)數(shù)的取值范圍是 5(2011年高考遼寧卷理科24)（本小題滿分10分）選修4-5：不等式選講已知函數(shù)f（x）=|x-2|-|x-5|.（I）證明：-3f（x）3；（II）求不等式f（x）x2-8x+15的解集.6. (2011年高考全國新課標(biāo)卷理科24)（本小題滿分10分） 選修4-5不等選講設(shè)函數(shù)（1）當(dāng)時，求不等式的解集；(2)如果不等式的解集為，求的值。7.

10、(2011年高考江蘇卷21)選修4-5：不等式選講（本小題滿分10分）解不等式：8(2011年高考福建卷理科21)（本小題滿分7分）選修4-5：不等式選講設(shè)不等式的解集為M（I）求集合M；（II）若a，bM，試比較ab+1與a+b的大小9（2010年高考陜西卷理科15）(不等式選做題)不等式的解集為.10（2010年高考福建卷理科21）（本小題滿分7分）選修4-5：不等式選講已知函數(shù)。（）若不等式的解集為，求實(shí)數(shù)的值；（）在（）的條件下，若對一切實(shí)數(shù)x恒成立，求實(shí)數(shù)m的取值范圍。12（2010年高考遼寧卷理科24）（本小題滿分10分）選修4-5：不等式選講已知均為正數(shù)，證明：，并確定為何值時，

11、等號成立。13（2008廣東，14）（不等式選講選做題）已知，若關(guān)于x的方程有實(shí)根，則a的取值范圍是 。14（2007廣東，14）（不等式選講選做題）設(shè)函數(shù)= ；若，則x的取值范圍是 。4設(shè)函數(shù)f（x）=|x+1|+|x-a|的圖象關(guān)于直線x=1對稱，則a的值為16（2007海南、寧夏，22C，10分）（選修4 5：不等式選講）設(shè)函數(shù) （1）解不等式； （2）求函數(shù)的最小值。17（2008山東高考題）若不等式的解集中的整數(shù)有且僅有1、2、3，則b的取值范圍為 。18. （2009廣東14)不等式的實(shí)數(shù)解為 .19（2009福建選考21（3） 解不等式2x-10的解集為_.4江西15.（2）（不

12、等式選做題）在實(shí)數(shù)范圍內(nèi)，不等式|2x-1|+|2x+1|6的解集為_。 5遼寧24. （本小題滿分10分）選修4-5：不等式選講已知，不等式的解集為（1）求的值（2）若恒成立，求的取值范圍6若存在實(shí)數(shù)使成立，則實(shí)數(shù)的取值范圍是 7新課標(biāo)（24）（本小題滿分10分）選修：不等式選講已知函數(shù)（1）當(dāng)時，求不等式的解集；（2）若的解集包含，求的取值范圍。技術(shù)官員村位于位于亞運(yùn)城東部，主干道二以東石樓涌以西的地塊，占地面積、m2，總建筑面積、m2，共包括地下室南區(qū)、地下室北區(qū)、地上部分1棟12棟、服務(wù)中心、室外工程等多個單體工程。其中住宅面積m2，共12棟，17棟建筑層數(shù)為11層，812棟11層（局

13、部復(fù)式頂層），首層局部架空，布置公建配套設(shè)施。integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the citys liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufactu

14、ring industry, changning district, shale gas production capacity reached 277 million cubic metres, built the countrys first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Y

15、ibin city, fiscal revenue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for water and the second land of the seven hills. 236 meters to 2000 meters above se

16、a level in the city, low mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building s

17、ituation, old town-the South Bank Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area s

18、pace is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, suspicious pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed,

19、 integrated energy, chemicals and textile Yibin city, are the three core pillars of the industry. In 2014, the wuliangye brand value to 73.58 billion yuan, the citys liquor industry slip to stabilise. Promoting deep development of integrated energy, advanced equipment manufacturing industry, changni

20、ng district, shale gas production capacity reached 277 million cubic metres, built the countrys first independent high-yield wells and pipelines in the first section, the lead in factory production and supply to the population. 2.1-3 GDP growth figure 2.1-4 Yibin, Yibin city, Yibin city, fiscal reve

21、nue growth 2.1.4 topography terrain overall is Southwest, North-Eastern State. Low mountains and hills in the city landscape as the main ridge-and-Valley, pingba small fragmented nature picture for water and the second land of the seven hills. 236 meters to 2000 meters above sea level in the city, l

22、ow mountain, 46.6% hills 45.3%, pingba only 8.1%. 2.1.5 development of Yibin landscapes and distinctive feature in the center of the city, with limitations, and spatial structure of typical zonal group, 2012-cities in building with an area of about 76.2km2. From city-building situation, old town-the

23、 South Bank Center construction is lagging behind, disintegration of the old city is slow, optimization and upgrading, quality public service resources are still heavily concentrated in the old town together. Southbank Centre has not been formed, functions of the service area space is missing. Meanwhile, peripheral group centres service was weak and inadequate accounting for city development, suspicious pattern could not be formed. As regards transport, with the outward expansion of cities, cities have been expanding, centripetal city traffic organization has not changed,