初中數(shù)學(xué)一元一次不等式及不等式組

/r/n13/r/n一元一次不等式及不等式組/r/n中考要求/r/n內(nèi)容/r/n基本要求/r/n略高要求/r/n較高要求/r/n不等式/r/n（/r/n組/r/n）/r/n能根據(jù)具體問(wèn)題中的大小關(guān)系了解不等式的意義./r/n能根據(jù)具體問(wèn)題中的數(shù)量關(guān)系列出不等式/r/n（/r/n組/r/n）/r/n./r/n不等式的性質(zhì)/r/n理解不等式的基本性質(zhì)./r/n會(huì)利用不等式的性質(zhì)比較兩個(gè)實(shí)數(shù)的大小./r/n解一元一次不/r/n等式/r/n（/r/n組/r/n）/r/n了解一元一次不等式/r/n（/r/n組/r/n）/r/n的解的意義，會(huì)在數(shù)軸上表示/r/n（/r/n確定/r/n）/r/n其解集./r/n會(huì)解一元一次不等式和由兩個(gè)一元一次不等式組成的不等式組，并會(huì)根據(jù)條件求整數(shù)解./r/n能根據(jù)具體問(wèn)題中的數(shù)量關(guān)系列出一元一次不等式解決簡(jiǎn)單問(wèn)題./r/n例題精講/r/n板塊一、一元一次不等式及不等式組的解法/r/n【例/r/n1/r/n】求不等式/r/nx/r/n+/r/n3/r/n凹〉/r/n1/r/n-口/r/n的解集./r/n82/r/n【解析】對(duì)本例，首先應(yīng)去分母，化成標(biāo)準(zhǔn)形式求解．/r/n去分母，得/r/n8/r/nx/r/n+/r/n3/r/n（/r/nx/r/n+/r/n1/r/n）/r/n>/r/n8/r/n一/r/n4/r/n（/r/nx/r/n-/r/n5/r/n）/r/n，/r/n去括號(hào)，得/r/n8x/r/n+/r/n3x/r/n+/r/n3/r/n〉/r/n8/r/n一/r/n4/r/nx/r/n+/r/n20/r/n移項(xiàng)，得/r/n8x/r/n+/r/n3x/r/n+/r/n4/r/nx/r/n〉/r/n8/r/n+/r/n20/r/n-/r/n3/r/n,/r/n合并同類項(xiàng)，得/r/n15x/r/n〉/r/n25/r/n系數(shù)化為/r/n1/r/n,得/r/nx/r/n>/r/n5/r/n3/r/n【答案】/r/nx/r/n>/r/n5/r/n3/r/n【鞏固】/r/n當(dāng)/r/nx/r/n為何值時(shí)，代數(shù)式凹/r/n-/r/n1/r/n的值不小于的值？/r/n34/r/n【解析】解決此類問(wèn)題首先應(yīng)理解/r/n“/r/n不小于/r/n”/r/n的意思，進(jìn)而再列出不等式，按照解一元一次不等式方法求解/r/n2/r/nx/r/n+/r/n1/r/n3/r/n+/r/n5/r/nx/r/n依題意，得土/r/n1/r/n-/r/n1/r/n三/r/n3/r/n^/r/n5/r/nx/r/n3/r/n4/r/n???/r/n4/r/n（/r/n???/r/n4/r/n（/r/n2/r/nx/r/n+/r/n1/r/n）/r/n-/r/n12/r/n三/r/n3/r/n（/r/n3/r/n+/r/n5/r/nx/r/n）/r/n,/r/n8/r/nx/r/n-/r/n15x/r/n2/r/n9/r/n+/r/n12/r/n-/r/n4/r/n,/r/n-/r/n7x/r/n2/r/n17/r/n???x/r/nW/r/n17/r/n7/r/n所以，當(dāng)/r/nx/r/n時(shí)，代數(shù)式竽/r/n一/r/n1/r/n的值不小于罟的值/r/n?/r/n17/r/n【答案】/r/nx/r/nW/r/n-/r/n17/r/n【例/r/n2/r/n】求不等式</r/n1/r/n的正整數(shù)解./r/n12/r/n解析】對(duì)于求不等式的正整數(shù)解，應(yīng)先不考慮這一限制條件，按解一元一次不等式的方法求解后，再研/r/n

/r/n究限制條件，便可達(dá)到目的．/r/n去分母，/r/n得/r/n4/r/nx/r/n-/r/n5/r/n</r/n12/r/n移項(xiàng)，合并，得/r/n4/r/nx/r/n</r/n17/r/n17/r/n系數(shù)化為/r/n1/r/n,得/r/nx/r/n</r/n17/r/n4/r/n??/r/n?求原不等式正整數(shù)解./r/n???x/r/n=/r/n1/r/n,/r/n2/r/n,/r/n3/r/n,/r/n4/r/n為原不等式正整解./r/n【答案】/r/nx/r/n=/r/n1,2,3,4/r/n【鞏固】/r/n不等式/r/nx/r/n+/r/n3/r/n3/r/n2/r/nx/r/n的負(fù)整數(shù)解是/r/n一/r/n__/r/n答案】/r/n-/r/n5/r/n、/r/n-/r/n4/r/n、/r/n-/r/n3/r/n、/r/n-/r/n2/r/n、/r/n-/r/n1/r/n丨/r/n3/r/nx/r/n—/r/n1/r/n〉一/r/n4/r/n【例/r/n3/r/n】解不等式組/r/n]/r/n4/r/n，并把它的解集表示在數(shù)軸上./r/n[/r/n2/r/nx/r/n</r/nx/r/n+/r/n2/r/n【解析】/r/nF/r/nx/r/n一/r/n1/r/n〉一/r/n4/r/nnF/r/n>-/r/n1/r/nn-/r/nl/r/n</r/nx/r/n</r/n2./r/n二/r/n[/r/n2/r/nx/r/n</r/nx/r/n+/r/n2/r/n[/r/nx/r/n</r/n2/r/n_3-2/r/n-1/r/no1/r/n23/r/n?/r/n原不等式組的解集是/r/n一/r/n1/r/n</r/nx/r/n</r/n2./r/n在數(shù)軸上表示為：答案】/r/n一/r/n1/r/n</r/nx/r/n</r/n2/r/n2/r/nx/r/n一/r/n3/r/n1/r/n【鞏固】/r/n解不等式：亠/r/n3/r/n</r/n2/r/n</r/n-/r/nx/r/n+/r/n1/r/n42/r/n解析】原不等式相當(dāng)于：/r/n解得/r/n解析】原不等式相當(dāng)于：/r/n解得/r/n2/r/n</r/nx/r/n</r/n丄/r/n【答案】/r/n2/r/n</r/nx/r/n</r/n11/r/n鞏固】/r/n解不等式組：/r/n1/r/nx/r/n鞏固】/r/n解不等式組：/r/n1/r/nx/r/n-/r/n10/r/n〉/r/n4/r/n+/r/nx/r/n-/r/n10/r/n【解/r/n析】原方程組的解為/r/nF/r/n>/r/n'/r/n且'^/r/n=/r/n10/r/n，綜合得/r/nx/r/n>/r/n8/r/n且/r/nx/r/n=/r/n10;/r/n[/r/nx/r/n>/r/n8/r/n【答案】/r/nx/r/n>/r/n8/r/n且/r/nx/r/n=/r/n10/r/n【鞏固】/r/n如果/r/n2m/r/n、/r/nm/r/n、/r/n1/r/n-/r/nm/r/n這三個(gè)實(shí)數(shù)在數(shù)軸上所對(duì)應(yīng)的點(diǎn)從左到右依次排列，求/r/nm/r/n的取值范圍./r/n{m/r/n</r/n1/r/n—/r/nm/r/n，解得/r/nm/r/n</r/n0/r/n./r/n2/r/nm/r/n</r/nm/r/n答案】/r/nm/r/n</r/n0/r/n

/r/n【例/r/n4/r/n】已知方程組爲(wèi)/r/ny/r/n=/r/n2/r/n+/r/n3/r/nm/r/n的解滿足/r/nx/r/n>/r/n0/r/n，/r/ny/r/n>/r/n0/r/n，試求/r/nm/r/n的取值范圍/r/n【答案】解方程組得/r/nJ/r/n1/r/n+/r/n3/r/nm/r/n【答案】解方程組得/r/nJ/r/n1/r/n+/r/n3/r/nm/r/nx/r/n=/r/n一/r/n2/r/n1/r/n—/r/n3/r/nmy/r/n=/r/n2/r/n->/r/n0/r/n1/r/n2/r/n，解得/r/nm/r/n>—/r/n1/r/n—/r/n3/r/nm/r/n3/r/n>/r/n0/r/n2/r/n【鞏固】求使方程組/r/n］/r/n4/r/n；/r/n+打=+/r/n：/r/n+/r/n3/r/n的解'/r/nx/r/n、/r/ny/r/n都是正數(shù)的/r/nm/r/n的取值范圍？/r/n【答案】解方程組得/r/nJ/r/nx/r/n=/r/n8/r/n—/r/nm/r/n,/r/n1/r/ny/r/n=/r/n2/r/nm/r/n—/r/n6/r/n?/r/n?x/r/n、/r/ny/r/n都是正數(shù)/r/n，/r/n【鞏固】求使方程組/r/n］/r/n4/r/n；/r/n+打=+/r/n：/r/n+/r/n3/r/n的解'/r/nx/r/n、/r/ny/r/n都是正數(shù)的/r/nm/r/n的取值范圍？/r/n【答案】解方程組得/r/nJ/r/nx/r/n=/r/n8/r/n—/r/nm/r/n,/r/n1/r/ny/r/n=/r/n2/r/nm/r/n—/r/n6/r/n?/r/n?x/r/n、/r/ny/r/n都是正數(shù)/r/n，/r/n???/r/nJ/r/n8/r/n—/r/nm/r/n>/r/n0/r/n，解得/r/n3/r/n</r/nm/r/n</r/n8/r/n1/r/n2/r/nm/r/n—/r/n6/r/n>/r/n0/r/n【鞏固】在方程組/r/nJ"/r/n+/r/ny/r/n=/r/n1/r/n—/r/nm/r/n中，若未知數(shù)/r/nx/r/n、/r/ny/r/n滿足/r/nx/r/n+/r/ny/r/n>/r/n0/r/n1/r/nx/r/n+/r/n2/r/ny/r/n=/r/n2/r/n，則/r/nm/r/n的取值范圍為./r/n答案】/r/n2/r/nx/r/n+/r/ny/r/n=/r/n1/r/n—/r/nm/r/n①/r/n，/r/n①/r/n+/r/n②/r/n得，/r/n3(/r/nx/r/n+/r/ny/r/n)/r/n=/r/n3/r/n-/r/nm,x/r/n+/r/n2/r/ny/r/n=/r/n2/r/n3/r/n—/r/nm/r/n??/r/n丁>/r/n0/r/n'解得/r/nm/r/n</r/n3/r/n例/r/n5/r/n】/r/n已知/r/nx/r/n、/r/ny/r/n、/r/nz/r/n為三個(gè)非負(fù)有理數(shù)，且滿足/r/n3/r/nx/r/n+/r/n2/r/ny/r/n+/r/nz/r/n=/r/n5/r/n，/r/nx/r/n+/r/ny/r/n—/r/nz/r/n=/r/n2/r/n，若/r/nS/r/n=/r/n2/r/nx/r/n+/r/ny/r/n—/r/nz/r/n的最大值和最小值之和是多少？/r/n解析】/r/n將/r/nx/r/n、/r/ny/r/n、/r/nz/r/n中的一個(gè)字母看做常數(shù)，解方程，然后將結(jié)果代入/r/nS/r/n2/r/nx/r/n+/r/ny/r/n-/r/nz/r/n進(jìn)行消元/r/n答案】/r/n方法一、由嚴(yán)/r/n+/r/n2y/r/n+/r/nz/r/n=/r/n5/r/n解得，/r/nJ/r/nx/r/n=/r/n1/r/n-/r/n3z/r/n,/r/n1/r/nx/r/n+/r/ny/r/n—/r/nz/r/n=/r/n2/r/n1/r/ny/r/n=/r/n4/r/nz/r/n+/r/n1/r/n1/r/n-/r/n3/r/nz/r/n>/r/n0/r/nx/r/n、/r/ny/r/n、/r/nz/r/n為三個(gè)非負(fù)有理數(shù)，/r/n???/r/n^/r/n4/r/nz/r/n+/r/n1/r/n>/r/n0/r/n，解得/r/n0/r/n</r/nz<T/r/n將/r/nJ/r/nx/r/n=/r/n1/r/n-/r/n3z/r/n1/r/ny/r/n=/r/n4/r/nz/r/n+/r/n1/r/n代入/r/nS/r/n=/r/n2/r/nx/r/n+/r/ny/r/n—/r/nz/r/n得，/r/nS/r/n=/r/n3/r/n—/r/n3/r/nz/r/n?/r/n?/r/n?/r/n2/r/n</r/nS/r/n</r/n3/r/n，/r/n?/r/n?/r/n?/r/nS/r/n的最大值與最小值之和為/r/n5/r/n方法二、根據(jù)題意得/r/nx/r/n=/r/nS/r/n—/r/n2/r/nS/r/n—/r/n2/r/n>/r/n0/r/n3/r/nx/r/n+/r/n2/r/ny/r/n+/r/nz/r/n=/r/n5/r/nJ/r/nx/r/n+/r/ny/r/n—/r/nz/r/n=/r/n2/r/n，解得/r/nJ/r/n2/r/nx/r/n+/r/ny/r/n—/r/nz/r/n=/r/nS/r/n15/r/n—/r/n4/r/nS/r/n，/r/ny/r/n=/r/n3/r/n3/r/n—/r/nS/r/nz=/r/n—/r/n3/r/n?*x/r/n、/r/ny/r/n、/r/nz/r/n都是非負(fù)數(shù)，?/r/n:/r/n◎/r/n>/r/n0/r/n

/r/n3/r/n3/r/n—/r/nS/r/n>/r/n0/r/n1/r/n3/r/nS/r/n>/r/n2/r/n???」/r/nS/r/n</r/n15/r/n?/r/n?/r/n?/r/n2/r/n</r/nS/r/n</r/n3/r/n,/r/n?/r/n?/r/n?/r/nS/r/n的最大值與最小值之和為/r/n5/r/n4/r/nS/r/n</r/n3/r/n板塊二、解含有參數(shù)的不等式/r/n【例/r/n6/r/n】解關(guān)于/r/nx/r/n的不等式/r/n2/r/nmx/r/n+/r/n3/r/nV/r/n3/r/nx/r/n+/r/nn/r/n【答案】由原不等式，得：/r/n(2/r/nm/r/n-/r/n3)/r/nx/r/nV/r/nn/r/n-/r/n3/r/n3/r/n./r/nn/r/n一/r/n3/r/n當(dāng)/r/n2/r/nm/r/n-/r/n3/r/n>/r/n0/r/n，即/r/nm/r/n〉一/r/n時(shí)，其解集為/r/nx/r/n</r/n2/r/n2/r/nm/r/n-/r/n3/r/n當(dāng)/r/n2/r/nm/r/n-/r/n3/r/n</r/n0/r/n，即/r/nm/r/n<—/r/n時(shí)，其解集為/r/nx/r/n>/r/n——/r/n2/r/nm/r/n-/r/n3/r/n3/r/n當(dāng)/r/n2/r/nm/r/n-/r/n3/r/n=/r/n0/r/n，即/r/nm=—/r/n時(shí)，/r/n2/r/n若/r/nn/r/n-/r/n3/r/n>/r/n0/r/n，即/r/nn/r/n>/r/n3/r/n，解集為所有數(shù)；若/r/nn/r/n-/r/n3/r/n</r/n0/r/n，即/r/nn/r/n</r/n3/r/n，原不等式無(wú)解./r/n【鞏固】/r/n解關(guān)于/r/nx/r/n的不等式：/r/na/r/n(/r/nx/r/n-/r/na/r/n)/r/n>/r/nb/r/n(/r/nx/r/n-/r/nb/r/n)/r/n【答案】由原不等式得：/r/n(/r/na/r/n-/r/nb/r/n)/r/nx/r/n>/r/n(/r/na/r/n-/r/nb/r/n)(/r/na/r/n+/r/nb/r/n)/r/n當(dāng)/r/na/r/n-/r/nb/r/n>/r/n0/r/n，即得不等式解集為/r/nx/r/n>/r/na/r/n+/r/nb/r/n;/r/n當(dāng)/r/na/r/n-/r/nb/r/n=/r/n0/r/n，即得/r/n0/r/n>/r/n0/r/n，不等式無(wú)解；/r/n當(dāng)/r/na/r/n-/r/nb/r/n</r/n0/r/n，即得不等式解集為/r/nx/r/n</r/na/r/n+/r/nb/r/n./r/n【例/r/n7/r/n】已知關(guān)于/r/nx/r/n的不等式/r/n3/r/n(/r/nx/r/n-/r/na/r/n)/r/n</r/n4/r/n(/r/nx/r/n-/r/n1/r/n)/r/n+/r/n9/r/n的解集是/r/nx/r/n>/r/n1/r/n，求/r/na/r/n的值。/r/n解析】解這個(gè)不等式：/r/n3/r/nx/r/n-/r/n3/r/na/r/n</r/n4/r/nx/r/n-/r/n4/r/n+/r/n9,3/r/nx/r/n-/r/n4/r/nx/r/n</r/n3/r/na/r/n-/r/n4/r/n+/r/n9,/r/nx/r/n>-/r/n3/r/na/r/n-/r/n5/r/n解集是/r/nx/r/n>/r/n1/r/n，/r/n-/r/n3/r/na/r/n-/r/n5/r/n=/r/n1/r/n，解得/r/na/r/n=/r/n-/r/n2/r/n。/r/n答案】/r/na/r/n=-/r/n2/r/n【例/r/n8/r/n】已知/r/nx/r/n=/r/n3/r/n是關(guān)于/r/nx/r/n的不等式/r/n3/r/nx/r/n-/r/n竺竺/r/n>/r/n弐的解，求/r/na/r/n的取值范圍。/r/n23/r/n【解/r/n析】將/r/nx/r/n=/r/n3/r/n代入不等式，得/r/n9/r/n-/r/n丸竺/r/n>/r/n2/r/n。解這個(gè)不等式，得/r/na/r/n</r/n4/r/n。/r/n2/r/n答案】/r/na/r/n</r/n4/r/n【例/r/n9/r/n】已知關(guān)于/r/nx/r/n的不等式/r/n(4/r/na/r/n-/r/n3/r/nb/r/n)/r/nx/r/n>/r/n2/r/nb/r/n-/r/na/r/n的解集為/r/nx/r/n</r/n|/r/n，求/r/nax/r/n>/r/nb/r/n的解集./r/n解析】根據(jù)題意可得：/r/n2/r/nb/r/n-/r/na/r/n4/r/n5/r/n4a/r/n一/r/n3b/r/n</r/n0/r/n且/r/n4/r/na/r/n-/r/n3/r/nb/r/n=/r/n9/r/n，可得/r/n6b/r/n=/r/n5a/r/n，/r/na/r/n</r/n0/r/n，/r/nax/r/n>/r/nb/r/n的解集為/r/nx/r/n</r/n6/r/n-/r/n【答案】/r/nx/r/n</r/n6/r/n解不等式/r/n3/r/nax/r/n+/r/n5/r/nb/r/n>/r/n0/r/n./r/n【鞏固】/r/n已知關(guān)于/r/nx/r/n的不等式/r/n(2/r/na/r/n-/r/nb/r/n)/r/nx/r/n+/r/na/r/n解不等式/r/n3/r/nax/r/n+/r/n5/r/nb/r/n>/r/n0/r/n./r/n【解析】/r/nV/r/n(2/r/na/r/n-/r/nb/r/n)/r/nx/r/n>/r/n5/r/nb/r/n-/r/na/r/n的解集為/r/nx/r/n</r/n10/r/n，可得/r/n2/r/na/r/n-/r/nb/r/n</r/n0/r/n，且/r/nx/r/n</r/n—/r/n，/r/n?/r/n:—/r/n=/r/n10/r/n,解得/r/nb/r/n=/r/n3/r/na/r/n,/r/nTOC\o"1-5"\h\z/r/n7/r/n2/r/na/r/n-/r/nb/r/n2/r/na/r/n-/r/nb/r/n7/r/n5/r/n2/r/na/r/n-/r/nb/r/n=/r/n2/r/na/r/n-/r/n3/r/na/r/n=/r/n—/r/n</r/n0/r/n，即/r/na/r/n</r/n0/r/n./r/n不等式/r/n3/r/nax/r/n+/r/n5/r/nb/r/n>/r/n0/r/n的解集為/r/nx/r/n</r/n-—=/r/n-/r/n1/r/n./r/n5/r/n5/r/n3/r/na/r/n答案】/r/nx/r/n</r/n-/r/n1/r/nx/r/n-/r/na/r/n</r/n0/r/n①/r/n【例/r/n10/r/n】求關(guān)于/r/nx/r/n的不等式組/r/nL/r/n-/r/n1/r/nx/r/n+/r/n2/r/n①的解集。/r/n+/r/n</r/nx/r/n②/r/n〔/r/n2/r/n3/r/n【答案】解/r/n①/r/n得/r/nx/r/n</r/na/r/n，由/r/n②/r/n得/r/nx/r/n>/r/n1/r/n。/r/n應(yīng)分情況討論：/r/n(⑴當(dāng)/r/na/r/nW/r/n1/r/n時(shí)，原不等式組無(wú)解。/r/n⑵/r/n當(dāng)/r/na/r/n>/r/n1/r/n時(shí)，原不等式組的解集為/r/n1/r/n</r/nx/r/n</r/na/r/n。/r/nxax/r/n+/r/n2/r/n>/r/n【鞏固】/r/n解關(guān)于/r/nx/r/n的不等式組：/r/n</r/n3/r/n2/r/n6/r/n2(/r/nx/r/n+/r/n1)/r/n>/r/n11/r/n—/r/nx/r/n【答案】原不等式組可化為/r/nI/r/nx/r/n>/r/n3a/r/n+/r/n2/r/n，/r/n[/r/nx/r/n>/r/n3/r/n當(dāng)/r/n3/r/na/r/n+/r/n2/r/n>/r/n3/r/n，即/r/na/r/n>/r/n-時(shí)，不等式組的解集為/r/nx/r/n>/r/n3/r/na/r/n+/r/n2/r/n；/r/n3/r/n當(dāng)/r/n3/r/na/r/n+/r/n2/r/n</r/n3/r/n，即/r/na/r/n</r/n-時(shí)，不等式組的解集為/r/nx/r/n>/r/n3/r/n./r/n3/r/nI/r/nx/r/n+/r/n9/r/n</r/n5/r/nx/r/n+/r/n1/r/n【例/r/n11/r/n】不等式組/r/nI/r/nx/r/n的解集是/r/nx/r/n>/r/n2/r/n，求/r/nm/r/n的取值范圍./r/nI/r/nx/r/n>/r/nm/r/n+/r/n1/r/n【答案】/r/na/r/n</r/n-/r/n1/r/n【例/r/n12/r/n】試確定/r/nc/r/n的范圍，使不等式組/r/nx/r/n-/r/n■_/r/n->3/r/n-/r/n(/r/n2/r/nx/r/n+/r/n5/r/n)/r/n5/r/n</r/n11/r/n1.5c/r/n—/r/n(x/r/n+/r/n1/r/n)/r/n>/r/n—/r/n(c/r/n-/r/nx/r/n)/r/n+/r/n0.5/r/n(/r/n2x/r/n-/r/n1/r/n)/r/nI/r/n2/r/n2/r/n⑴只有一個(gè)整數(shù)解；/r/n⑵沒(méi)有整數(shù)解．/r/n【解/r/n析】/r/n⑴/r/n解不等式/r/n①/r/n得/r/nx/r/n>/r/n1.7/r/n，解不等式/r/n②/r/n得/r/nx/r/n</r/nc/r/n./r/n(1)/r/n要使不等式組只有一個(gè)整數(shù)解，則不等式的解/r/n集為/r/n-/r/n1.7/r/n</r/nx/r/n</r/nc/r/n，且這個(gè)惟一的整數(shù)必為/r/n-/r/n1/r/n，故/r/n-/r/n1/r/n</r/nc/r/nW/r/n0/r/n./r/n⑵/r/n要使不等式組沒(méi)有整數(shù)解，則/r/nc/r/nW/r/n-/r/n1/r/n./r/n【答案】/r/n⑴/r/n-/r/n1/r/n</r/nc/r/nW/r/n0/r/n;/r/n⑵/r/nc/r/nW/r/n-/r/n1/r/n板塊三、不等式及不等式組的應(yīng)用/r/n【例/r/n13/r/n】若干名學(xué)生合影留念，需交照像費(fèi)/r/n20/r/n元/r/n(/r/n有兩張照片/r/n)/r/n，如果另外加洗一張照片，又需收費(fèi)/r/n1.5/r/n元，要使每人平均出錢不超過(guò)/r/n4/r/n元錢，并都分到一張照片，至少應(yīng)有幾名同學(xué)參加照像？/r/n【解析】設(shè)有/r/nx/r/n位同學(xué)參加照像，根據(jù)題意得：/r/n20/r/n+/r/n1.5(/r/nx/r/n-/r/n2)/r/nW/r/n4/r/nx/r/n，解得/r/nx/r/n2/r/n6.8/r/n，所以至少應(yīng)有/r/n7/r/n名同學(xué)參加照像．/r/n【答案】/r/n7/r/n【例/r/n14/r/n】商業(yè)大廈購(gòu)進(jìn)某種商品/r/n1000/r/n件，售價(jià)定為進(jìn)價(jià)的/r/n125/r/n%.現(xiàn)計(jì)劃節(jié)日期間按原售價(jià)讓利/r/n10/r/n%，至多售出/r/n100/r/n件商品；而在銷售淡季按原定價(jià)的/r/n60/r/n%大甩賣.為使全部商品售完后贏利，在節(jié)日和淡季之外要按原定價(jià)銷售出至少多少件商品？/r/n【解析】設(shè)進(jìn)價(jià)為/r/na/r/n元，按原定價(jià)售出/r/nx/r/n件，節(jié)日讓利售出/r/ny/r/n件/r/n(0/r/n</r/ny/r/n</r/n100)/r/n./r/n依題意有/r/na/r/n-/r/nx/r/n-/r/n125%/r/n+/r/na/r/n-/r/ny/r/n?/r/n125%/r/n-/r/n(1/r/n-/r/n10%)/r/n+/r/n(1000/r/n-/r/nx/r/n-/r/ny/r/n)/r/n-/r/na/r/n-/r/n125%/r/n-/r/n60%/r/n>/r/n1000/r/na/r/n，/r/n

/r/n整理得/r/n4/r/nx/r/n+/r/n3/r/ny/r/n>/r/n2000/r/n，由于/r/n0/r/n</r/ny/r/n</r/n100/r/n，所以/r/nx/r/n>/r/n425/r/n，因此按原定價(jià)至少銷售/r/n426/r/n件./r/n【答案】/r/n426/r/n件/r/n【例/r/n15/r/n】某高速公路收費(fèi)站有/r/nm/r/n(/r/nm/r/n>/r/n0/r/n)輛汽車排隊(duì)等候通過(guò)，假設(shè)通過(guò)收費(fèi)站得車流量保持不變，每個(gè)收費(fèi)窗口的收費(fèi)檢票的速度也是不變的，若開(kāi)放一個(gè)收費(fèi)窗口，則需/r/n20/r/nmin/r/n才能將原來(lái)排隊(duì)等候的汽車以及后來(lái)到的汽車全部收費(fèi)通過(guò)。若同時(shí)開(kāi)放兩個(gè)收費(fèi)窗口，只需/r/n8/r/nmin/r/n就可以將原來(lái)排隊(duì)等候的汽車以及后來(lái)到的汽車全部收費(fèi)通過(guò)，若要求在/r/n3/r/nmin/r/n內(nèi)將排隊(duì)等候收費(fèi)的汽車全部通過(guò)，并使后來(lái)到站的汽車也隨到隨時(shí)收費(fèi)通過(guò)，請(qǐng)問(wèn)至少要同時(shí)開(kāi)放幾個(gè)收費(fèi)窗口？/r/n【答案】解：設(shè)一個(gè)收費(fèi)窗口每分鐘可通過(guò)/r/nx/r/n輛汽車，車流量為/r/ny/r/n輛/r/n//r/nmin/r/n,需開(kāi)放/r/nn/r/n個(gè)收費(fèi)窗口才能在/r/n3/r/nmin/r/n內(nèi)將排隊(duì)等候及后來(lái)到的汽車全部通過(guò)，根據(jù)題意得/r/nm/r/n+/r/n20/r/ny/r/n=/r/n20/r/nx/r/n①/r/n,/r/n最小正整數(shù)/r/nn/r/n=/r/n5/r/n即至少需要開(kāi)/r/nm/r/n+/r/n8/r/ny/r/n=/r/n2/r/nx/r/n8/r/nx/r/n②/r/n，由/r/n①②/r/n得/r/nj/r/n歹，代入/r/n③/r/n得/r/nn/r/n最小正整數(shù)/r/nn/r/n=/r/n5/r/n即至少需要開(kāi)/r/n—/r/nI/r/nm/r/n=/r/n40/r/ny/r/n9/r/nm/r/n+/r/n3/r/ny/r/n</r/nn/r/n?/r/n3/r/nx/r/n③/r/n

/r/n放/r/n5/r/n個(gè)收費(fèi)窗口/r/n例/r/n16/r/n】為加強(qiáng)公民的節(jié)水意識(shí)，某市制定了以下用水收費(fèi)標(biāo)準(zhǔn)：每戶每月用水未超過(guò)/r/n7/r/n立方米時(shí)，每立方米收費(fèi)/r/n1.0/r/n元并加收/r/n0.2/r/n元的城市污水處理費(fèi)，超過(guò)/r/n7/r/n立方米的部分每立方米收費(fèi)/r/n1/r/n．/r/n5/r/n元并加收/r/n0/r/n．/r/n4/r/n元的城市污水處理費(fèi)，如果某單元共有用戶/r/n50/r/n戶，某月共交水費(fèi)/r/n541.6/r/n元，且每戶的用水量均未超過(guò)/r/n10/r/n立方米，求這個(gè)月用水未超過(guò)/r/n7/r/n立方米的最多有多少戶？/r/n【解/r/n析】若月用水量恰為/r/n7/r/n立方米，則應(yīng)交費(fèi)/r/n7/r/nx/r/n(1/r/n+/r/n0.2)/r/n=/r/n8.4(/r/n元/r/n)/r/n，因?yàn)?r/n〉/r/n8.4/r/n，即每戶平均用水超/r/n過(guò)/r/n7/r/n立方米，所以設(shè)有/r/nx/r/n戶用水未超過(guò)/r/n7/r/n立方米，要使/r/nx/r/n取值最大，則需另夕卜/r/n(50/r/n-/r/nx/r/n)/r/n戶用水量盡/r/n可能大，故若這/r/nx/r/n戶每戶用水均為/r/n7/r/n立方米及另外/r/n(50/r/n-/r/nx/r/n)/r/n戶每戶用水為/r/n10/r/n立方米，則總水費(fèi)應(yīng)不少于/r/n541.6/r/n，/r/n2/r/n即：/r/n8.4/r/nx/r/n+/r/n(50/r/n-/r/nx/r/n)/r/nx/r/n[8.4/r/n+/r/n(10/r/n-/r/n7)/r/nx/r/n(1.5/r/n+/r/n0.4)]/r/n>/r/n541.6/r/n，解得:/r/nx/r/n</r/n28—/r/n，所以這個(gè)月用水未超過(guò)/r/n7/r/n3/r/n立方米的最多有/r/n28/r/n戶．/r/n答案】/r/n28/r/n戶/r/n例/r/n17/r/n】為了加強(qiáng)學(xué)生的交通安全意識(shí)，某中學(xué)和交警大隊(duì)聯(lián)合舉行了/r/n“/r/n我當(dāng)一日小警察/r/n”/r/n活動(dòng)，星期天選派部分學(xué)生到交通路口執(zhí)勤，協(xié)助交警維護(hù)交通秩序。若每個(gè)路口安排/r/n4/r/n人，那么還剩下/r/n78/r/n人；若每個(gè)路口安排/r/n8/r/n人，那么最后一個(gè)路口不足/r/n8/r/n人，但不少于/r/n4/r/n人，求這個(gè)中學(xué)共選派執(zhí)勤學(xué)生多少人，在幾個(gè)交通路口安排執(zhí)勤/r/n【答案】設(shè)選派執(zhí)勤學(xué)生/r/nx/r/n人，在/r/ny/r/n個(gè)交通路口執(zhí)勤，則根據(jù)題意得/r/nx/r/n-/r/n4/r/ny/r/nx/r/n-/r/n4/r/ny/r/n=/r/n78/r/n4/r/n</r/nx/r/n-/r/n8(/r/ny/r/n-/r/n1)/r/n</r/n8/r/n解得/r/n19.5/r/n</r/ny/r/n</r/n20.5/r/n。/r/n???/r/n整數(shù)/r/ny/r/n=/r/n20/r/n，/r/nx/r/n=/r/n158/r/n，/r/n?/r/n學(xué)校選派了/r/n158/r/n名學(xué)生在/r/n20/r/n個(gè)路口安排執(zhí)勤/r/n例/r/n18/r/n】/r/n2008/r/n年/r/n8/r/n月，北京奧運(yùn)會(huì)帆船比賽將在青島國(guó)際帆船中心舉行．觀看帆船比賽的船票分為兩種/r/nA/r/n種船票/r/n600/r/n元/r/n//r/n張，/r/nB/r/n種船票/r/n120/r/n元/r/n//r/n張.某旅行社要為一個(gè)旅行團(tuán)代購(gòu)部分船票，在購(gòu)票費(fèi)不超過(guò)/r/n5000/r/n元的情況下，購(gòu)買/r/nA,B/r/n兩種船票共/r/n15/r/n張，要求/r/nA/r/n種船票的數(shù)量不少于/r/nB/r/n種船票數(shù)量的一半.若設(shè)購(gòu)買/r/nA/r/n種船票/r/nx/r/n張，請(qǐng)你解答下列問(wèn)題：/r/n共有幾種符合題意的購(gòu)票方案？寫出解答過(guò)程；/r/n根據(jù)計(jì)算判斷：哪種購(gòu)票方案更省錢？/r/n解析】/r/n(1)/r/n由題意：/r/n600/r/nx/r/n+/r/n120/r/n(/r/n15/r/n-/r/nx/r/n)/r/nW/r/n5000/r/nx/r/n三丄/r/n(/r/n15/r/n-/r/nx/r/n)/r/n〔/r/n2/r/n

/r/n方案一：/r/nA/r/n種船票/r/n5/r/n張，/r/nB/r/n種船票/r/n10/r/n張/r/n;/r/n方案二：/r/nA/r/n種船票/r/n6/r/n張，/r/nB/r/n種船票/r/n9/r/n張/r/n（2）/r/n因?yàn)?r/nB/r/n種船票價(jià)格便宜，因此/r/nB/r/n種船票越多，總購(gòu)票費(fèi)用少./r/n??.第一種方案省錢，為/r/n5/r/nx/r/n600/r/n+/r/n120/r/nx/r/n10/r/n=/r/n4200/r/n（/r/n元/r/n）/r/n答案】（/r/n1/r/n）共兩種購(gòu)票方案：/r/n方案一：/r/nA/r/n種船票/r/n5/r/n張，/r/nB/r/n種船票/r/n10/r/n張/r/n;/r/n方案二：/r/nA/r/n種船票/r/n6/r/n張，/r/nB/r/n種船票/r/n9/r/n張（/r/n2/r/n）第一種方案省錢/r/n例/r/n19/r/n】/r/n2007/r/n年我市某縣籌備/r/n20/r/n周年縣慶，園林部門決定利用現(xiàn)有的/r/n3490/r/n盆甲種花卉和/r/n2950/r/n盆乙種花卉搭配/r/nA/r/n,/r/nB/r/n兩種園藝造型共/r/n50/r/n個(gè)擺放在迎賓大道兩側(cè)，已知搭配一個(gè)/r/nA/r/n種造型需甲種花卉/r/n80/r/n盆，乙種花卉/r/n40/r/n盆，搭配一個(gè)/r/nB/r/n種造型需甲種花卉/r/n50/r/n盆，乙種花卉/r/n90/r/n盆./r/n⑴某校九年級(jí)⑴班課外活動(dòng)小組承接了這個(gè)園藝造型搭配方案的設(shè)計(jì)，問(wèn)符合題意的搭配方案有幾種？請(qǐng)你幫助設(shè)計(jì)出來(lái)./r/n⑵若搭配一個(gè)/r/nA/r/n種造型的成本是/r/n800/r/n元，搭配一個(gè)/r/nB/r/n種造型的成本是/r/n960/r/n元，試說(shuō)明/r/n（1）/r/n中哪種方案成本最低？最低成本是多少元？/r/n【解析】/r/n⑴/r/n設(shè)搭配/r/nA/r/n種造型/r/nx/r/n個(gè)，則/r/nB/r/n種造型為/r/n（50/r/n-/r/nx/r/n）/r/n個(gè)，/r/n依題意，得：/r/n80/r/nx/r/n+/r/n50(50/r/n依題意，得：/r/n80/r/nx/r/n+/r/n50(50/r/n-/r/nx/r/n)/r/n</r/n3490/r/n40/r/nx/r/n+/r/n90(50/r/n-/r/nx/r/n)/r/n</r/n2950/r/n，解得：/r/nx/r/n</r/n33/r/nx/r/n>/r/n31/r/n?/r/n31/r/n</r/nx/r/n</r/n33/r/n???x/r/n是整數(shù)，/r/n???/r/nx/r/n可取/r/n31/r/n，/r/n32/r/n，/r/n33/r/n，??.可設(shè)計(jì)三種搭配方案：/r/nA/r/n種園藝造型/r/n31/r/n個(gè)，/r/nB/r/n種園藝造型/r/n19/r/n個(gè)；/r/nA/r/n種園藝造型/r/n32/r/n個(gè)，/r/nB/r/n種園藝造型/r/n18/r/n個(gè)；/r/nA/r/n種園藝造型/r/n33/r/n個(gè)，/r/nB/r/n種園藝造型/r/n17/r/n個(gè)./r/n（2）（/r/n法/r/n1）/r/n：由于/r/nB/r/n種造型的造價(jià)成本高于/r/nA/r/n種造型成本.所以/r/nB/r/n種造型越少，成本越低，故應(yīng)選擇方案/r/n③/r/n，成本最低，最低成本為：/r/n33/r/nx/r/n800/r/n+/r/n17/r/nx/r/n960/r/n=/r/n42720/r/n（/r/n元/r/n）/r/n（/r/n法/r/n2）/r/n:方案/r/n①/r/n需成本：/r/n31/r/nx/r/n800/r/n+/r/n19/r/nx/r/n960/r/n=/r/n43040/r/n（/r/n元/r/n）/r/n方案/r/n②/r/n需成本：/r/n32/r/nx/r/n800/r/n+/r/n18/r/nx/r/n960/r/n=/r/n42880/r/n（/r/n元/r/n）/r/n方案/r/n③/r/n需成本：/r/n33/r/nx/r/n800/r/n+/r/n17/r/nx/r/n960/r/n=/r/n42720/r/n（/r/n元/r/n）/r/n答案】（/r/n1/r/n）可設(shè)計(jì)三種搭配方案：/r/nA/r/n種園藝造型/r/n31/r/n個(gè)，/r/nB/r/n種園藝造型/r/n19/r/n個(gè)；/r/nA/r/n種園藝造型/r/n32/r/n個(gè)，/r/nB/r/n種園藝造型/r/n18/r/n個(gè)；/r/nA/r/n種園藝造型/r/n33/r/n個(gè)，/r/nB/r/n種園藝造型/r/n17/r/n個(gè)./r/n（/r/n2/r/n）方案/r/n③/r/n成本最低，最低成本為：/r/n42720/r/n（/r/n元/r/n）/r/n例/r/n20/r/n】某園林的門票每張/r/n10/r/n元，一次使用，考慮到人們的不同需求，也為了吸引更多的游客，該園林除保留原來(lái)的售票方法外，還推出了一種/r/n“/r/n購(gòu)買個(gè)人年票/r/n”/r/n的售票方法（個(gè)人年票從購(gòu)買日起，可供持票者使用一年）。年票分/r/nA/r/n、/r/nB/r/n、/r/nC/r/n三類：/r/nA/r/n類年票每張/r/n120/r/n元，持票者進(jìn)入園林時(shí)，無(wú)需再購(gòu)買門票；/r/nB/r/n類年票每張/r/n60/r/n元，/r/n持票者進(jìn)入園林時(shí)，需購(gòu)買門票，每次/r/n2/r/n元；/r/nC/r/n類門票每張/r/n40/r/n元，持票者進(jìn)入該園林時(shí)，需在購(gòu)買門票，每次/r/n3/r/n元/r/n

/r/n⑴如果你只選擇一種購(gòu)買門票的方式，并且你計(jì)劃在一年中用/r/n80/r/n元花在該園林的門票上，試通過(guò)/r/n計(jì)算，找出可使進(jìn)入該園林的次數(shù)最多的購(gòu)票方式/r/n⑵求一年中進(jìn)入該園林至少超過(guò)多少次時(shí)，購(gòu)買/r/nA/r/n類年票比較合算/r/n【答案】/r/n⑴/r/n不可能選/r/nA/r/n類年票/r/n若選/r/nB/r/n類年票，/r/n80/r/n二/r/n60/r/n=/r/n10/r/n次/r/n2/r/n若選/r/nC/r/n類年票，則/r/n80/r/n-/r/n40/r/n=/r/n13-/r/n次/r/n33/r/n若不購(gòu)買年票，則夕/r/n=/r/n8/r/n次/r/n10/r/n所以計(jì)劃用/r/n80/r/n元花在該園林的門票上時(shí)，選擇購(gòu)買/r/nC/r/n類年票的方法進(jìn)入園林的次數(shù)最多，為/r/n13/r/n次/r/nx/r/n>/r/n30/r/n2/r/n解得/r/n</r/nx/r/n>/r/nx/r/n>/r/n30/r/n2/r/n解得/r/n</r/nx/r/n>/r/n26—/r/n3/r/nx/r/n>/r/n12/r/n60/r/n+/r/n2/r/nx/r/n>/r/n120/r/n則/r/n”0/r/n+/r/n3/r/nx/r/n>/r/n120/r/n、/r/n10/r/nx/r/n>/r/n120/r/n所以，一年中進(jìn)入該園林至少超過(guò)/r/n30/r/n次時(shí)，購(gòu)買/r/nA/r/n類年票比較合算/r/n課堂檢測(cè)/r/n討論/r/nax/r/n</r/nb/r/n的解集./r/nb/r/n【答案】當(dāng)/r/na/r/n>/r/n0/r/n時(shí)，解集為/r/nx/r/n</r/n-/r/n；/r/na/r/nb/r/n當(dāng)/r/na/r/n</r/n0/r/n時(shí)，解集為/r/nx/r/n>—/r/n；/r/na/r/n當(dāng)/r/na/r/n=/r/n0/r/n時(shí)/r/n若/r/nb/r/n>/r/n0/r/n，則解集為所有數(shù)；/r/n若/r/n—/r/n</r/n0/r/n，不等式無(wú)解./r/n已知/r/nx/r/n、/r/ny/r/n同時(shí)滿足三個(gè)條件：①/r/n3/r/nx/r/n-/r/n2/r/ny/r/n=/r/n4/r/n-/r/np/r/n:②/r/n4/r/nx/r/n-/r/n3/r/ny/r/n=/r/n2/r/n+/r/np/r/n:③/r/nx/r/n>/r/ny/r/n則/r/np/r/n的取值范圍/r/n是/r/n【答案】/r/n②/r/n—/r/n①/r/n得/r/nx/r/n—/r/ny/r/n=/r/n2/r/np/r/n—/r/n2/r/n>/r/n0/r/n，?/r/n：/r/np/r/n>/r/n1/r/n已知非負(fù)數(shù)/r/na/r/n、/r/n—/r/n、/r/nc/r/n滿足條件：/r/n3a/r/n+/r/n2—/r/n+/r/nc/r/n=/r/n4/r/n,/r/n2a/r/n+/r/n—/r/n+/r/n3c/r/n=/r/n5/r/n，設(shè)/r/nS/r/n=/r/n5a/r/n+/r/n4—/r/n+/r/n7c/r/n的最小值為/r/nm/r/n，最大值為/r/nn/r/n，求/r/nm/r/n—/r/nn/r/n的值/r/n【答案】/r/n丄/r/n2/r/n開(kāi)學(xué)初，小芳和小亮去學(xué)校商店購(gòu)買學(xué)習(xí)用品，小芳用/r/n18/r/n元錢買了/r/n1/r/n支鋼筆和/r/n3/r/n本筆記本；小亮用/r/n

/r/n31/r/n元買了同樣的鋼筆/r/n2/r/n支和筆記本/r/n5/r/n本．/r/n⑴求每支鋼筆和每本筆記本的價(jià)格；/r/n⑵校運(yùn)會(huì)后，班主任拿出/r/n200/r/n元學(xué)校獎(jiǎng)勵(lì)基金交給班長(zhǎng)，購(gòu)買上述價(jià)格的鋼筆和筆記本共/r/n48/r/n件作為獎(jiǎng)品，獎(jiǎng)給校運(yùn)會(huì)中表現(xiàn)突出的同學(xué)，要求筆記本數(shù)不少于鋼筆數(shù)，共有多少種購(gòu)買方案？請(qǐng)你一一寫出．/r/n【解/r/n析】/r/n⑴/r/n設(shè)每支鋼筆/r/nx/r/n元，每支筆記本/r/ny/r/n本./r/nx/r/n+/r/n3/r/ny/r/n=/r/n18/r/n./r/n\/r/nx/r/n=/r/n3/r/n2/r/nx/r/n+/r/n5/r/ny/r/n=/r/n31/r/n，/r/n??/r/nj/r/ny/r/n=/r/n5/r/n⑵/r/n設(shè)購(gòu)買鋼筆/r/na/r/n支，筆記本/r/nb/r/n個(gè)./r/na/r/n+/r/nb/r/n=/r/n48/r/n</r/n3/r/na/r/n+/r/n5/r/nb/r/nW/r/n200,/r/nb/r/n三/r/na/r/n+/r/nb/r/n=/r/n48/r/n</r/n3/r/na/r/n+/r/n5/r/nb/r/nW/r/n200,/r/nb/r/n三/r/na/r/na/r/n三/r/n20/r/nb/r/nW/r/n28/r/n則共有五種購(gòu)買方案/r/na/r/n=/r/n20,21,22,23,24/r/nb/r/n=/r/n28,27,26,25,24/r/n答案】(/r/n1/r/n)每支鋼筆/r/n3/r/n元，每支筆記本/r/n5/r/n本./r/n5/r/n)五種方案：/r/na/r/n=/r/n20,21,22,23,24/r/nb/r/n=/r/n28,27,26,25,24/r/n課后作業(yè)/r/n若不等式/r/n(/r/na/r/n+/r/nb/r/n)/r/nx/r/n+/r/n(2/r/na/r/n-/r/n3/r/nb/r/n)/r/n</r/n0/r/n的解集為/r/nx/r/n>-/r/n1/r/n，求不等式/r/n(/r/na/r/n-/r/n3/r/nb/r/n)/r/nx/r/n+/r/n(/r/nb/r/n-/r/n2/r/na/r/n)/r/n>/r/n0/r/n的解集./r/n【解析】原的解集為/r/nx/r/n>-/r