11.3 一元一次不等式组 (2课时,学生版+答案版) 2025-2026学年数学人教版七年级下册

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11.3　一元一次不等式组
第1课时　一元一次不等式组的解法
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1.了解一元一次不等式组的概念．2．理解一元一次不等式组的解集的意义，掌握求一元一次不等式组的解集的方法．3．会用数轴确定两个一元一次不等式组成的不等式组的解集．
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知识点一　一元一次不等式组
1．把两个含有同一个未知数的不等式合起来，组成一个一元一次不等式组．
2．一般地，几个不等式的解集的公共部分，叫作由它们所组成的不等式组的解集．
练习1　下列不等式组为一元一次不等式组的是(　B　).
A．
B．
C．
D．
知识点二　一元一次不等式组的解法
3．一元一次不等式组的解集的求法：解一元一次不等式组时，一般先求出其中各不等式的解集，再求出这些解集的公共部分．利用数轴可以直观地确定不等式组的解集．
4．不等式组的解的情况与图示．
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(1)同大取大：图示：如图1，解集为x＞b．
(2)同小取小：图示：如图2，解集为x＜a．
(3)大小小大中间找：图示：如图3，解集为a＜x＜b．
(4)大大小小无处找：图示：如图4，解集为无解．
练习2　(教材P139例1变式)解下列不等式组并把解集在数轴上表示出来：
(1)
(2)
【答案】(1)－1＜x≤2
(2)x≤－1
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1．将不等式组的解集表示在数轴上，下列正确的是(　A　).
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2．已知关于x的不等式组无解，则a的取值范围是(　B　).
A．a＞2 B．a≥2
C．a＜2 D．a≤2
3．如图，一个运算程序，若需要经过两次运算才能输出结果，则x的取值范围为(　D　).
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A．x>8 B．8C．8≤x≤13 D．8≤x<13
4．若关于x的不等式组的解集为x＜3，则m的取值范围是(　C　).
A．m＞2 B．m＜2
C．m≥2 D．m≤2
5．解不等式组：并把解集在数轴上表示出来，再写出所有的非正整数解．
【答案】不等式组的解集为－＜x≤，图略．
不等式组的非正整数解为－2，－1，0.
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6．已知关于x的不等式组的整数解共有6个，则a的取值范围是(　A　).
A．－5C．－5≤a≤－4 D．－57．已知关于x，y的方程组
(1)若此方程组的解满足－1＜x＋y≤3，求a的取值范围；
(2)在(1)的条件下，若关于m的不等式2am－m＞2a－1的解集为m＜1，求满足条件的a的整数值．
【答案】(1)－2＜a≤2
(2)－1，0
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8．若某一元一次方程的解在某一元一次不等式组的解集范围内，则称该一元一次方程为该不等式组的“关联方程”．例如：方程x－1＝3的解为x＝4，而不等式组的解集为3＜x＜5，不难发现x＝4在3＜x＜5的范围内，所以方程x－1＝3是不等式组的“关联方程”．
(1)在方程①2(x＋1)－x＝5；②2x－4＝0；③＋1＝x中，关于x的不等式组的“关联方程”是________(填序号)；
(2)若关于x的方程2x－k＝6是不等式组的“关联方程”，求k的取值范围．
【答案】(1)①②
(2)－8＜k≤8第2课时　一元一次不等式组的应用
1.会求一元一次不等式组的特殊解．2．会利用一元一次不等式组解决实际问题．
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知识点一　由解集情况确定字母的取值范围
练习1　若不等式组的解集为x＜m，则m的取值范围为 ．
知识点二　一元一次不等式组的实际应用
练习2　用若干辆装载质量为8吨的货车运一批货物，若每辆货车装4吨，则剩下20吨货物；若每辆货车装8吨，则最后一辆车装的货物不满也不空，若设有x辆货车，则x应满足的不等式组是( ).
A.
B．
C．
D．
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基础巩固
1．已知不等式组的解集是22．小慧购买了两种糕点当伴手礼，其中桂圆糕点一盒12个，售价70元；蜜枣糕点一盒6个，售价40元．已知小慧共购买10盒糕点，花费的金额不超过500元．若她将糕点分给75位同事，每人至少能拿到一个糕点，则小慧花了________元购买糕点．
3．下图是某运行程序，从“输入整数x”到“结果是否>18”为一次程序操作，①若输入整数11，输出结果为27；②若输入整数x后程序操作仅进行了两次就停止，则x的最大值是8；③若操作停止时输出结果为21，则输入的整数x是9；④输入整数x后，该操作永不停止，则x≤3.以上结论正确的是________(填序号).
4．如图，用长为40 m的铁栅栏一边靠墙围成两个相同的长方形菜地(靠墙部分不使用铁栅栏)，墙的长度MN＝34 m，要使靠墙的AC边不小于25 m，那么与墙垂直的一边AB的长度范围为__________________．
5．小红家开了一家糕点店，现有11 kg面粉，9.4 kg鸡蛋，计划加工一般糕点和精制糕点两种产品共50盒．已知加工1盒一般糕点需0.3 kg面粉和0.1 kg鸡蛋；加工1盒精制糕点需0.1 kg面粉和0.3 kg鸡蛋．
(1)有哪几种加工方案？
(2)如果销售1盒一般糕点和1盒精制糕点的利润分别为3元和4元，那么按哪一种方案加工小红家可获得最大利润？最大利润是多少？
6．为迎接暑假旅游高峰的到来，七年级(5)班劳动实践小组的同学决定购进A，B两种纪念品售卖给到当地来旅游的游客．若购进A种纪念品7件，B种纪念品4件，需要760元；若购进A种纪念品5件，B种纪念品8件，需要800元．
(1)购进A，B两种纪念品每件各需要多少元？
(2)若该实践小组决定购进这两种纪念品共100件，考虑市场需求和资金周转，这100件纪念品的资金不少于7 100元，但不超过7 200元，那么该实践小组共有几种进货方案？
(3)若销售A种纪念品每件可获利润30元，B种纪念品每件可获利润20元，用(2)中的进货方案，哪一种方案获利最大？最大利润是多少元？第2课时　一元一次不等式组的应用
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1.会求一元一次不等式组的特殊解．2．会利用一元一次不等式组解决实际问题．
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知识点一　由解集情况确定字母的取值范围
练习1　若不等式组的解集为x＜m，则m的取值范围为m≤1．
知识点二　一元一次不等式组的实际应用
练习2　用若干辆装载质量为8吨的货车运一批货物，若每辆货车装4吨，则剩下20吨货物；若每辆货车装8吨，则最后一辆车装的货物不满也不空，若设有x辆货车，则x应满足的不等式组是(　B　).
A.
B．
C．
D．
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1．已知不等式组的解集是2【答案】x＝－
2．小慧购买了两种糕点当伴手礼，其中桂圆糕点一盒12个，售价70元；蜜枣糕点一盒6个，售价40元．已知小慧共购买10盒糕点，花费的金额不超过500元．若她将糕点分给75位同事，每人至少能拿到一个糕点，则小慧花了________元购买糕点．
【答案】490
3．下图是某运行程序，从“输入整数x”到“结果是否>18”为一次程序操作，①若输入整数11，输出结果为27；②若输入整数x后程序操作仅进行了两次就停止，则x的最大值是8；③若操作停止时输出结果为21，则输入的整数x是9；④输入整数x后，该操作永不停止，则x≤3.以上结论正确的是________(填序号).
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【答案】①②④
4．如图，用长为40 m的铁栅栏一边靠墙围成两个相同的长方形菜地(靠墙部分不使用铁栅栏)，墙的长度MN＝34 m，要使靠墙的AC边不小于25 m，那么与墙垂直的一边AB的长度范围为__________________．
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【答案】2≤AB≤5
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5．小红家开了一家糕点店，现有11 kg面粉，9.4 kg鸡蛋，计划加工一般糕点和精制糕点两种产品共50盒．已知加工1盒一般糕点需0.3 kg面粉和0.1 kg鸡蛋；加工1盒精制糕点需0.1 kg面粉和0.3 kg鸡蛋．
(1)有哪几种加工方案？
(2)如果销售1盒一般糕点和1盒精制糕点的利润分别为3元和4元，那么按哪一种方案加工小红家可获得最大利润？最大利润是多少？
【解】(1)设加工一般糕点x盒，则加工精制糕点(50－x)盒．
根据题意，得
解得28≤x≤30.
因为x为整数，
所以x可取28，29，30，
因此加工方案有三种：
①加工一般糕点28盒，精制糕点22盒；
②加工一般糕点29盒，精制糕点21盒；
③加工一般糕点30盒，精制糕点20盒．
(2)由题意知，精制糕点数量越多利润越大，故当加工一般糕点28盒、精制糕点22盒时，可获得最大利润，最大利润为28×3＋22×4＝172(元).
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6．为迎接暑假旅游高峰的到来，七年级(5)班劳动实践小组的同学决定购进A，B两种纪念品售卖给到当地来旅游的游客．若购进A种纪念品7件，B种纪念品4件，需要760元；若购进A种纪念品5件，B种纪念品8件，需要800元．
(1)购进A，B两种纪念品每件各需要多少元？
(2)若该实践小组决定购进这两种纪念品共100件，考虑市场需求和资金周转，这100件纪念品的资金不少于7 100元，但不超过7 200元，那么该实践小组共有几种进货方案？
(3)若销售A种纪念品每件可获利润30元，B种纪念品每件可获利润20元，用(2)中的进货方案，哪一种方案获利最大？最大利润是多少元？
【解】(1)设购进A种纪念品每件x元，B种纪念品每件y元．
根据题意，得
解得
答：购进A，B两种纪念品每件各需要80元、50元．
(2)设购进A种纪念品a件，则购进B种纪念品(100－a)件．
根据题意，得
解得70≤a≤.
因为a为正整数，
所以a的值为70或71或72或73，
故该实践小组共有4种进货方案．
方案1：购进A种纪念品70件，B种纪念品30件．
方案2：购进A种纪念品71件，B种纪念品29件．
方案3：购进A种纪念品72件，B种纪念品28件．
方案4：购进A种纪念品73件，B种纪念品27件．
(3)方案1的利润为30×70＋20×30＝2 700(元)，
方案2的利润为30×71＋20×29＝2 710(元)，
方案3的利润为30×72＋20×28＝2 720(元)，
方案4的利润为30×73＋20×27＝2 730(元)，
故方案4获利最大，最大利润是2 730元．11.3　一元一次不等式组
第1课时　一元一次不等式组的解法
1.了解一元一次不等式组的概念．2．理解一元一次不等式组的解集的意义，掌握求一元一次不等式组的解集的方法．3．会用数轴确定两个一元一次不等式组成的不等式组的解集．
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知识点一　一元一次不等式组
1．把两个含有 未知数的不等式合起来，组成一个一元一次不等式组．
2．一般地，几个不等式的解集的 ，叫作由它们所组成的不等式组的解集．
练习1　下列不等式组为一元一次不等式组的是( ).
A．
B．
C．
D．
知识点二　一元一次不等式组的解法
3．一元一次不等式组的解集的求法：解一元一次不等式组时，一般先求出其中各不等式的解集，再求出这些解集的 ．利用数轴可以直观地确定不等式组的解集．
4．不等式组的解的情况与图示．
eq \o(\s\up7(),\s\do5(图1))　 eq \o(\s\up7(),\s\do5(图2))
eq \o(\s\up7(),\s\do5(图3))　 eq \o(\s\up7(),\s\do5(图4))
(1)同大取大：图示：如图1，解集为 ．
(2)同小取小：图示：如图2，解集为 ．
(3)大小小大中间找：图示：如图3，解集为 ．
(4)大大小小无处找：图示：如图4，解集为 ．
练习2　(教材P139例1变式)解下列不等式组并把解集在数轴上表示出来：
(1)
(2)
1．将不等式组的解集表示在数轴上，下列正确的是( ).
A． B．
C． D．
2．已知关于x的不等式组无解，则a的取值范围是( ).
A．a＞2 B．a≥2
C．a＜2 D．a≤2
3．如图，一个运算程序，若需要经过两次运算才能输出结果，则x的取值范围为( ).
A．x>8 B．8C．8≤x≤13 D．8≤x<13
4．若关于x的不等式组的解集为x＜3，则m的取值范围是( ).
A．m＞2 B．m＜2
C．m≥2 D．m≤2
5．解不等式组：并把解集在数轴上表示出来，再写出所有的非正整数解．
6．已知关于x的不等式组的整数解共有6个，则a的取值范围是( ).
A．－5C．－5≤a≤－4 D．－57．已知关于x，y的方程组
(1)若此方程组的解满足－1＜x＋y≤3，求a的取值范围；
(2)在(1)的条件下，若关于m的不等式2am－m＞2a－1的解集为m＜1，求满足条件的a的整数值．
8．若某一元一次方程的解在某一元一次不等式组的解集范围内，则称该一元一次方程为该不等式组的“关联方程”．例如：方程x－1＝3的解为x＝4，而不等式组的解集为3＜x＜5，不难发现x＝4在3＜x＜5的范围内，所以方程x－1＝3是不等式组的“关联方程”．
(1)在方程①2(x＋1)－x＝5；②2x－4＝0；③＋1＝x中，关于x的不等式组的“关联方程”是________(填序号)；
(2)若关于x的方程2x－k＝6是不等式组的“关联方程”，求k的取值范围．