11.1 不等式 (2课时,学生版+答案版)2025-2026学年数学人教版七年级下册

11.1　不等式
11．1.1　不等式及其解集
1.结合具体问题，了解不等式的意义．2．学会推理不等式的解与理解解集的意义．3．能在数轴上表示出不等式的解集．
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知识点一　不等式的概念
1．不等式：用符号“＜”或“＞”表示 关系的式子，叫作不等式．用“≠”表示 关系的式子也是不等式．
练习1　下列式子：①3>0；②4x＋y<2；③x＋5＝0；④y－7；⑤m－2.5＞3；⑥x≠－3.其中不等式有( ).
A．2个 B．3个
C．4个 D．5个
知识点二　不等式的解与解集
2．把使不等式成立的未知数的 叫作不等式的解．它是一个具体的数值．
3．一般地，一个含有未知数的不等式的 ，组成这个不等式的解集．它是一个集合，包含了不等式所有的解．
练习2　下列说法正确的是( ).
A．y＝4是不等式y＋4<5的解
B．y＝2是不等式2y<7的解集
C．不等式2x<6的解集是x＝3
D．y＝2是不等式3y≥6的一个解
知识点三　用数轴表示不等式的解集
练习3　在数轴上表示不等式x＞－2的解集，正确的是( ).
A． B．
C． D．
知识点四　列不等式
4．列不等式：审清题意，弄清关键词的含义，找出已知量与未知量以及它们之间存在的关系，然后用 将不等关系表示出来．
练习4　(教材P121例1变式)x与5的和大于6，用不等式表示为( ).
A．x＋5＜6 B．x＋5＞6
C．x－5＞6 D．x－5＜6
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基础巩固
1．下列数学表达式中：①－8＜0；②2x＋3y＞6；③x＝11；④m2－4mn＋n2；⑤x≠8；⑥x＋1＜7.其中不等式有( ).
A．3个 B．4个
C．5个 D．6个
2．下列说法正确的是( ).
A．x＝1是不等式x<2的一个解
B．x＝2是不等式3x>5的解集
C．不等式3x>9的解集是x＝4
D．x<5是不等式x－5>0的解集
3．北斗高精导航能够实时显示当前路口的信号灯颜色及时长，一辆小车匀速行驶在限速60 km/h的路段上，当距离下一路口800 m时，发现导航显示下一路口的信号灯为绿灯，且剩余时间为64 s，此时导航提示：按照当前速度行驶能通过下一路口，则小车当前行驶速度x km/h的取值范围是____________．
4．下列数中，能使不等式5x－1＜6成立的x的值为( ).
A．1 B．2
C．3 D．4
5．在数轴上表示下列不等式：
(1)x＞－3; (2)x＜4.
6．根据下列关系列出不等式．
(1)x2是非负数；
(2)x的相反数与1的差小于2；
(3)x与7的和比x的2倍小；
(4)x的2倍与5的和是正数；
(5)a，b两数的平方的差不小于1；
(6)a的绝对值不小于它本身．
7．用不等式表示下列数量之间的关系：
(1)哥哥存款x元，弟弟存款y元，兄弟二人的存款总数少于2 000元；
(2)长为a＋5，宽为a－3的长方形的面积小于边长为a＋2的正方形的面积；
(3)一列动车有n节车厢，每节车厢有100个座位．在五一期间，这列动车上有m个人，其中有一些人没有座位；
(4)某市身高不超过1.2 m的儿童可免费乘坐公共汽车．记可以免费乘坐公共汽车的儿童的身高为h(m).11．1.2　不等式的性质
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1.探索不等式的基本性质．2．掌握不等式的三个性质并且能正确应用．3．理解解不等式的概念．4．熟练运用不等式的性质解不等式．5．感受不等式在实际生活中的应用．
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知识点一　不等式的性质
1．不等式的性质1：不等式两边加(或减)同一个数(或式子)，不等号的方向不变．如果a＞b，那么a±c＞b±c．
2．不等式的性质2：不等式两边乘(或除以)同一个正数，不等号的方向不变．如果a＞b，c＞0，那么ac＞bc(或＞).
3．不等式的性质3：不等式两边乘(或除以)同一个负数，不等号的方向改变．如果a＞b，c＜0，那么ac＜bc(或＜).
练习1　(教材P125例2变式)已知a＜b，则下列各式中，错误的是(　D　).
A．a＋4＜b＋4 B．a－4＜b－4
C．＜ D．－4a＜－4b
知识点二　利用不等式的性质解不等式
4．与解方程类似，解不等式要借助不等式的性质，将不等式逐步化为x＞m或x＜m(m为常数)的形式．还可以在数轴上直观地表示不等式的解集．
练习2　(教材P126例3变式)将下列不等式化成“x＞a”或“x＜a”的形式：
(1)5x＞4x－6；　(2)－x－2＜6.
【答案】(1)x＞－6
(2)x＞－8
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1．若a＜b，则下列式子中一定正确的是(　D　).
A.ac＜bc B．am2＜bm2
C．2－a＜2－b D．a＋2＜b＋2
2．下列变形正确的是(　D　).
A．由2x＞－x＋1，得2x－x＞1
B．由2－x＜3，得－x＞3－2
C．由2x≥3，得x≥
D．由－3x≥－6，得x≤2
3．在不等式2x＋4＞0两边同时________得不等式2x＞－4，在不等式2x＞－4两边同时________得________，则原不等式的解集为________．
【答案】减4　除以2　x＞－2　x＞－2
4．用“>”或“<”填空：
(1)a＋3________a＋5；
(2)－a－1________－a－3；
(3)若m＋3(4)若ac2>bc2，则a________b，－a－4________－b－4.
【答案】(1)＜　(2)>　(3)＜　>
(4)＞　<
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5．某种商品的进价为500元，标价为750元，商店要求以利润率不低于5%的售价打折出售．设商店在标价的基础上打x折出售商品，那么x满足的条件是(　B　).
A．750·≥500×5%
B．750·≥500×(1＋5%)
C．750·≤500×5%
D．750·≤500×(1＋5%)
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7．若不等式组(2m－3)x>2m－3的解集为x<1，则符合条件的正整数m的值为________．
【答案】1
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8．问题：已知x－y＝2，且x>1，y<0，试确定x＋y的取值范围．
解决此问题的过程如下：
解：因为x－y＝2，x>1，
所以x－2>－1.所以y>－1.
又y<0，
所以－1同理得1由①＋②，得－1＋1所以0请按照上述方法，解答下列问题：
(1)若a－b＝4，且a>1，b<2，求a＋b的取值范围；
(2)若a－b＝10，且a>1，b≤1，求2a＋3b的取值范围及其最大值．
【解】(1)因为a－b＝4，a>1，
所以a＝b＋4>1，
所以b>－3.
又因为b<2，
所以－3同理可得1由①＋②，得－3＋1所以－2(2)因为a－b＝10，a>1，
所以a＝b＋10>1，
所以b>－9.
又因为b≤1，
所以－9所以－27<3b≤3.③
同理可得2<2a≤22，④
由③＋④，得－27＋2<2a＋3b≤3＋22，
所以－25<2a＋3b≤25，
所以2a＋3b的最大值为25.INCLUDEPICTURE "第十一章章头.TIF" INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\第十一章章头.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\第十一章章头.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\第十一章章头.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\第十一章章头.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\第十一章章头.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\第十一章章头.TIF" \* MERGEFORMATINET
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11.1　不等式
11．1.1　不等式及其解集
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1.结合具体问题，了解不等式的意义．2．学会推理不等式的解与理解解集的意义．3．能在数轴上表示出不等式的解集．
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知识点一　不等式的概念
1．不等式：用符号“＜”或“＞”表示不等关系的式子，叫作不等式．用“≠”表示不等关系的式子也是不等式．
练习1　下列式子：①3>0；②4x＋y<2；③x＋5＝0；④y－7；⑤m－2.5＞3；⑥x≠－3.其中不等式有(　C　).
A．2个 B．3个
C．4个 D．5个
知识点二　不等式的解与解集
2．把使不等式成立的未知数的值叫作不等式的解．它是一个具体的数值．
3．一般地，一个含有未知数的不等式的所有的解，组成这个不等式的解集．它是一个集合，包含了不等式所有的解．
练习2　下列说法正确的是(　D　).
A．y＝4是不等式y＋4<5的解
B．y＝2是不等式2y<7的解集
C．不等式2x<6的解集是x＝3
D．y＝2是不等式3y≥6的一个解
知识点三　用数轴表示不等式的解集
练习3　在数轴上表示不等式x＞－2的解集，正确的是(　C　).
A． INCLUDEPICTURE "25W21.TIF" INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21.TIF" \* MERGEFORMATINET B． INCLUDEPICTURE "25W21A.TIF" INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21A.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21A.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21A.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21A.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21A.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21A.TIF" \* MERGEFORMATINET
C． INCLUDEPICTURE "25W21B.TIF" INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21B.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21B.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21B.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21B.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21B.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21B.TIF" \* MERGEFORMATINET D． INCLUDEPICTURE "25W21C.TIF" INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21C.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21C.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21C.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21C.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21C.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\AppData\\Local\\Microsoft\\Windows\\INetCache\\Content.Word\\25W21C.TIF" \* MERGEFORMATINET
知识点四　列不等式
4．列不等式：审清题意，弄清关键词的含义，找出已知量与未知量以及它们之间存在的关系，然后用不等式将不等关系表示出来．
练习4　(教材P121例1变式)x与5的和大于6，用不等式表示为(　B　).
A．x＋5＜6 B．x＋5＞6
C．x－5＞6 D．x－5＜6
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1．下列数学表达式中：①－8＜0；②2x＋3y＞6；③x＝11；④m2－4mn＋n2；⑤x≠8；⑥x＋1＜7.其中不等式有(　B　).
A．3个 B．4个
C．5个 D．6个
2．下列说法正确的是(　A　).
A．x＝1是不等式x<2的一个解
B．x＝2是不等式3x>5的解集
C．不等式3x>9的解集是x＝4
D．x<5是不等式x－5>0的解集
3．北斗高精导航能够实时显示当前路口的信号灯颜色及时长，一辆小车匀速行驶在限速60 km/h的路段上，当距离下一路口800 m时，发现导航显示下一路口的信号灯为绿灯，且剩余时间为64 s，此时导航提示：按照当前速度行驶能通过下一路口，则小车当前行驶速度x km/h的取值范围是____________．
【答案】45≤x≤60
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4．下列数中，能使不等式5x－1＜6成立的x的值为(　A　).
A．1 B．2
C．3 D．4
5．在数轴上表示下列不等式：
(1)x＞－3; (2)x＜4.
【答案】图略
6．根据下列关系列出不等式．
(1)x2是非负数；
(2)x的相反数与1的差小于2；
(3)x与7的和比x的2倍小；
(4)x的2倍与5的和是正数；
(5)a，b两数的平方的差不小于1；
(6)a的绝对值不小于它本身．
【解】(1)x2≥0.
(2)－x－1<2.
(3)x＋7<2x.
(4)2x＋5>0.
(5)a2－b2≥1.
(6)|a|≥a.
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7．用不等式表示下列数量之间的关系：
(1)哥哥存款x元，弟弟存款y元，兄弟二人的存款总数少于2 000元；
(2)长为a＋5，宽为a－3的长方形的面积小于边长为a＋2的正方形的面积；
(3)一列动车有n节车厢，每节车厢有100个座位．在五一期间，这列动车上有m个人，其中有一些人没有座位；
(4)某市身高不超过1.2 m的儿童可免费乘坐公共汽车．记可以免费乘坐公共汽车的儿童的身高为h(m).
【解】(1)x＋y<2 000.
(2)(a＋5)(a－3)<(a＋2)2.
(3)100n(4)h≤1.2.11．1.2　不等式的性质
1.探索不等式的基本性质．2．掌握不等式的三个性质并且能正确应用．3．理解解不等式的概念．4．熟练运用不等式的性质解不等式．5．感受不等式在实际生活中的应用．
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知识点一　不等式的性质
1．不等式的性质1：不等式两边加(或减) 数(或式子)，不等号的方向 ．如果a＞b，那么 ．
2．不等式的性质2：不等式两边乘(或除以) ，不等号的方向 ．如果a＞b，c＞0，那么 (或 ).
3．不等式的性质3：不等式两边乘(或除以) ，不等号的方向 ．如果a＞b，c＜0，那么 (或 ).
练习1　(教材P125例2变式)已知a＜b，则下列各式中，错误的是( ).
A．a＋4＜b＋4 B．a－4＜b－4
C．＜ D．－4a＜－4b
知识点二　利用不等式的性质解不等式
4．与解方程类似，解不等式要借助不等式的性质，将不等式逐步化为 或 (m为常数)的形式．还可以在数轴上直观地表示不等式的解集．
练习2　(教材P126例3变式)将下列不等式化成“x＞a”或“x＜a”的形式：
(1)5x＞4x－6；　(2)－x－2＜6.
1．若a＜b，则下列式子中一定正确的是( ).
A.ac＜bc B．am2＜bm2
C．2－a＜2－b D．a＋2＜b＋2
2．下列变形正确的是( ).
A．由2x＞－x＋1，得2x－x＞1
B．由2－x＜3，得－x＞3－2
C．由2x≥3，得x≥
D．由－3x≥－6，得x≤2
3．在不等式2x＋4＞0两边同时________得不等式2x＞－4，在不等式2x＞－4两边同时________得________，则原不等式的解集为________．
4．用“>”或“<”填空：
(1)a＋3________a＋5；
(2)－a－1________－a－3；
(3)若m＋3(4)若ac2>bc2，则a________b，－a－4________－b－4.
5．某种商品的进价为500元，标价为750元，商店要求以利润率不低于5%的售价打折出售．设商店在标价的基础上打x折出售商品，那么x满足的条件是( ).
A．750·≥500×5%
B．750·≥500×(1＋5%)
C．750·≤500×5%
D．750·≤500×(1＋5%)
6．(跨学科融合)设，，分别表示三种不同的物体，现用天平称两次，情况如图所示，那么，，这三种物体按质量从大到小排列应为( ).
A．，， B．，，
C．，， D．，，
7．若不等式组(2m－3)x>2m－3的解集为x<1，则符合条件的正整数m的值为________．
8．问题：已知x－y＝2，且x>1，y<0，试确定x＋y的取值范围．
解决此问题的过程如下：
解：因为x－y＝2，x>1，
所以x－2>－1.所以y>－1.
又y<0，
所以－1同理得1由①＋②，得－1＋1所以0请按照上述方法，解答下列问题：
(1)若a－b＝4，且a>1，b<2，求a＋b的取值范围；
(2)若a－b＝10，且a>1，b≤1，求2a＋3b的取值范围及其最大值．