1
00:00:00,590 --> 00:00:03,880
让我们来学习解绝对值方程式

2
00:00:03,880 --> 00:00:05,119
先来复习下

3
00:00:05,119 --> 00:00:07,650
当计算一个数值的绝对值时

4
00:00:07,650 --> 00:00:10,680
以-1为例 来计算绝对值

5
00:00:10,680 --> 00:00:12,263
你需要做的是

6
00:00:12,263 --> 00:00:16,090
计算出1与0之间的距离

7
00:00:16,090 --> 00:00:20,620
对-1来说 下面以数轴示例

8
00:00:20,620 --> 00:00:23,310
这个没画好

9
00:00:23,310 --> 00:00:26,230
这是数轴 这是0

10
00:00:26,230 --> 00:00:28,470
这是-1

11
00:00:28,470 --> 00:00:30,230
-1与0之间的距离是1

12
00:00:30,230 --> 00:00:33,250
那-1的绝对值就是1

13
00:00:33,250 --> 00:00:38,850
1和0之间的距离也是1

14
00:00:38,850 --> 00:00:40,610
所以1的绝对值同样是1

15
00:00:40,610 --> 00:00:43,500
也就是说 绝对值为与0之间的距离

16
00:00:43,500 --> 00:00:45,587
来换一种更简单的方式来理解

17
00:00:45,587 --> 00:00:48,600
绝对值其实就是取数值的正值

18
00:00:48,600 --> 00:00:59,360
-7346的绝对值是7346.

19
00:00:59,360 --> 00:01:00,779
理解这点以后 下面

20
00:01:00,779 --> 00:01:05,050
开始学习解绝对值方程式

21
00:01:05,050 --> 00:01:06,675
来看这个方程式

22
00:01:06,675 --> 00:01:14,500
x-5的绝对值等于10

23
00:01:14,500 --> 00:01:15,895
可以这样解读这个等式

24
00:01:15,895 --> 00:01:18,161
也希望能够这样理解 这就是说

25
00:01:18,161 --> 00:01:23,120
x与5之间的距离为10

26
00:01:23,120 --> 00:01:26,750
与5之间的距离是10的数值有几个

27
00:01:26,750 --> 00:01:29,430
大家应该已经想到解法了

28
00:01:29,430 --> 00:01:31,960
不过下面要学习更系统的解法

29
00:01:31,960 --> 00:01:36,510
这个方程式有两种情况

30
00:01:36,510 --> 00:01:41,800
一种是x-5等于10

31
00:01:41,800 --> 00:01:44,630
如果假定绝对值里面的值为10

32
00:01:44,630 --> 00:01:46,610
取绝对值后

33
00:01:46,610 --> 00:01:48,380
会得到10

34
00:01:48,380 --> 00:01:53,130
或者假设x-5等于-10

35
00:01:53,130 --> 00:01:58,700
如果x-5等于-10 取绝对值后

36
00:01:58,700 --> 00:01:59,950
也是10

37
00:01:59,950 --> 00:02:04,280
所以第二种情况是 x-5等于-10

38
00:02:04,280 --> 00:02:07,730
两种情况同时满足这个方程式

39
00:02:07,730 --> 00:02:08,958
先解这个

40
00:02:08,958 --> 00:02:11,500
等式两边加5

41
00:02:11,500 --> 00:02:14,160
得到x=15

42
00:02:14,160 --> 00:02:17,830
再解这个 等式两边加5

43
00:02:17,830 --> 00:02:20,900
x等于-5

44
00:02:20,900 --> 00:02:21,963
得出计算结果

45
00:02:21,963 --> 00:02:24,910
有两个x值满足这个方程式

46
00:02:24,910 --> 00:02:26,890
x可以是15

47
00:02:26,890 --> 00:02:29,502
15-5等于10 取绝对值后

48
00:02:29,502 --> 00:02:32,690
得到10 或者x为-5

49
00:02:32,690 --> 00:02:36,060
-5减5等于-10

50
00:02:36,060 --> 00:02:39,020
取绝对值后 还是10

51
00:02:39,020 --> 00:02:41,632
注意 这两个值

52
00:02:41,632 --> 00:02:45,750
与5之间的距离都是10

53
00:02:45,750 --> 00:02:48,050
再解一个

54
00:02:48,050 --> 00:02:51,130
再来一个

55
00:02:51,130 --> 00:02:52,182
现在有

56
00:02:52,182 --> 00:02:58,580
x+2的绝对值等于6

57
00:02:58,580 --> 00:02:59,610
这说明什么呢

58
00:02:59,610 --> 00:03:03,132
也就是说 要么x+2

59
00:03:03,132 --> 00:03:07,030
也就是绝对值符号里的这个值 它等于6

60
00:03:07,030 --> 00:03:10,380
或者绝对值符号里的这个值

61
00:03:10,380 --> 00:03:12,050
x+2 它等于-6

62
00:03:12,050 --> 00:03:13,910
假设等于-6时

63
00:03:13,910 --> 00:03:16,210
取绝对值后 得到6

64
00:03:16,210 --> 00:03:20,340
那么 假设x+2等于-6

65
00:03:20,340 --> 00:03:22,880
然后 等式两边减2

66
00:03:22,880 --> 00:03:25,850
得到 x等于4

67
00:03:25,850 --> 00:03:29,780
再看这个方程式 两边减2

68
00:03:29,780 --> 00:03:33,690
得到x等于-8

69
00:03:33,690 --> 00:03:37,240
这就是方程式的两种解法

70
00:03:37,240 --> 00:03:39,740
为了加深理解

71
00:03:39,740 --> 00:03:42,500
对于绝对值 可以把它看成是距离

72
00:03:42,500 --> 00:03:43,940
这样重写这个绝对值方程式

73
00:03:43,940 --> 00:03:50,410
x减-2的绝对值等于6

74
00:03:50,410 --> 00:03:52,759
其实就是说

75
00:03:52,759 --> 00:03:57,590
哪些值与-2之间的距离刚好是6呢

76
00:03:57,590 --> 00:03:59,168
还记得吗 在这里讲过

77
00:03:59,168 --> 00:04:03,560
哪些值与5之间的距离是10

78
00:04:03,560 --> 00:04:05,990
不管被5减的值是几

79
00:04:05,990 --> 00:04:08,560
它们与5之间的距离肯定是10

80
00:04:08,560 --> 00:04:09,515
这边呢

81
00:04:09,515 --> 00:04:13,080
哪些值与-2之间的距离是6

82
00:04:13,080 --> 00:04:15,510
那就是4 或-8

83
00:04:15,510 --> 00:04:17,959
自己试着验证下结果

84
00:04:17,959 --> 00:04:20,459
再做一个

85
00:04:20,459 --> 00:04:25,330
再来一个 这次用紫色

86
00:04:25,330 --> 00:04:30,190
这有4x的绝对值

87
00:04:30,190 --> 00:04:31,430
来计算一个有难度的

88
00:04:31,430 --> 00:04:33,390
4x减1

89
00:04:33,390 --> 00:04:36,583
4x-1的绝对值 等于--

90
00:04:36,583 --> 00:04:40,200
这里让它等于19

91
00:04:40,200 --> 00:04:41,769
这样 跟之前的解法一样

92
00:04:41,769 --> 00:04:47,640
一种情况是4x减1等于19

93
00:04:47,640 --> 00:04:51,670
另一种情况是4x减1等于-19

94
00:04:51,670 --> 00:04:53,130
因为当取绝对值以后

95
00:04:53,130 --> 00:04:54,800
都会得到19

96
00:04:54,800 --> 00:04:59,100
或者4x减1等于-19

97
00:04:59,100 --> 00:05:00,970
现在只需要解这两个方程式

98
00:05:00,970 --> 00:05:02,945
等式两边加1

99
00:05:02,945 --> 00:05:04,274
两个同时做

100
00:05:04,274 --> 00:05:08,510
两边加1后 得到4x等于20

101
00:05:08,510 --> 00:05:11,005
这两边加1后

102
00:05:11,005 --> 00:05:15,340
得到4x等于-18

103
00:05:15,340 --> 00:05:20,210
等式两边除4 得到x等于5

104
00:05:20,210 --> 00:05:23,920
这个等式两边除4 得到x等于-18/4

105
00:05:23,920 --> 00:05:31,770
也等于-9/2

106
00:05:31,770 --> 00:05:35,730
这两个x值同时满足这个绝对值方程式

107
00:05:35,730 --> 00:05:36,587
试试看

108
00:05:36,587 --> 00:05:39,580
-9/2乘4

109
00:05:39,580 --> 00:05:41,570
等于-18

110
00:05:41,570 --> 00:05:44,200
-18再减1就是-19

111
00:05:44,200 --> 00:05:46,740
取绝对值后 得到19

112
00:05:46,740 --> 00:05:49,920
等于5时 4乘5是20

113
00:05:49,920 --> 00:05:51,960
减1等于19

114
00:05:51,960 --> 00:05:53,260
再取绝对值以后

115
00:05:53,260 --> 00:05:55,920
还是等于19

116
00:05:55,920 --> 00:05:58,580
下面来画绝对值方程式的图形 增加趣味性

117
00:05:58,580 --> 00:05:59,283
现在有

118
00:05:59,283 --> 00:06:04,990
y等于x+3的绝对值

119
00:06:04,990 --> 00:06:07,840
这是个函数 或者说图形

120
00:06:07,840 --> 00:06:09,410
并包含绝对值在里面

121
00:06:09,410 --> 00:06:11,820
这要考虑两种情况

122
00:06:11,820 --> 00:06:13,136
一种情况是

123
00:06:13,136 --> 00:06:16,430
绝对值符号里的这个值为正

124
00:06:16,430 --> 00:06:18,873
也就是x+3

125
00:06:18,873 --> 00:06:23,420
写这边 x+3>0

126
00:06:23,420 --> 00:06:29,370
另一种情况是x+3<0

127
00:06:29,370 --> 00:06:32,658
当x+3>0时

128
00:06:32,658 --> 00:06:36,490
这个图形 或这条直线 这里可能还不能称为一条直线

129
00:06:36,490 --> 00:06:41,690
这个函数 就等同于y=x+3

130
00:06:41,690 --> 00:06:44,370
如果这边这个值是大于0的话

131
00:06:44,370 --> 00:06:46,750
那么绝对值符号就形同虚设

132
00:06:46,750 --> 00:06:48,780
那么这个就完全等同于

133
00:06:48,780 --> 00:06:50,280
y=x+3

134
00:06:50,280 --> 00:06:52,590
那么什么情况下x+3>0

135
00:06:52,590 --> 00:06:56,366
不等式两边减3

136
00:06:56,366 --> 00:06:59,910
得到x>-3

137
00:06:59,910 --> 00:07:02,249
那么当x>-3时

138
00:07:02,249 --> 00:07:08,460
这个图形与y=x+3是等同的

139
00:07:08,460 --> 00:07:11,500
当x+3<0时

140
00:07:11,500 --> 00:07:13,328
在这种情况下

141
00:07:13,328 --> 00:07:16,509
绝对值符号里的值为负

142
00:07:16,509 --> 00:07:20,356
在这种情况下等式就变成

143
00:07:20,356 --> 00:07:26,250
y等于-（x+3）

144
00:07:26,250 --> 00:07:27,540
为什么呢

145
00:07:27,540 --> 00:07:30,520
仔细听 现在这里是负值 如果

146
00:07:30,520 --> 00:07:33,060
x+3是负值

147
00:07:33,060 --> 00:07:36,010
在这个前提下 如果它是负值的话

148
00:07:36,010 --> 00:07:38,090
那么当取负值的绝对值后

149
00:07:38,090 --> 00:07:40,050
它肯定是得到正值

150
00:07:40,050 --> 00:07:43,280
就相当于把它与-1相乘

151
00:07:43,280 --> 00:07:45,870
如果知道是取负值的绝对值

152
00:07:45,870 --> 00:07:48,890
就相当于用这个值乘-1

153
00:07:48,890 --> 00:07:51,010
因为绝对值都是正值

154
00:07:51,010 --> 00:07:53,870
也就是这里所讨论的情况

155
00:07:53,870 --> 00:07:55,840
x+3小于0

156
00:07:55,840 --> 00:07:59,850
两边减3 当x小于-3

157
00:07:59,850 --> 00:08:01,280
当x小于-3

158
00:08:01,280 --> 00:08:03,920
那么当x<-3时

159
00:08:03,920 --> 00:08:05,040
图形应该是这样

160
00:08:05,040 --> 00:08:08,280
当x>-3

161
00:08:08,280 --> 00:08:09,600
图形是这样

162
00:08:09,600 --> 00:08:11,300
下面一起来看

163
00:08:11,300 --> 00:08:13,670
完整的图形

164
00:08:13,670 --> 00:08:21,520
先画数轴

165
00:08:21,520 --> 00:08:26,070
这是x轴 这是y轴

166
00:08:26,070 --> 00:08:29,090
先把这个函数乘出来 这样可以得到mx

167
00:08:29,090 --> 00:08:29,870
加b的函数格式

168
00:08:29,870 --> 00:08:36,070
这个等于-x-3

169
00:08:36,070 --> 00:08:37,409
先看看这种函数图形

170
00:08:37,409 --> 00:08:38,620
的一般形式

171
00:08:38,620 --> 00:08:42,020
-x-3

172
00:08:42,020 --> 00:08:47,380
y轴截距是-3 所以1 2 3

173
00:08:47,380 --> 00:08:51,060
-x那就应该是向下倾斜 也就是

174
00:08:51,060 --> 00:08:52,290
向下斜率为1

175
00:08:52,290 --> 00:08:53,540
看起来就是这样

176
00:08:56,840 --> 00:09:02,830
x轴的截距应该是x等于

177
00:09:02,830 --> 00:09:07,740
当假设y等于0时 这边x

178
00:09:07,740 --> 00:09:08,575
等于-3

179
00:09:08,575 --> 00:09:10,380
穿过这条线

180
00:09:10,380 --> 00:09:11,920
这个点

181
00:09:11,920 --> 00:09:14,190
这个图形 如果没有这里这个限制条件

182
00:09:14,190 --> 00:09:15,600
应该是这样

183
00:09:19,890 --> 00:09:22,760
如果没有区间限制条件

184
00:09:22,760 --> 00:09:23,880
x轴的这个区间限制条件

185
00:09:23,880 --> 00:09:27,080
那么这个图形 应该是怎样的

186
00:09:27,080 --> 00:09:27,480
来看看

187
00:09:27,480 --> 00:09:31,810
y轴截距是3

188
00:09:31,810 --> 00:09:33,230
像这样

189
00:09:33,230 --> 00:09:35,260
x轴的截距呢

190
00:09:35,260 --> 00:09:37,970
当y等于0时 x等于-3

191
00:09:37,970 --> 00:09:39,760
还是一样穿过这个点

192
00:09:39,760 --> 00:09:40,620
斜率为1

193
00:09:40,620 --> 00:09:43,710
那就应该是这样的

194
00:09:43,710 --> 00:09:45,330
这就是这个函数的图形

195
00:09:45,330 --> 00:09:48,100
现在 可以得出结论 绝对值

196
00:09:48,100 --> 00:09:52,030
函数为紫色直线

197
00:09:52,030 --> 00:09:53,830
当x<-3时

198
00:09:53,830 --> 00:09:57,070
当x<-3 这里是x等于

199
00:09:57,070 --> 00:09:59,593
-3 当x<-3

200
00:09:59,593 --> 00:10:03,170
它就是这条紫线

201
00:10:03,170 --> 00:10:04,570
就是这条

202
00:10:04,570 --> 00:10:07,390
上面是当x<-3时的图形

203
00:10:07,390 --> 00:10:10,830
但是如果x>-3时 就应该是

204
00:10:10,830 --> 00:10:12,160
这条绿线了

205
00:10:12,160 --> 00:10:14,640
就是这条

206
00:10:14,640 --> 00:10:17,480
图形看起来很像奇怪的V

207
00:10:17,480 --> 00:10:21,430
当x>-3时 应该是正值

208
00:10:21,430 --> 00:10:24,950
所以这个图形 斜率为正

209
00:10:24,950 --> 00:10:28,270
但当x<-3时 本质上

210
00:10:28,270 --> 00:10:30,550
其实是取这个函数的负值 如果想要

211
00:10:30,550 --> 00:10:32,280
这么理解的话 所以这里是负斜率

212
00:10:32,280 --> 00:10:35,060
如果看到这种V型函数 或是

213
00:10:35,060 --> 00:10:38,250
V型图 那就表明它是

214
00:10:38,250 --> 00:10:39,950
绝对值函数