6.10 绝对价值不平等

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• 选择节绝对价值不平等

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  绝对价值不平等

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  Absolute Value Inequalities 
  ::绝对价值不平等

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  Absolute value inequalities are solved in a similar way to . In both cases, you must consider the same two options:
  ::绝对价值不平等的解决方式与 .在这两种情况下,你必须考虑同样的两种选择:

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  1. The expression inside the absolute value is not negative.
     ::绝对值中的表达式不是负值。
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  2. The expression inside the absolute value is negative.
     ::绝对值内的表达式为负值。

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  Then you must solve each inequality separately.
  ::然后,你必须分别解决每一个不平等问题。

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  Solve Absolute Value Inequalities
  ::解决绝对价值不平等

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  Consider the inequality $\begin{array}{r}|x|\le 3\end{array}$ . Since the absolute value of $\begin{array}{r}x\end{array}$ represents the distance from zero, the solutions to this inequality are those numbers whose distance from zero is less than or equal to 3. The following graph shows this solution:
  ::考虑不平等 x3。 由于x的绝对值代表距离零的距离,解决这种不平等的办法就是那些与零的距离小于或等于3的数字。 下图显示了这一解决办法:

  

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  Notice that this is also the graph for the compound inequality $\begin{array}{r}-3\le x\le 3\end{array}$ .
  ::请注意,这也是化合物不平等的图表-3x_3。

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  Now consider the inequality $\begin{array}{r}|x|>2\end{array}$ . Since the absolute value of $\begin{array}{r}x\end{array}$ represents the distance from zero, the solutions to this inequality are those numbers whose distance from zero are more than 2. The following graph shows this solution.
  ::现在考虑不平等 x2. 由于x的绝对值代表与零的距离,解决这种不平等的办法就是那些与零的距离大于2的数字。 下图显示了这一解决办法。

  

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  Notice that this is also the graph for the compound inequality $\begin{array}{r}x<-2\end{array}$ or $\begin{array}{r}x>2\end{array}$ .
  ::请注意,这也是化合物不平等的图形 x2 或 x>2 。

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  Solving Inequalities 
  ::解决不平等

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  Solve the following inequalities and show the solution graph.
  ::解决以下不平等问题,并显示解析图。

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  a)  $\begin{array}{r}|x|<5\end{array}$
  :a) x5

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    $\begin{array}{r}|x|<5\end{array}$ represents all numbers whose distance from zero is less than 5.
  ::x==================================================================

  

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  This answer can be written as “ $\begin{array}{r}-5<x<5\end{array}$ ”
  ::此答案可以写为“ - 5 < x < 5” 。

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  b)  $\begin{array}{r}|x|\ge 2.5\end{array}$
  :b) x2.5

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  $\begin{array}{r}|x|\ge 2.5\end{array}$ represents all numbers whose distance from zero is more than or equal to 2.5
  ::=============================================================================================

  

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  This answer can be written as “ $\begin{array}{r}x\le -2.5\end{array}$ or $\begin{array}{r}x\ge 2.5\end{array}$ ”
  ::此答案可以写为“ x2.5 或 x2.5 ”

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  Rewrite and Solve Absolute Value Inequalities as Compound Inequalities
  ::重写和解决作为复合不平等的绝对价值不平等

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  In the last section you saw that absolute value inequalities are .
  ::在最后一节,你看到绝对价值不平等。

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  Inequalities of the type $\begin{array}{r}|x|<a\end{array}$ can be rewritten as “ $\begin{array}{r}-a<x<a\end{array}$ ”
  ::xa类型中的不平等可以改写为“-a<x<a”

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  Inequalities of the type $\begin{array}{r}|x|>b\end{array}$ can be rewritten as “ $\begin{array}{r}x<-b\end{array}$ or $\begin{array}{r}x>b\end{array}$ ”
  ::xb类型的不平等可以改写为“xb或 x>b”

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  To solve an absolute value inequality, we separate the expression into two inequalities and solve each of them individually.
  ::为了解决绝对价值不平等问题,我们将表达方式分为两种不平等,并单独解决每一种不平等。

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  1. Solve the inequality $\begin{array}{r}|x-3|<7\end{array}$ and show the solution graph.
  ::1. 解决不平等问题*x-37,并显示解决方案图。

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  Re-write as a compound inequality: $\begin{array}{r}-7<x-3<7\end{array}$
  ::重写为复合不平等: - 7 < x-3 < 7

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  Write as two separate inequalities: $\begin{array}{r}x-3<7\end{array}$ and $\begin{array}{r}x-3>-7\end{array}$
  ::以两种不同的不平等形式写作: x-3<7 和 x-37

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  Solve each inequality: $\begin{array}{r}x<10\end{array}$ and $\begin{array}{r}x>-4\end{array}$
  ::解决每一种不平等: x<10 和 x4

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  Re-write solution: $\begin{array}{r}-4<x<10\end{array}$
  ::重写解决方案 : - 4 < x < 10

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  The solution graph is
  ::解决方案图

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  We can think of the question being asked here as “What numbers are within 7 units of 3?” The answer can then be expressed as “All the numbers between -4 and 10.”
  ::我们可以将这里提出的问题视为“7个单位中的7个单位在3个中是多少?” 然后,答案可以表述为“4个和10个之间的所有数字”。

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  2. Solve the inequality $\begin{array}{r}|4x+5|\le 13\end{array}$ and show the solution graph.
  ::2. 消除不平等+4x+5+13,并显示解析图。

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  Re-write as a compound inequality: $\begin{array}{r}-13\le 4x+5\le 13\end{array}$
  ::重写为一种复合不平等: - 134x+513

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  Write as two separate inequalities: $\begin{array}{r}4x+5\le 13\end{array}$ and $\begin{array}{r}4x+5\ge -13\end{array}$
  ::写成两种不同的不平等:4x+513和4x+513。

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  Solve each inequality: $\begin{array}{r}4x\le 8\end{array}$ and $\begin{array}{r}4x\ge -18\end{array}$
  ::解决每一种不平等:4x8和4x18

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  $\begin{array}{r}x\le 2\end{array}$ and $\begin{array}{r}x\ge -\frac{9}{2}\end{array}$
  ::x% 2 和 x% 92

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  Re-write solution: $\begin{array}{r}-\frac{9}{2}\le x\le 2\end{array}$
  ::重写解决方案:- 92x% 2

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  The solution graph is
  ::解决方案图

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  3. Solve the inequality $\begin{array}{r}|x+12|>2\end{array}$ and show the solution graph.
  ::3. 解决不平等问题*x+122,并显示解析图。

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  Re-write as a compound inequality: $\begin{array}{r}x+12<-2\end{array}$ or $\begin{array}{r}x+12>2\end{array}$
  ::重写为复合不平等: x+122 或 x+12>2

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  Solve each inequality: $\begin{array}{r}x<-14\end{array}$ or $\begin{array}{r}x>-10\end{array}$
  ::解决每一种不平等: x14 或 x10

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  The solution graph is
  ::解决方案图

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  Solve Real-World Problems Using Absolute Value Inequalities
  ::利用绝对价值不平等解决现实世界问题

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  Absolute value inequalities are useful in problems where we are dealing with a range of values.
  ::绝对价值不平等在我们处理一系列价值问题时是有益的。

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  Real-World Application: Velocity 
  ::现实世界应用:速度

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  The velocity of an object is given by the formula $\begin{array}{r}v=25t-80\end{array}$ , where the time is expressed in seconds and the velocity is expressed in feet per second. Find the times for which the magnitude of the velocity is greater than or equal to 60 feet per second.
  ::公式 v=25t- 80给定物体的速度,其中时间以秒表示,速度以每秒英尺表示。查找速度大小大于或等于每秒60英尺的时间。

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  The magnitude of the velocity is the absolute value of the velocity. If the velocity is $\begin{array}{r}25t-80\end{array}$ feet per second, then its magnitude is $\begin{array}{r}|25t-80|\end{array}$ feet per second. We want to find out when that magnitude is greater than or equal to 60, so we need to solve $\begin{array}{r}|25t-80|\ge 60\end{array}$ for $\begin{array}{r}t\end{array}$ .
  ::速度的大小是速度的绝对值。 如果速度为 25t- 80 英尺/ 秒, 那么速度为 25t- 80 英寸/ 秒 。 我们想找出该数量在什么时候大于或等于 60, 所以我们需要解决 25t- 80 ° 60 t 。

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  First we have to split it up: $\begin{array}{r}25t-80\ge 60\end{array}$ or $\begin{array}{r}25t-80\le -60\end{array}$
  ::首先我们得分头行动: 25t-8060或25t-8060

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  Then solve: $\begin{array}{r}25t\ge 140\end{array}$ or $\begin{array}{r}25t\le 20\end{array}$
  ::然后解答: 25t140 或 25t20

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  $\begin{array}{r}t\ge 5.6\end{array}$ or $\begin{array}{r}t\le 0.8\end{array}$
  ::t5.6 或 t0.8

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  The magnitude of the velocity is greater than 60 ft/sec for times less than 0.8 seconds and for times greater than 5.6 seconds.
  ::速度的大小大于60平方英尺/秒,时速小于0.8秒,时速大于5.6秒。

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  When $\begin{array}{r}t=0.8seconds,v=25(0.8)-80=-60ft/sec.\end{array}$ The magnitude of the velocity is 60 ft/sec. (The negative sign in the answer means that the object is moving backwards.)
  ::当 t=0. 8 秒时, v= 25( 0.8)- 8060 ft/sec. 速度的大小为 60 ft/sec. (回答中的负符号表示对象向后移动 。)

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  When $\begin{array}{r}t=5.6seconds,v=25(5.6)-80=60ft/sec.\end{array}$
  ::当 t=5.6 秒时, v= 25( 5.6)- 80= 60 ft/sec.

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  To find where the magnitude of the velocity is greater than 60 ft/sec, check some arbitrary values in each of the following time intervals: $\begin{array}{r}t\le 0.8,0.8\le t\le 5.6\end{array}$ and $\begin{array}{r}t\ge 5.6\end{array}$ .
  ::要找到速度大于60英尺/秒的大小,请在以下每个时间间隔中检查一些任意值:t0.8、0.8°t5.6和t5.6。

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  Check $\begin{array}{r}t=0.5:v=25(0.5)-80=-67.5ft/sec\end{array}$
  ::检查 t= 0.5: v= 25( 0. 5) - 80\\\\\\ 67.5 ft/sec

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  Check $\begin{array}{r}t=2:v=25(2)-80=-30ft/sec\end{array}$
  ::检查 t=2: v=25 (2)- 80\\\ 30 ft/sec

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  Check $\begin{array}{r}t=6:v=25(6)-80=-70ft/sec\end{array}$
  ::检查 t= 6: v= 25(6)- 80_ 70 ft/sec

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  You can see that the magnitude of the velocity is greater than 60 ft/sec only when $\begin{array}{r}t\ge 5.6\end{array}$ or when $\begin{array}{r}t\le 0.8\end{array}$ .
  ::只有在 t5.6 或 t0.8 时, 速度的大小才会大于 60 英尺/ sec 。

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  The answer checks out.
  ::答案检查出来。

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  Example 
  ::示例示例示例示例

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  Solve the inequality $\begin{array}{r}|8x-15|\ge 9\end{array}$ and show the solution graph.
  ::解决不平等 Q8x- 159, 并显示解析图 。

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  Re-write as a compound inequality: $\begin{array}{r}8x-15\le -9\end{array}$ or $\begin{array}{r}8x-15\ge 9\end{array}$
  ::重写为复合不平等:8x-159或8x-159

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  Solve each inequality: $\begin{array}{r}8x\le 6\end{array}$ or $\begin{array}{r}8x\ge 24\end{array}$
  ::解决每一种不平等:8x6或8x%24

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  $\begin{array}{r}x\le \frac{3}{4}\end{array}$ or $\begin{array}{r}x\ge 3\end{array}$
  ::x34 或 x3

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  The solution graph is
  ::解决方案图

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  Review 
  ::回顾

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  Solve the following inequalities and show the solution graph.
  ::解决以下不平等问题,并显示解析图。

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  1. $\begin{array}{r}|x|\le 6\end{array}$
     ::
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  2. $\begin{array}{r}|x|>3.5\end{array}$
     ::
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  3. $\begin{array}{r}|x|<12\end{array}$
     ::x12
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  4. $\begin{array}{r}|x|>10\end{array}$
     ::
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  5. $\begin{array}{r}|7x|\ge 21\end{array}$
     ::7x21号
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  6. $\begin{array}{r}|x-5|>8\end{array}$
     ::- 58
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  7. $\begin{array}{r}|x+7|<3\end{array}$
     ::X+73
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  8. $\begin{array}{r}\left|x-\frac{3}{4}\right|\le \frac{1}{2}\end{array}$
     ::X - 3412
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  9. $\begin{array}{r}|2x-5|\ge 13\end{array}$
     ::2x-513
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  10. $\begin{array}{r}|5x+3|<7\end{array}$
      ::5x+3+7
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  11. $\begin{array}{r}\left|\frac{x}{3}-4\right|\le 2\end{array}$
      ::X3 -42
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  12. $\begin{array}{r}\left|\frac{2x}{7}+9\right|>\frac{5}{7}\end{array}$
      ::2x7+9 57
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  13. .
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      1. How many solutions does the inequality $\begin{array}{r}|x|\le 0\end{array}$ have?
         ::不平等有多少解决方案?
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      2. How about the inequality $\begin{array}{r}|x|\ge 0\end{array}$ ?
         ::不平等如何?
      
      ::。 不平等有多少解决办法? 不平等如何? 不平等如何? 不平等如何?
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  14. A company manufactures rulers. Their 12-inch rulers pass quality control if they are within $\begin{array}{r}\frac{1}{32}inches\end{array}$ of the ideal length. What is the longest and shortest ruler that can leave the factory?
      ::公司制造统治者。 他们的12英寸统治者如果在理想长度的132英寸之内,就会通过质量控制。 能够离开工厂的最长和最短的统治者是什么?
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  15. A three month old baby boy weighs an average of 13 pounds. He is considered healthy if he is at most 2.5 lbs. more or less than the average weight. Find the weight range that is considered healthy for three month old boys.
      ::3个月大的男婴平均体重为13磅,如果最大体重为2.5磅,则被视为健康,或大于或小于平均体重,为3个月大的男孩确定被认为是健康的体重范围。

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。