1.12 多重不平等

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  多重不平等

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  Mr. Garcia, the Spanish teacher, announces that the students' final grades will consist of 40% projects (0-100 score). The remaining 60% (0-100) will come from the final exam. Going into the final exam, Madison has an 84 in the project score. Within what grade range must she fall on the final exam to raise her score to an A (90-100 overall score)?
  ::西班牙教师Garcia先生宣布,学生的最后一年级将包括40%的项目(0-100分),其余的60%(0-100分)来自期末考试,进入期末考试时,麦迪逊在期末考试中得分为84分,她必须在哪个年级范围内参加期末考试,才能将得分提高到A(90-100分)?

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  Compound Inequalities
  ::多重不平等

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  are inequalities that have been joined by the words “and” or “or.” For example:
  ::“和”或“或”等字加上“和”或“或”。

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  $\begin{array}{r}-2<x\le 5\end{array}$ Read, “ $\begin{array}{r}x\end{array}$ is greater than -2 and less than or equal to 5.”
  ::− 2<x=5 读“x大于-2,小于或等于5”

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  $\begin{array}{r}x\ge 3\end{array}$ or $\begin{array}{r}x<-4\end{array}$ Read, “ $\begin{array}{r}x\end{array}$ is greater than or equal to 3 or less than -4.”
  ::读“x大于或等于3或小于-4。”

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  Notice that both of these inequalities have two inequality signs . So, it is like solving or graphing two inequalities at the same time. When graphing, look at the inequality to help you. The first compound inequality above, $\begin{array}{r}-2<x\le 5\end{array}$ , has the $\begin{array}{r}x\end{array}$ in between -2 and 5, so the shading will also be between the two numbers.
  ::注意这两种不平等都有两个不平等迹象。 因此, 这就像同时解决或图形化两种不平等。 当图形化时, 查看不平等来帮助您。 上面的第一种复合不平等, 即 - 2 < x% 5 , 在 - 2 和 5 之间有 x , 所以阴影也会在这两个数字之间 。

  

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  And, with the “or” statement, the shading will go in opposite directions.
  ::而用“或”的语句,阴影会朝着相反的方向发展。

  

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  Let's w rite the inequality statement given by the graph below.
  ::让我们写下下图给出的不平等声明。

  

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  Because the shading goes in opposite directions, we know this is an “or” statement. Therefore , the statement is $\begin{array}{r}x<-2\end{array}$ or $\begin{array}{r}x>1\end{array}$ .
  ::因为阴影的方向相反, 我们知道这是一个“ 或” 的语句。 因此, 语句是 x2 或 x> 1 。

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  Now, let's solve the following compound inequalities.
  ::现在,让我们解决以下的 多重不平等。

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  1. Solve and graph $\begin{array}{r}-3<2x+5\le 11\end{array}$ .
     ::解决和图形 - 3 < 2x+511。

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  This is like solving two inequalities at the same time. You can split the statement apart to have two inequalities, $\begin{array}{r}-3<2x+5\end{array}$ and $\begin{array}{r}2x+5\le 11\end{array}$ and solve. You can also leave the compound inequality whole to solve.
  ::这就像同时解决两个不平等。 您可以将语句分割为两个不平等, 即 - 3 < 2x+5 和 2x+5+11 , 并解决。 您也可以将复合不平等全部留待解决 。

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  $$\begin{array}{rl}& -3<2x+\bcancel{5}\le 11\\ & \underset{\_}{-5-\bcancel{5}-5}\\ & \frac{-8}{2}<\frac{\bcancel{2}x}{\bcancel{2}}\le \frac{6}{2}\\ & -4<x\le 3\end{array}$$

  
  ::- 3 < 2x+511 - 5 - 5 - 5 - 582 < 2x2_ 62 - 4 < x% 3

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  Test a solution, $\begin{array}{r}x=0:\end{array}$
  ::测试溶液, x=0 :

  $$\begin{array}{rl}-3<2(0)+5& \le 11\\ -3<5& \le 11\end{array}$$

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  Here is the graph:
  ::以下是图表:

  

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  2. Solve and graph $\begin{array}{r}-32>-5x+3\end{array}$ or $\begin{array}{r}x-4\le 2\end{array}$ .
     ::解析和图形 - 325x+3 或 x- 42。

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  When solving an “or” inequality, solve the two inequalities separately, but show the solution on the same number line.
  ::在解决“或”不平等时,分别解决两种不平等,但在同一数字线上显示解决办法。

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  $$\begin{array}{rl}& -32>-5x+\bcancel{3}orx-\bcancel{4}\le 2\\ & \underset{\_}{-3-\bcancel{3}}\underset{\_}{+\bcancel{4}+4}\\ & \frac{-35}{-5}>\frac{-\bcancel{5}x}{-\bcancel{5}}x\le 6\\ & 7<x\end{array}$$

  
  ::- 325x+3 或 x-42 - 3 - 3_ +4+4+435 - 55x-5x-5x_6 7 < x

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  Notice that in the first inequality, we had to flip the inequality sign because we divided by -5. Also, it is a little more complicated to test a solution for these types of inequalities. You still test one point, but it will only work for one of the inequalities. Let’s test $\begin{array}{r}x=10\end{array}$ . First inequality: $\begin{array}{r}-32>-5(10)+3\to -32>-47\end{array}$ . Second inequality: $\begin{array}{r}10-4\le 2\to 5\cancel{\le }2\end{array}$ . Because $\begin{array}{r}x=10\end{array}$ works for the first inequality, is a solution. Here is the graph.
  ::请注意,在第一个不平等中,我们不得不翻转不平等标志,因为我们除以 -5. 。 另外,检验这些类型的不平等的解决方案要复杂得多。 您仍然要测试一个点, 但只能测试一个不平等。 让我们测试 x=10。 第一个不平等: - 325(10)+33247。 第二个不平等: 10-42。 因为 x=10为第一个不平等工作, 是一个解决方案。 下面是图表 。

  

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to find the grade range  Madison  must fall within on the final exam to raise her score to an A. 
  ::早些时候,你被要求找到 麦迪逊在期末考试中必须属于的级别范围 才能将分数提高到A

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  Writing the grading as an expression , we get $\begin{array}{r}0.4(84)+0.6x\end{array}$ where x is the final exam score. Madison wants to get an A, so we will have a compound inequality that ranges between 90 and 100.
  ::将评分写成表达式, 我们得到 0.4( 84) +0. 6x, 其中x是最后考试分数。 麦迪逊想要获得 A 。 因此我们将会有一个在 90 和 100 之间不等的复合不平等。

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  $$\begin{array}{r}90\le 33.6+0.6x\le 100\\ 56.4\le 0.6x\le 66.4\\ 94\le x\le 110.67\end{array}$$

  
  ::9033.6+0.6x10056.40.6x66.494x110.67

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  Unless Mr. Garcia offers extra credit, Madison can't score higher than 100. So, she has to score at least 94 or more, up to 100, to get an A.
  ::除非Garcia先生提供额外信贷 否则麦迪逊分数不能超过100 所以她得分至少94分或以上 最多100分 才能拿到A

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  Example 2
  ::例2

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  Graph $\begin{array}{r}-7\le x\le -1\end{array}$ on a number line.
  ::图7*x**1 数字行。

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  This is an “and” inequality, so the shading will be between the two numbers.
  ::这是一个“和”的不平等,所以阴影将存在于这两个数字之间。

  

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  Example 3
  ::例3

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  Solve and graph the following compound inequality:  $\begin{array}{r}5\le -\frac{2}{3}x+1\le 15\end{array}$ .
  ::解决并绘制以下复合不平等情况图:523x+115。

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  $$\begin{array}{rl}& 5\le -\frac{2}{3}x+\bcancel{1}\le 15\\ & \underset{\_}{-1-\bcancel{1}-1}\\ & 4\le -\frac{2}{3}x\le 14\\ & -\frac{3}{2}\left(4\le -\frac{2}{3}x\le 14\right)\\ & -6\ge x\ge -21\end{array}$$

  
  ::523x+115-1-1-1-1-1-1-423x}14-32(423x__14)-6x21

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  Test a solution, $\begin{array}{r}x=-10:\end{array}$
  ::测试一个溶液, x% 10 :

  $$\begin{array}{r}5\le -\frac{2}{3}(-12)+1\le 15\\ 5\le 9\le 15\end{array}$$

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  This solution can also be written $\begin{array}{r}-21\le x\le -6\end{array}$ .
  ::此解决方案也可以写入 - 21x6 。

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  The graph is:
  ::图表是:

  

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  Example 4
  ::例4

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  Solve and graph the following compound inequality:  $\begin{array}{r}\frac{x}{4}-7>5\end{array}$ or $\begin{array}{r}\frac{8}{5}x+2\le 18\end{array}$ .
  ::解决并绘制以下化合物不平等图:x4-7>5或85x+218。

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  This is an “or” compound inequality. Solve the two inequalities separately.
  ::这是一种“或”的复合不平等,分别解决这两种不平等。

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  $$\begin{array}{rl}& \frac{x}{4}-\bcancel{7}>5or\frac{8}{5}x+\bcancel{2}\le 18\\ & \underset{\_}{+\bcancel{7}+7}\underset{\_}{-\bcancel{2}-2}\\ & \bcancel{4}\cdot \frac{x}{4}>12\cdot 4or\xcancel{\frac{5}{8}\cdot \frac{8}{5}}x\le 16\cdot \frac{5}{8}\\ & x>48orx\le 10\end{array}$$

  
  ::x4-7>5 或 85x+218+7+7_-2-2-2_ 4x4>124 或 5885x16_58 x48 或 x__10

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  Test a solution, $\begin{array}{r}x=0:\end{array}$
  ::测试溶液, x=0 :

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  $$\begin{array}{rl}& \frac{0}{4}-7>5or\frac{8}{5}(0)+2\le 18\\ & -7\bcancel{\ge }5or2\le 18\end{array}$$

  
  ::04-7>5或85(0)+218-7□5或218

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  Notice that $\begin{array}{r}x=0\end{array}$ is a solution for the second inequality, which makes it a solution for the entire compound inequality. Here is the graph:
  ::注意 x=0 是第二个不平等的解决方案, 这使得它成为整个化合物不平等的解决方案。 以下是图表 :

  

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  In Examples  3 and 4 we changed the scale of the number line to accommodate the solution.
  ::在例3和例4中,我们改变了数字线的规模,以适应解决办法。

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  Review
  ::回顾

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  Graph the following compound inequalities. Use an appropriate scale.
  ::如下图所示化合物不平等。使用适当的比例尺。

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  1. $\begin{array}{r}-1<x<8\end{array}$
     ::-1 <x <8
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  2. $\begin{array}{r}x>5\end{array}$ or $\begin{array}{r}x\le 3\end{array}$
     ::x5 或 x%3 的 x5 或 x%3
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  3. $\begin{array}{r}-4\le x\le 0\end{array}$
     ::-4x0

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  Write the compound inequality that best fits each graph below.
  ::写下最符合下图各图的复合不平等。

  4. 
  5. 
  6. 

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  Solve each compound inequality and graph the solution.
  ::解决每种复合不平等,并绘制解决方案图。

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  7. $\begin{array}{r}-11<x-9\le 2\end{array}$
     ::- 11 <x-92
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  8. $\begin{array}{r}8\le 3-5x<28\end{array}$
     ::83-5x<28
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  9. $\begin{array}{r}2x-7>-13\end{array}$ or $\begin{array}{r}\frac{1}{3}x+5\le 1\end{array}$
     ::2 - 713 或 13x+51
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  10. $\begin{array}{r}0<\frac{x}{5}<4\end{array}$
      ::0 <x5 < 4
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  11. $\begin{array}{r}-4x+9<35\end{array}$ or $\begin{array}{r}3x-7\le -16\end{array}$
      ::- 4x+9 < 35 或 3x-716
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  12. $\begin{array}{r}\frac{3}{4}x+7\ge -29\end{array}$ or $\begin{array}{r}16-x>2\end{array}$
      ::34x+729 或 16-x>2
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  13. $\begin{array}{r}3\le 6x-15<51\end{array}$
      ::36x-15<51
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  14. $\begin{array}{r}-20<-\frac{3}{2}x+1<16\end{array}$
      ::-2032x+1 <16
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  15. Challenge Write a compound inequality whose solutions are all real numbers. Show why this is true.
      ::挑战 写一个复杂的不平等 其解决方案都是真实数字。 显示为什么这是真实的 。

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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 。