2.8 解决复合物不平等

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  解决复合物不平等问题

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  Suppose that a company's budget requires it to spend at least $20,000, but no more than $30,000, on training for its employees. The cost of training is the combination of a flat fee of $5,000 plus $500 per employee. If the company has n employees, how much is it required to spend per employee? In this section, you will learn how to solve problems such as this one by using .
  ::假设一个公司的预算要求它至少花费20,000美元,但不超过30,000美元,用于其雇员的培训。培训费用是5,000美元加上每个雇员500美元的固定费用。如果公司有n雇员,那么每个雇员需要花费多少钱?在本节中,你将学会如何通过使用(......)解决这样的问题。

  

   

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  Compound Inequalities
  ::多重不平等

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

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  • $\begin{align*}-2 < x \le 5\end{align*}$  is read, “-2 is less than $\begin{align*}x\end{align*}$    and $\begin{align*}x\end{align*}$  is  less than or equal to 5.”
    ::-2<x=5 改为:“-2小于x,x小于或等于5。”
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  • $\begin{align*}x \ge 3\end{align*}$ or $\begin{align*}x<-4\end{align*}$  is read, “ $\begin{align*}x\end{align*}$ is greater than or equal to 3 or  $\begin{align*}x\end{align*}$   is  less than -4.”
    ::改为“x大于或等于3,或x小于-4。”

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  Notice that both of these inequality statements have two inequality signs . So, we are solving or graphing two inequalities at the same time.
  ::这些不平等声明都有两个不平等迹象。 因此,我们正在同时解决或描述两种不平等。

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  Inequalities usually describe intervals of numbers. Recall from Chapter 1, the words "and" and "or" described the intersection or union of two sets, respectively. Intervals are sets, so from the bullets above we have:
  ::不平等通常描述数字的间隔。 回顾第一章, “ 和” 和 “ 或” 等词分别描述两组的交叉或结合。 间隔是一组, 从上面的子弹中我们可以看到:

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  • The interval  $\begin{align*}(-2,\infty )\end{align*}$   intersected with the interval where  $\begin{align*}(-\infty, 5]\end{align*}$ : $\begin{align*}(-\infty,5] \cap (-2,\infty)= (-2,5]\end{align*}$ .  
    ::间隔(-2, ) 与以下间隔( , ,5) 的间隔( , 5) 的间隔(-2, ) = (-2, 5) 的间隔(-2, ) 相交 。
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  • The interval  $\begin{align*}[3,\infty)\end{align*}$  has a  union  with the interval where  $\begin{align*}(-\infty,-4)\end{align*}$ : $\begin{align*}(-\infty,-4)\cup [3,\infty)\end{align*}$  . 
    ::间隔[3, ] 具有以下间隔的结合(, − 4): (, − 4) , [3, ) 。

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  When graphing, focus on the inequality symbol. In the first compound inequality above the $\begin{align*}x\end{align*}$  is in between -2 and 5, so the shading will also be between the two numbers to indicate that the numbers in this interval are greater than -2 and less than or equal to 5. 
  ::当图形化时, 聚焦于不平等符号。 在 x 上的第一个复合不平等值介于 - 2 和 5 之间, 因此阴影也介于这两个数字之间, 以显示此间隔中的数字大于 - 2 或小于 5 或等于 5 。

  

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  With the “or” statement, graph each inequality separately regardless of the other one. The numbers in this compound inequality are less than -4  or greater than or equal to 3. 
  ::使用“ 或” 语句, 将每一种不平等分别列出, 不论另一种不平等。 这种复合不平等中的数字小于 - 4 或大于 3 或等于 3 。

  

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  When it comes to solving compound inequalities, let's start with solving "or" statements first.
  ::在解决多重不平等问题时, 让我们先解决“ 或” 声明。

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  Solving  Compound Inequalities—OR
  ::解决复合物不平等——OR

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  To solve compound inequalities of the OR-type, solve each inequality separately and then combine the solutions into one set . 
  ::为解决OR型的多重不平等,分别解决每一种不平等,然后将解决办法合并为一套。

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  by Mathispower4u demonstrates how to solve a compound inequality involving OR or the union of two in tervals . The solution is also written using interval notation. 
  ::由 Mathispower4u 演示如何解决涉及 OR 或 双间隔结合的复合不平等。 解决方案也使用间隔符号书写 。

   

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  Example 1
  ::例1

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

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  Solution:  Solve the two inequalities separately. T he solution will include both inequalities and you will graph  both on the same number line.
  ::解决方案: 分别解决两个不平等问题。 解决方案将包含两个不平等问题, 您将在同一数字线上绘制两个图表 。

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  $$\begin{align*}-32 > -5x + \cancel{3} \quad &\text{or} \quad x- \cancel{4} \le 2\\ \underline{\ -3 \quad \ \qquad -\cancel{3}} \quad & \ \ \ \quad \underline{\quad +\cancel{4} \ +4}\\\ \frac{-35}{-5} > \frac{-\cancel{5}x}{-\cancel{5}} \qquad &\qquad \quad x \le 6\\ 7<x \qquad \qquad & \ \end{align*}$$
  ::- 325x+3orx-42 - 3- 3_ +4+4_-35-5_5x-5x-5x-5x_67 < x

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  Notice that in the first inequality, we had to flip the inequality sign because we divided by -5.
  ::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。

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  To check, you need to check the boundary points and pick a test point in the solution set for each inequality. 
  ::要检查, 您需要检查边界点, 并在为每种不平等设定的解决方案中选择一个测试点 。

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  $$\begin{align*}&x=7: \ -32=-5(7)+3 \implies -32=-35+3 \implies -32=-32 && \color{gray}{\text{boundary points}} \\ \\ &x=6: \ 6-4=2 \implies 2=2 \\ &x=8: \ -32>-5(8)+3 \implies -32>-40+3 \implies -32>-37 && \color{gray}{\text{test points}} \\ \\ &x=0: \ 0-4<2 \implies -4 \le 2 \\\end{align*}$$
  ::x=7: - 32_5(7)+3_32_35+3_32_32_32_32_32-32边界点x=6: 6-4=2_2=2x8: - 32_5(8)+3_32_40+3_32_32_37试验点x=0: 0-4<2_4}2x=8: - 32_5(8)+3_32_40+3_32_32_37试验点x=0: 0- 4>_2}

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  Since the boundary points and the test points check,  the solution set  $\begin{align*}(-\infty, 6] \cup (7,\infty)\end{align*}$  is correct.
  ::由于边界点和测试点的检查,所设定的解决方案(,6, , , , , )是正确的。

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  Graphing both intervals on the same number line, we have:
  ::在同一数字线上绘制两个间距的图,我们有:

  

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  Example 2
  ::例2

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  Solve and graph  $\begin{align*}\frac{x}{4}-7>5\end{align*}$    or $\begin{align*}\frac{8}{5}x+2\le 18\end{align*}$ .
  ::解决和图x4-7>5或85x+218。

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  Solution:  Again, we solve the two inequalities separately.
  ::解决方案:再次,我们分别解决两个不平等问题。

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  $$\begin{align*}\frac{x}{4}- \cancel{7} > 5 \quad \quad &\text{or} \quad \ \frac{8}{5}x+ \cancel{2} \le 18\\ \underline{\quad \ \ + \cancel{7} \ +7 \;} \qquad &\qquad \underline{\qquad \ - \cancel{2} \ \ -2 \; \; \; \;}\\ \ \cancel{4} \cdot \frac{x}{\cancel{4}} > 12 \cdot 4 \ &\quad \ \frac{\cancel{5}}{\cancel{8}} \cdot \frac{\cancel {8}}{\cancel{5}}x \le 16 \times \frac{5}{8}\\ \ \quad \ x > 48 \quad \ \ &\qquad \quad \ \ x \le 10\end{align*}$$
  ::x4-7>5or 85x+218+7+7+7_-2-2-2_ 4x4>12}4 5885x16×58 x>48x_10

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  To check, you need to check the boundary points and pick a test point in the solution set for each inequality. 
  ::要检查, 您需要检查边界点, 并在为每种不平等设定的解决方案中选择一个测试点 。

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  $$\begin{align*}& x=48: \ \frac{48}{4} -7 = 5 \implies 12-7=5 \implies 5=5 && \color{gray}{\text{boundary points}} \\ \\ &x=10: \ \frac{8}{5}(10)+2 = 18 \implies 18 = 18 \\ & x=50: \ \frac{50}{4} -7 > 5 \implies 12.5-7>5 \implies 5.5>5 && \color{gray}{\text{test points}} \\ \\ &x=0: \ \frac{8}{5}(0)+2 \le 18 \implies 2 \le 18 \end{align*}$$
  ::x=48: 484-7=512-7=55=5边端点x=10: 85(10)+2=1818=18x=50: 504-7>512.5-7>55.5>5试验点x=0: 85(0)+218=218

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  The boundary points and the test points check . Therefore , the solution set  $\begin{align*}(-\infty,10] \cup (48,\infty)\end{align*}$  is correct. Here is the graph:
  ::边界点和测试点检查。 因此, 解决方案集( , 10] ( 48, ) 正确 。 以下是图表 :

  

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  Solving Compound Inequalities—AND
  ::解决复合物不平等 -- -- 和

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  Now let's consider compound inequalities of the AND type.
  ::现在让我们来考虑 多种不平等的 类型和类型。

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  Example 3
  ::例3

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

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  Solution:  There are two ways to solve compound inequalities involving "and." You can split the statement apart to have two inequalities, $\begin{align*}-3<2x+5\end{align*}$ and $\begin{align*}2x+5\le 11\end{align*}$ and solve.
  ::解决方案: 解决与“ 和” 相关的多重不平等有两种方法。 您可以将声明分割为两个不平等, 即 3 < 2x+5 和 2x+5+11 并解决 。

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  $$\begin{align*}-3<2x+\cancel{5} \quad &\text{and} \quad 2x+5 \le 11 \\ \underline{-5 \qquad \ -\cancel{5}} \quad &\ \ \ \quad \underline{\qquad -\cancel{5} \ -5} \\ \frac{-8}{2}<\frac{\cancel{2}x}{\cancel{2}} \qquad &\ \qquad \frac{\cancel{2}x}{\cancel{2}} \le \frac{6}{\cancel{2}} \\ -4<x \qquad &\ \qquad x \le 3\\ \qquad -4<&x \le 3 \end{align*}$$
  ::- 3 < 2x+5和2x+5=5*11-5-5-5-5-5-5-5-582 < 2x2 2x2x2_62-4 < xx3-4<x3_4>

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  Since both inequalities have to be true for all of the solutions, the final solution is the intersection of the two intervals:  $\begin{align*}(-4,\infty) \cap (-\infty,3] = (-4,3]\end{align*}$ . 
  ::由于两种不平等必须适用于所有解决办法,最后的解决办法是两个间隔的交叉-4) (-4, 3) 。

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  Notice as we were solving we did the same operations in each inequality—we subtracted by 5 and then divided by 2. Another way to approach solving inequalities of this type is to leave the compound inequality as is and solve simultaneously.
  ::在我们解决这种不平等时,我们采取了同样的行动——我们减去5,然后除以2。 解决这种不平等的另一种办法是,使复杂的不平等问题与现在一样,并同时解决。

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  $$\begin{align*}-3<2x+\cancel{5} \le 11\\ \underline{\ -5 \qquad \ -\cancel{5} \ \ -5}\\ \frac{-8}{2}< \frac{\cancel{2}x}{\cancel{2}} \le \frac{6}{2}\\ -4<x \le 3 \qquad\end{align*}$$
  ::- 3 < 2x+511 - 5 - 5 - 5 - 582 < 2x2_ 62 - 4 < x% 3

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  Remember, you have to perform the operations on each side of the inequality signs to keep the inequalities balanced. 
  ::记住,你必须在不平等标志的两侧执行操作 以保持不平等的平衡。

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  To check, you need to check the two boundary points and pick a test point in the solution set for each inequality. Let's test  $\begin{align*}x = -4, 0, 3:\end{align*}$
  ::要检查, 您需要检查两个边界点, 并在为每种不平等设定的解决方案中选择一个测试点 。 让我们测试 x+4, 0, 3 :

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  $$\begin{align*}-3=2(-4)+5 & = 11 && \color{gray}{\text{boundary points}} \\ -3=-3 & = 11&& \color{gray}{x=-4 \Longrightarrow -3=-3} \\ -3=2(3)+5 & = 11 \\ -3\neq 11 & = 11 && \color{gray}{x=3 \Longrightarrow 11=11}\\ -3<2(0)+5 & \le 11 && \color{gray}{\text{test points}}\\ -3<5 & \le 11 \end{align*}$$
  ::-3=2(-4)+5=11个边界点 -33=11x4}3=11=11x=3}11=11x=3}11=11_3<2(0)+5}11=3测试点3<5}11

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  All of these  points check.  Therefore, the solution set  $\begin{align*}(-4,3]\end{align*}$  is correct.
  ::所有这些点数都核对,因此,所设定的解决方案(-4.3)是正确的。

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  Here is the graph:
  ::以下是图表:

  

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  by CK-12 demonstrates how to solve and graph compound inequalities.  
  ::CK-12展示了如何解决和勾画复合不平等。

   

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  Example 4
  ::例4

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  Solve  and graph  $\begin{align*}5 \le - \frac{2}{3}x+1 \le 15\end{align*}$ .
  ::解析和图 523x+115。

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  Solution:  We will solve the inequality simultaneously.
  ::解决方案:我们将同时解决不平等问题。

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  $$\begin{align*}& \ \ 5 \le - \frac{2}{3}x+ \cancel{1} \le 15\\ & \underline{-1 \qquad \quad \ \ -\cancel{1} \ \ -1}\\ & \qquad \ 4 \le - \frac{2}{3}x \le 14\\ & -\frac{3}{2} \left(4 \le -\frac{2}{3}x \le 14\right)\\ & \quad \ \ -6 \ge x \ge -21\end{align*}$$
  ::523x+115-1-1-1-1-1-1-423x}14-32(423x__14)-6x21

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  Here, since we multiplied all sides of the inequalities by  $\begin{align*}-\frac{3}{2}\end{align*}$ , we need to switch the direction of both of the inequality signs. This solution can also be written $\begin{align*}-21 \le x \le -6\end{align*}$ . In interval notation,  the solution is  $\begin{align*}[-21,-6]\end{align*}$ . 
  ::在这里,由于我们把不平等的方方面面乘以-32,我们需要改变不平等迹象的两面方向,这种解决办法也可以写为-211x_#6。 在间隙符号中,解决办法是[-21,-6]。

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  To check, you need to check the boundary points and pick a test point in the solution set for each inequality. Let's test , $\begin{align*}x = -21, -12, -6.\end{align*}$
  ::要检查, 您需要检查边界点, 并在为每种不平等设定的解决方案中选择一个测试点 。 让我们测试 x21, - 12, - 6 。

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  $$\begin{align*}5 = - \frac{2}{3}(-21)+1 = 15 && \color{gray}{\text{boundary points}} \\ 5 \neq 15 = 15 && \color{gray}{x=-21 \Longrightarrow 15=15} \\ 5 = - \frac{2}{3}(-6)+1 = 15\\ 5 = 5 \neq 15 && \color{gray}{x=-6 \Longrightarrow 5=5} \\ 5 \le - \frac{2}{3}(-12)+1 \le 15 && \color{gray}{\text{test points}} \\ 5 \le 9 \le 15 \end{align*}$$
  ::523(-21)+1=15 边界点515=15x_21}15=155}23(-6)+1=155=5}15x6}5=55}23(12)+1}15试验点5}9}15

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  All of our test points satisfy the necessary conditions . Therefore, the solution set [-21,-6] is correct.  The graph is:
  ::我们所有的测试点都符合必要的条件。因此,设定的解决方案[21,-6]是正确的。

  

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     WARNING
  ::警告

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  1. Do not try to combine compound inequalities of the OR-type like compound inequalities of the AND-type. The following is incorrect:
  ::1. 不试图将 " 或 " 类 " 的复合不平等与 " 和 " 类 " 的复合不平等相结合。

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  $$\begin{align*}4>x \ \text{or} \ x\le 1 \\ 4>x \le 1\end{align*}$$
  ::4>x 或 x%14>x%1

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  The last inequality says that 4 is greater than  x AND  x is less than or equal to 1. This is impossible. 
  ::最后一种不平等表示,4大于x,x小于或等于1,这是不可能的。

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  2. Do not separate the expression in the middle of a compound inequality of the AND type. The expression is part of both of the inequalities. The following is incorrect:
  ::2. 不要在类型和类型的复合不平等中将表达方式分开,这是两种不平等的一部分,以下不正确:

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  $$\begin{align*}-3<2x+5 \le 11 \\ -3<2x \qquad 5 \le 11 \\\end{align*}$$
  ::- 3 < 2x+511 - 3 < 2x511

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  Feature: Earth's Layers
  ::地貌图层

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  by Jen Kershaw
  ::由Jen Kershaw著

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  Have you ever wondered what makes up the Earth's interior? Like an onion, the Earth is actually composed of layers, and each layer has unique characteristics. Using your knowledge of compound inequalities, you can describe various depths of the Earth's crust.
  ::您是否想知道地球内部的成分是什么?像洋葱一样,地球实际上是由层构成的,每个层都有独特的特征。使用您对化合物不平等的知识,您可以描述地球地壳的不同深度。

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  The Earth is composed of four basic layers: a solid inner core, a liquid outer core, a thick mantle, and the outermost crust. The crust is solid and made up of various types of rock. The depth of the Earth's crust ranges from about 5 to 70 kilometers (km). Continental crust, which is where there are landmasses, tends to be thicker than the oceanic crust found beneath the oceans. The majority of continental crust is between 30 to 50 km thick. There are portions that are greater than 50 km thick, but they are rare and account for less than 10% of the Earth's continental crust. Oceanic crust is usually between 5 to 10 km thick.
  ::地球由四个基本层组成:一个坚固的内核、一个液体外核、一个厚厚的地壳和最外层。地壳是固体的,由各种类型的岩石组成。地壳的深度在5至70公里之间。地壳的深度在5至70公里之间。大陆地壳,即有陆地资产的地方,往往比在海洋之下发现的海洋地壳厚得多。大陆地壳大部分厚30至50公里。有些地壳厚50公里以上,但很少见,占地球大陆地壳的不到10%。海洋地壳通常厚5至10公里。

  

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  We can use compound inequalities to describe the above characteristics of the Earth's crust. Why compound inequalities? Well, a compound inequality can show that something is greater than one value but less than another value—both values are needed to get the whole picture. As far as the depth of Earth's crust is concerned, we have the following setup: $\begin{align*}5 \le d \le 70\end{align*}$ , where $\begin{align*}d\end{align*}$ represents depth in kilometers. This compound inequality shows us that the crust's depth is greater than or equal to 5 km but less than or equal to 70 km. Likewise, the compound inequalities $\begin{align*}30 \le t_{continental} \le 50\end{align*}$ and $\begin{align*}5 \le t_{oceanic} \le 10\end{align*}$ can be used to represent the typical thicknesses of continental and oceanic crust, respectively.
  ::我们可以用复合不平等来描述地球地壳的上述特征。为什么是复合不平等? 复合不平等可以表明某物大于一个价值,但少于另一个价值——需要两种价值才能获得全貌。就地球地壳的深度而言,我们设置了以下结构:5d70, 以公里表示深度。这种复合不平等向我们表明,地壳的深度大于或等于5公里,但小于或等于70公里。同样地壳的复合不平等可以分别用来代表大陆和大洋地壳的典型厚度。

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   by Darrel Luck discusses the depths of layers of the Earth.
  ::达勒勒勒克讨论地球的深度

   

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  Keep an eye out for more properties of the Earth's layers that can be expressed as compound inequalities.
  ::注意地球层层的更多特性,这些特性可以表现为复合不平等。

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    by MITK12Videos  also focuses on the layers of the Earth.
  ::麻省理工学院12Videos也关注地球的层层。

   

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  Summary
  ::摘要

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  • To solve compound inequalities of the OR-type, solve each separately and take the union of the solution intervals.
    ::解决OR型的多重不平等问题,分别解决每个问题,并采用解决方案的间隔间隔。
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  • To solve inequalities of the AND-type, you can either solve the inequalities separately and take the intersection of the solution intervals or you can solve the inequalities simultaneously.
    ::要解决AND型的不平等,您可以单独解决不平等问题,在解决方案间隔的交叉点上走过,也可以同时解决不平等问题。

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  Review
  ::回顾

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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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  1. $\begin{align*}-4x+9 >35 \ \mathrm{or} \ 3x-7 \ge -16\end{align*}$
     ::-4x+9>35或3x-716
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  2. $\begin{align*}2x-7>-13 \ \mathrm{or} \ \tfrac{2}{3}x -4 \le -8\end{align*}$
     ::2 - 713 或 23x - 48
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  3. $\begin{align*}\tfrac{3}{4}x + 7 \le-29 \ \mathrm{or} \ 16-x>2\end{align*}$
     ::34x+729 或 16-x>2
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  4. $\begin{align*}-6.32x+1.15>4.31 \ \mathrm{or} \ x+\sqrt{3} > 2\sqrt{3}\end{align*}$
     ::-6.32x+1.15>4.31或 x3>23
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  5. $\begin{align*}-11 < x-9\le2\end{align*}$
     ::- 11 <x-92
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  6. $\begin{align*}0<\tfrac{x}{5}<4\end{align*}$
     ::0 <x5 < 4
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  7. $\begin{align*}3\le6x-15<51\end{align*}$
     ::36x-15<51
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  8. $\begin{align*}8\le3-5x<28\end{align*}$
     ::83-5x<28
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  9. $\begin{align*}-20<-\tfrac{3}{2}x+1<16\end{align*}$
     ::-2032x+1 <16
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  10. $\begin{align*}-4.2<-0.5x<-1.3\end{align*}$
      ::-4.20.5x1.3

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  Explore More
  ::探索更多

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  1. William’s pick-up truck gets between 18 and 22 miles per gallon of gasoline. His gas tank can hold 15 gallons of gasoline. If he drives at an average speed of 40 miles per hour, how much driving time does he get on a full tank of gas?
  ::1. 威廉的皮卡车每加仑汽油在18至22英里之间,他的油罐可容纳15加仑汽油,如果他每小时平均开车40英里,他有多少驾驶时间用满油罐?

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  (Hint: Use dimensional analysis to convert from time per tank to miles per gallon. Since the truck gets between 18 to 22 miles/gallon, you can write a compound inequality:  $\begin{align*}18 \le \frac{40t}{15} \le 22.)\end{align*}$
  :提示: 使用维度分析将每辆坦克的时间转换为每加仑英里。 由于卡车每加仑在18到22英里之间, 您可以写出一种复合不平等: 1840t1522。 )

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  2. On average, most college students get between 6 and 7 hours of sleep each night. 20% of that time is in Rapid Eye Movement (REM). Not getting enough REM can cause lack of concentration, which can lead to problems in college. ¹ How much time is spent in REM each night for the average college student?
  ::2. 平均而言,大多数大学生每晚睡6至7小时。 20%的时间在快速眼运动(快速眼运动)中,得不到足够的REM可能导致注意力不够集中,这可能导致大学出现问题。 1 平均大学生每晚在REM花多少时间?

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  3. You want to walk between 4 and 6 miles each day. You walk at a rate of 3 mi/hr.   Using, d = r t, w hat is the least and most amount of time you will need each day?
  ::3. 你想每天走4至6英里的路程,走速为3米/小时。使用,d=r t,你每天需要多少最少和最多的时间?

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  4.  You hope to make between $110 and $130 each week. You have 10 hours each week to work. What is the least and most amount of money you need to make per hour?
  ::4. 你希望每周挣110美元至130美元,每周工作10小时,每小时最少、最多需要多少钱?

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  5. Write a compound inequality whose solution is all real numbers. 
  ::5. 写出一种复合不平等,其解决办法都是真实数字。

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  Answers for Review and Explore More Problems
  ::回顾和探讨更多问题的答复

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  Please see the Appendix. 
  ::请参看附录。

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  PLIX
  ::PLIX

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  Try this interactive that reinforces the concepts explored in this section:
  ::尝试这一互动,强化本节所探讨的概念:

   

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  Re ferences
  ::参考参考资料

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  1.  “Sleep and College Life,” by James R. Oelschlager, Psy.D, pamphlet Florida Institute of Technology Counseling and Psychological Services, .
  ::1. " 睡眠和大学生活 " ,由James R. Oelschlager、Psy.D、佛罗里达技术咨询和心理服务研究所小册子出版。