1.5 间隔

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  间间数

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  A ride at the zoo requires children to be at least 8 years old to ride without an adult. We can describe this set of children as an interval of numbers. We discuss intervals and how to represent them in this section. 
  ::动物园的骑行要求儿童至少8岁才能没有成人的骑行。我们可以将这组儿童描述为数字间隔。我们讨论间隔以及如何在本节中代表他们。

  

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  Bounded Intervals
  ::间距

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  An interval on the number line is a connected set of numbers. A bounded interval is an interval with two endpoints or one that does not continue forever in either direction. For example,
  ::数字行的间隔是一组连接的数字。一个连接的间隔是带有两个端点的间隔,或者一个不会在两个端点上永久延续的间隔。例如,

  

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  is all of the numbers that satisfy $\begin{array}{r}0\le x\le 4\end{array}$ : all of the numbers that are greater than or equal to 0 and less than or equal to 4. This set includes 1, 2, and 3, but it also includes 0 and 4, and all fractions and irrational numbers in between 0 and 4 too. We denote that the endpoints are included as part of the interval on the graph by using closed or filled in circles over them. An interval where both of the endpoints are included is called a closed interval .
  ::是满足 0x4 的所有数字: 大于或等于 0 和小于或等于 4 的所有数字。 这组数字包括 1, 2 和 3 , 但也包括 0 和 4 , 以及 0 和 4 之间的所有分数和非理性数字。 我们表示, 端点是作为图中间隔的一部分, 使用封闭或填充的圆圈来显示 。 包括两个端点的间隔被称为封闭间隔 。

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  To indicate that the endpoints are not included, we use open circles or holes.
  ::为了表明未包括终点,我们使用开圈或孔。

  

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  This graph can be described by the inequality $\begin{array}{r}0<x<4\end{array}$ : all of the numbers that are greater than 0 and less than 4. Zero and 4 are not included in this set unlike in the previous example. An interval where the endpoints are not included is called an open interval .
  ::此图可描述为 0 < x < 4 的不平等值 : 所有大于 0 和 小于 4. 零 和 4 的数字都不包含在此组中, 与前一例不同 。 不包含端点的间隔称为开放间隔 。

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  Other Notations
  ::其他说明

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  Inequality notation is one way to indicate an interval. We can also indicate intervals on the number line with interval notation and set-builder notation.
  ::不平等标记是表示间隔的一种方式。我们也可以用间隔标记和设置-构建标记来表示数字行的间隔。

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  Interval Notation
  ::间间点

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  Interval notation uses the endpoints to indicate the smallest and largest values in the interval and we assume that all numbers in between them are included. For example, we express the first interval above as [0,4]. Zero and 4 are the endpoints of the interval. To indicate they are included in the set, we use brackets . Lastly, we express the numbers from smallest to largest.
  ::中间点使用端点来表示间隔中最小和最大的值,我们假定它们之间的所有数字都包括在内。例如,我们表示以上第一个间隔为[0/4]。0和4是间隔的端点。为了表明它们是否包括在数据集中,我们使用括号。最后,我们表示从最小到最大的数字。

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  The second interval above is (0,4). The endpoints are the same as the first interval, but in this case, they are not included, so we use " data-term="Parentheses" role="term" tabindex="0"> parentheses .
  ::以上第二个间隔是(0,4),终点与第一个间隔相同,但在此情况下,没有包括在内,因此我们使用括号。

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  Example 1
  ::例1

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  Express the following interval in interval notation.
  ::以间隔编号表示以下间隔。

  

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  Solution: The endpoints are 0 and 4 as before. In this case, 0 is not included, so it will get a parenthesis next to it in interval notation to indicate that. Four is included, so that will get a bracket next to it. Starting with the smallest number in the interval, we have (0, 4].
  ::解答 : 终点是 0 和 4 。 在此情况下, 0 不包括在内, 因此它会在其旁边有一个括号, 以间距符号表示 。 包括了 4 个, 这样就可以在它旁边有一个括号 。 从间隔中最小的数字开始, 我们有( 0 、 4 ) 。

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     WARNING
  ::警告

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  Intervals are always described by smallest number or quantity first, then the larger number or quantity.
  ::间隔总是先以最小数量或数量,然后以较大数量或数量来描述。

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  In the example above, if we wrote [4, 0), that would be meaningless because 4 is larger than 0. 
  ::在上述例子中,如果我们写[4,0],那将是毫无意义的,因为4大于0。

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  Set-Builder Notation
  ::Set- 建筑符号

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  An interval is a set of numbers, so we can use a notation called set-builder notation to represent an interval. In set-builder notation, the first part indicates the number system that we are using and the second part indicates the condition or restriction on that number system.
  ::间距是一组数字, 所以我们可以使用叫做 设置- 构建者 符号的标记来表示一个间隔。 在设置- 构建者 符号中, 第一部分表示我们使用的数字系统, 第二部分表示该数字系统的条件或限制 。

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  $$\begin{array}{r}\{\text{number system}|\text{condition}\}\\ \\ \text{For example}\\ \{x\in \mathbb{R}|0<x\le 4\}\end{array}$$

  
  ::{数系统\ { { matter} 例如{ xR { 0 < x} 4}

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  Notice how we use curly brackets, { and }, to surround our set as we did previously. 
  ::注意我们如何使用卷轴括号 {和} 来环绕我们之前的组合。

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  To indicate our number system,  $\begin{array}{r}x\end{array}$  represents a number that is an element of this set,  $\begin{array}{r}\in \end{array}$  is the symbol for "element in," and  $\begin{array}{r}\mathbb{R}\end{array}$  is the set of real numbers. Together,  x ∈ R   is read: "x is an element of ."
  ::要表示我们的数字系统, x 代表一个是此集元素的一个数字,\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  The bar that divides the two parts,   | , is read as "such that."
  ::将两个部分分开的栏, , 读为“ 如此 ” 。

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  Lastly, the condition on the real numbers is that they are between 0 and 4, particularly that they satisfy the inequality $\begin{array}{r}0<x\le 4\end{array}$ .
  ::最后,实际数字的条件是他们介于0到4之间,特别是满足了0<x4的不平等。

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  by MathWOEs provides an explanation of the symbols used in set-builder notation.       
  ::数学世界职业介绍所对设置建筑工符号标记中使用的符号作了解释。

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  Unbounded Intervals
  
  ::无约束间数

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  An interval may not have two endpoints. Sometimes, it has only one or none. Intervals with only one endpoint or no endpoints, those that continue forever in at least one direction, are called unbounded intervals . For example,
  ::间距可能没有两个端点。 有时, 间距只有一个端点或没有端点。 间距只有一个端点或没有端点, 至少一个端点会一直持续到一个端点, 间距会被称为无界间隔。 例如,

  

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  $\begin{array}{r}x\ge 4\end{array}$ are all of the numbers that are greater than or equal to 4. This set continues without bound forever.
  ::x 4 是所有大于或等于 4 的数值, 此组连续不设永久约束 。

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  We can express this set in interval notation and set-builder notation. In set-builder notation, $\begin{array}{r}x\ge 4\end{array}$ is the condition, so $\begin{array}{r}\{x\in \mathbb{R}|x\ge 4\}\end{array}$ . In interval notation, we need a symbol to indicate that this set continues forever without bound. We use the infinity symbol,  $\begin{array}{r}\mathrm{\infty }\end{array}$ , to indicate this. In interval notation, this set is $\begin{array}{r}[4,\mathrm{\infty })\end{array}$ . The "endpoint" infinity is not an actual point or number, so it gets a parenthesis.
  ::我们可以用间距标记和设置构建器的标记来表达此组。 在设置- 构建器的标记中, x% 4 是条件, 所以 {xR x4} 。 在间距标记中, 我们需要一个符号来表示该组将无限期持续, 而没有约束 。 我们使用无限符号来表示这一点 。 在间距标记中, 此组是 [4,\\\\\\\\] 。 “ 终点” 的无限性不是一个实际的点或数字, 所以它会得到括号 。

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  by Mathispower4u provides four examples of how to graph linear inequalities in one variable . Each interval is also expressed using interval notation. 
  ::由 Mathispower4u 提供四个例子,说明如何用一个变量来显示线性不平等。每个间隔也用间距符号表示。

  
   

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  Example 2
  ::例2

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  Express the following interval in inequality, interval, and set-builder notations.
  ::表示以下不平等、间隔和建造工序标记的间隔。

  

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  Solution: This set is all of the real numbers less than 4. We have
  ::解决方案:这一套是所有实际数字 少于4,我们有。

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  $$\begin{array}{rl}& \text{inequality notation:}x<4\\ \\ & \text{interval notation:}(-\mathrm{\infty },4)\\ \\ & \text{set-builder notation:}\{x\in \mathbb{R}|x<4\}\end{array}$$

  
  ::等值符号 : x < 4interval 符号 : (, 4) set- builder 符号 : {xR x < 4}

   

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     WARNING
  ::警告

  $\begin{array}{r}-\mathrm{\infty }\end{array}$  is always first in interval notation and  $\begin{array}{r}\mathrm{\infty }\end{array}$  is always last in interval notation.

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  To summarize, the possibilities we have are:
  ::简言之,我们拥有的可能性是:

  Bounded Intervals	Unbounded Intervals
  播放段落 Inequality:  $\begin{array}{r}a\le x\le b\end{array}$ ::不平等:axxb 播放段落 Interval:  $\begin{array}{r}[a,b]\end{array}$ ::间间:[a,b] 播放段落 Set-Builder:  $\begin{array}{r}\{x\in \mathbb{R}|a\le x\le b\}\end{array}$ ::设置为 {x{R} {x} {a} x} b} 。	播放段落 Inequality:  $\begin{array}{r}a<x\end{array}$ ::不平等:a<x 播放段落 Interval:  $\begin{array}{r}(a,\mathrm{\infty })\end{array}$ ::间隔a,) 播放段落 Set-Builder:  $\begin{array}{r}\{x\in \mathbb{R}|a<x\}\end{array}$ ::设置构建器 : {x{R} a<x}
  播放段落 Inequality:  $\begin{array}{r}a<x<b\end{array}$ ::不平等:a<x<b> 播放段落 Interval:  $\begin{array}{r}(a,b)\end{array}$ ::间隔a,b) 播放段落 Set-Builder:  $\begin{array}{r}\{x\in \mathbb{R}|a<x<b\}\end{array}$ ::设置构建器 : {x{R} a <x<b}	播放段落 Inequality:  $\begin{array}{r}a\le x\end{array}$ ::不平等:xx 播放段落 Interval:  $\begin{array}{r}[a,\mathrm{\infty })\end{array}$ ::间歇性:[a,] 播放段落 Set-Builder:  $\begin{array}{r}\{x\in \mathbb{R}|a\le x\}\end{array}$ ::设置- 构建器 : {x} R { ax}
  播放段落 Inequality:  $\begin{array}{r}a<x\le b\end{array}$ ::不平等: a<xb 播放段落 Interval:  $\begin{array}{r}(a,b]\end{array}$ ::间隔a,b) 播放段落 Set-Builder:  $\begin{array}{r}\{x\in \mathbb{R}|a<x\le b\}\end{array}$ ::设置构建器 : {x{R} a<x}	播放段落 Inequality:  $\begin{array}{r}a>x\end{array}$ ::不平等:a>x 播放段落 Interval:  $\begin{array}{r}(-\mathrm{\infty },a)\end{array}$ ::间隔,a) 播放段落 Set-Builder:  $\begin{array}{r}\{x\in \mathbb{R}|a>x\}\end{array}$ ::设置- 构建器 : {x{R} {a>x}
  播放段落 Inequality:  $\begin{array}{r}a\le x<b\end{array}$ ::不平等:ax<b 播放段落 Interval:  $\begin{array}{r}[a,b)\end{array}$ ::间间:[a,b] 播放段落 Set-Builder:  $\begin{array}{r}\{x\in \mathbb{R}|a\le x<b\}\end{array}$ ::设置构建器 : {xR } ax<b}	播放段落 Inequality:  $\begin{array}{r}a\ge x\end{array}$ ::不平等:xx 播放段落 Interval:  $\begin{array}{r}(-\mathrm{\infty },a]\end{array}$ ::间隔,a) 播放段落 Set-Builder:  $\begin{array}{r}\{x\in \mathbb{R}|a\ge x\}\end{array}$ ::设置- 构建器 : {x} R { ax}

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  Example 3
  ::例3

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  A ride at the zoo requires children to be at least 8 years old to ride without an adult. Let's consider the set of children to be people whose age is less than 18 years old. Describe this set of ages of children in inequality, interval, and set-builder notation.
  ::动物园的骑行要求儿童至少8岁才能没有成人骑行。让我们把一组儿童视为18岁以下者。描述这组不平等、间隔和建筑工名式的儿童年龄。

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  Solution: The children who can go on this ride have to be 8 years old. So, 8 is the smallest number in our interval. The children who can ride without an adult are between 8 and 17 years old, or $\begin{array}{r}8\le c\le 17\end{array}$ . In interval notation, we have [8, 17]. Lastly, in set-builder notation, we have  $\begin{array}{r}\{c\in \mathbb{R}|8\le c\le 17\}\end{array}$ .
  ::解决方案: 能够坐上这趟旅程的儿童必须年满8岁。 因此, 8 是我们间隔时间中最小的。 没有成年人可以坐8-17岁或817岁。 在间隔时间, 我们有[ 8, 17] 。 最后, 在固定建筑工名中, 我们有 {cR 8c17} 。

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  Summary
  ::摘要

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  • Intervals on the number line can be bounded, with two endpoints, or unbounded, with either one endpoint or no endpoints.
    ::数字行的中间点可以被捆绑,有两个端点,或者没有绑定,要么一个端点,要么没有端点。
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  • We have several notations for describing intervals, including inequality notation, interval notation, and set-builder notation.
    ::我们有几个标记来描述间隔,包括不平等标记、间隔标记和建筑工的标记。

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  Review
  ::回顾

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  Describe the following intervals with inequality, interval, and set-builder notation.
  ::以不平等、间隔和建筑工的标记说明以下间隔。

  1. 
     

     

  2. 
     

     

  3. 
     

     

  4. 
     

     

  5. 
     

     

  6. 
     

     

  7. 
     

     

  8. 
     

     

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  10. 
      

      

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  Explore More
  ::探索更多

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  1. Describe {1,2,3,4,5} in set-builder notation with an inequality as the condition.
  ::1. 在以不平等为条件的固定建筑工标记中描述 {1,2,3,4,5}。

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  2. Describe {-6, -4, -2, 0, 2, 4, 6} in set-builder notation with an inequality as the condition.
  ::2. 在以不平等为条件的固定建筑工标记中描述 {-6, 4, 2, 0, 2, 4, 6} 。

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  3. Describe {-5, -3, -1, 1, 3, 5} in set-builder notation with an inequality as the condition.
  ::3. 在以不平等为条件的固定建筑工标记中描述 {5,5,3,3,1,1,3,5}。

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  4. Describe {35, 45, 55} in set-builder notation with an inequality as the condition.
  ::4. 在以不平等为条件的固定建筑工标记中描述{35、45、55}。

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  5. Describe {33, 39, 45, 51} in set-builder notation with an inequality as the condition.
  ::5. 在以不平等为条件的固定建筑工标记中描述{33、39、45、51}。

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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: