4.3整数与绝对值的比较

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  整数与绝对值的比较

  

  [Figure 1]

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  Josh and his four friends are comparing how much money they each have. Josh has $23. His friend Krista has $25. His friend Mark has $8. His friends Luis and Tessa each owe their parents money so they basically have negative money. Luis has -$5 and Tessa has -$10. How could Josh use a  number line  to compare and  order  the  amount  of money he and his friends have from least to greatest?
  ::乔希和他的四个朋友在比较他们每人有多少钱。乔什有23美元。他的朋友克里斯塔有25美元。他的朋友马克有8美元。他的朋友路易斯和泰萨每人欠他们父母的钱,所以他们基本上都有负数的钱。路易斯有5美元,苔莎有10美元。乔什如何用数字线来比较和订购他和他的朋友从最少到最多的钱?

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  In this concept, you will learn how to order   and how to compare integers using inequality symbols.
  ::在此概念中,您将学会如何使用不平等符号排序和比较整数。

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  Comparing Integers with Absolute Value
  ::将整数与绝对值进行比较

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  Recall that integers are a set of numbers that include the  positive   whole numbers  (1, 2, 3, 4, 5, ...), their opposites (-1, -2, -3, -4, -5, ...) and  zero . Sometimes you will want to compare different integers or put them in order from smallest to largest. A number line can help you to do this.
  ::回顾整数是一组数字,包括正整数(1, 2, 3, 4, 4, 5, ......)、正反数(1, 2, 3, 4, 4, 5, 5)和零。 有时您会想要比较不同的整数, 或者将整数从最小到最大排序。 一个数字行可以帮助您做到这一点 。

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  On a number line, the further a number is to the right, the larger its value. Conversely, the further a number is to the left, the smaller its value.
  ::在数字行中,一个数字越向右,其值越大。相反,数字向左越远,其值就越小。

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  Let's look at an example.
  ::让我们举个例子。

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  Plot these numbers on a number line. Then determine which one is the largest and which is the smallest: -12, 5, -6, -9, 10, 7
  ::在数字行中绘制这些数字。 然后确定哪一个最大,哪个最小: -12, 5, 6, 9, 10, 7

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  First, draw a number line and plot the numbers on the number line.
  ::首先,绘制数字线,并在数字线上绘制数字。

  

  [Figure 2]

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  Now look at the values on this number line and where they are located.
  ::现在看看这个数字行的值, 以及它们的位置 。

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  You can see that the number furthest to the right is 10. That is the largest number. The number furthest to the left is -12. That is the smallest number.
  ::您可以看到最远的右侧数字是 10。 这是最大的数字。 最远的左侧数字是 - 12。 这是最小的数字 。

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  Let's look at another example.
  ::让我们再举一个例子。

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  Order these integers from least to greatest: -6, 0, 5, -1.
  ::将这些整数从最小排列为最大: -6, 0, 5, 1。

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  Again, to help you to order these integers, draw a number line. Then, plot points for -6, 0, 5 and -1.
  ::再次,为了帮助您命令这些整数, 请绘制一个数字行。 然后, 绘图点为 - 6, 0, 5 和 - 1 。

  

  [Figure 3]

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  You can see that -6 is furthest to the left. This means -6 is the smallest number in your set. Moving to the right, the next plotted point is -1. This means -1 is the next smallest number in your set. Continuing in this way you see that the next smallest number is 0 and then the largest number is 6.
  ::您可以看到 - 6 距离左方最远。 这意味着 - 6 是您组中最小的数字。 向右移动, 下一个绘图点是 - 1 。 这意味着 - 1 是您组中下一个最小的数字。 继续这样看, 下一个最小的数字是 0, 然后最大的数字是 6 。

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  Ordered from least to greatest, the numbers are -6, -1, 0, 5.
  ::从最低到最高顺序排列,数字为 -6,1,0,5。

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  Once you know how to order integers from smallest to largest, you can compare integers using symbols. You can use the following inequality symbols to compare integers:
  ::一旦知道如何将整数从最小点到最大点,您就可以使用符号比较整数。您可以使用以下不平等符号来比较整数:

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  • > means  is greater than.
    ::> 手段大于。
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  • < means  is less than.
    ::< 手段小于 。
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  • = means  is equal to.
    ::手段等于。
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  • ≠ means  is not equal to.
    ::手段不等于。

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  Let's look at an example.
  ::让我们举个例子。

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  Choose the inequality symbol that goes in the blank to make the statement true:  $\begin{array}{r}-2\underset{\_}{}-4\end{array}$
  ::选择空白中的不平等符号, 使声明变为事实: - 2 _ - 4

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  First, draw a number line.
  ::首先,绘制数字线。

  

  [Figure 4]

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  Next, notice that -2 is to the right of -4 on the number line. This means that -2 is greater than -4 so the symbol  $\begin{array}{r}>\end{array}$  should go in the blank.
  ::下一步请注意 -2 是数字行 -4 右侧的 -2。 这意味着 - 2 大于 - 4 , 所以符号 > 应该在空白中输入 。

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  The answer is  $\begin{array}{r}\text{-}2>\text{-}4\end{array}$ .
  ::答案是 - 2 > - 4。

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  Let's look at another example.
  ::让我们再举一个例子。

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  Choose the inequality symbol that goes in the blank to make the statement true:  $\begin{array}{r}|\text{-}4|\underset{\_}{}4\end{array}$
  ::选择空白中的不平等符号, 使声明变为事实 : @ - 4 @ _ 4

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  First, evaluate the absolute value.  $\begin{array}{r}|\text{-}4|\end{array}$  is 4, because -4 is 4 units from zero on the number line. This means  $\begin{array}{r}|\text{-}4|\end{array}$  is equal to  $\begin{array}{r}|4|\end{array}$  so the symbol = goes in the blank.
  ::首先,评价绝对值。\\\\\\\\\\\\\\4\\\\\\\\\\\\\\\\\\\\4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  The answer is  $\begin{array}{r}|\text{-}4|=4\end{array}$ .
  ::答案是 4 = 4 = 4 = 4 = = 4 = = 4 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = == = == = = == == == === == == ==== === ====== === ==== ===== ===== === ======== =================================================================================================================================================================================================================================================================================================================================================

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about Josh and his four friends.
  ::早些时候,你得到一个问题 关于乔什和他的四个朋友。

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  Josh has $23, his friend Krista has $25, his friend Mark has $8, his friend Luis has -$5 (he owes his parents), and his friend Tessa has -$10 (she also owes her parents). Josh wanted to compare and order the amount of money he and his friends have.
  ::Josh有23美元,他的朋友Krista有25美元,他的朋友Mark有8美元,他的朋友Luis有5美元(他欠他的父母),他的朋友Tessa有10美元(她也欠她的父母 ) 。 Josh想比较和订购他和他的朋友所拥有的金额。

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  First, draw a number line. Make sure it is big enough to include all the numbers you need to plot. Plot all the money values on the number line.
  ::首先, 绘制一个数字行。 请确定它足够大, 包括您需要绘制的所有数字。 绘制数字行上的所有货币值 。

  

  [Figure 5]

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  Next, figure out which number is furthest to the left on the number line. This will be your smallest number. -10 is the smallest number which means Tessa has the least amount of money.
  ::下一步, 找出数字线左侧最远的数字。 这将是你最小的数字 。 - 10 是最小的数字 , 表示泰萨的钱最少 。

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  Then, look to see which plotted number is to the right of -10. -5 comes next so it is the second smallest number. Luis has the next smallest amount of money.
  ::然后,看看哪个绘图数字在 - 10. - 5 的右侧, 下一个是第二个最小的数字。 Luis 拥有下一个最小的金额 。

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  Continue by reading the plotted points off the number line from left to right. This will give you the numbers ordered from least to greatest.
  ::继续读取数字线从左到右的绘图点。 这将为您提供从最小到最大排序的数字 。

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  The answer is that the money amounts in order from least to greatest are -$10, -$5, $8, $23, $25. Tessa has the smallest amount of money, then Luis, then Mark, then Josh. Krista has the most amount of money.
  ::答案是,从最低到最高数额的金额是 — — 10美元,5美元,8美元,23美元,25美元。泰萨的金额最小,然后是路易斯,然后是马克,然后是乔希。克里斯塔的金额最多。

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  Example 2
  ::例2

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  Choose the inequality symbol that goes in the blank to make the statement true:  $\begin{array}{r}-22\underset{\_}{}-32\end{array}$
  ::选择空白中的不平等符号,使声明变为事实: - 22 _ - 32

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  First, visualize a number line and imagine plotting the points -22 and -32.
  ::首先,想象一个数字线, 并想象绘制点 - 22 和 - 32 。

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  Next, determine where -22 is on the number line compared to -32. -22 will be further to the right. This means -22 is greater than -32.
  ::下一步, 确定 - 22 在数字线上的位置, 与 - 32 相比 。 - 22 将更进一步。 这意味着 - 22 大于 - 32 。

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  The answer is  $\begin{array}{r}-22>-32\end{array}$ .
  ::答案是-22 >-32。

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  Example 3
  ::例3

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  Use a number line and write these numbers in order from least to greatest.
  ::使用数字行并写入这些数字,以便从最小到最大顺序。

  −4, 2, 8, 9, −11, −5

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  First, draw a number line. Make sure it is big enough to include all the numbers you need to plot.
  ::首先, 绘制数字行。 请确定它足够大, 包括您需要绘制的所有数字 。

  

  [Figure 6]

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  Next, figure out which number is furthest to the left on the number line. This will be your smallest number. -11 is the smallest number.
  ::下一步, 找出数字行左侧最远的数字。 这将是你最小的数字 。 - 11 是最小的数字 。

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  Then, look to see which plotted number is to the right of -11. -5 comes next so it is the second smallest number.
  ::然后,看看哪个绘图数是 -11。 - 5 的右侧, 下一个是第二个最小的数字 。

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  Continue by reading the plotted points off the number line from left to right. This will give you the numbers ordered from least to greatest.
  ::继续读取数字线从左到右的绘图点。 这将为您提供从最小到最大排序的数字 。

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  The answer is -11, -5, -4, 2, 8, 9.
  ::答案是 -11,5,5,4,2,8,9

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  Example 4
  ::例4

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  Use a number line and write these numbers in order from least to greatest.
  ::使用数字行并写入这些数字,以便从最小到最大顺序。

  5, −16, 6, -12, 1, 4

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  First, draw a number line. Make sure it is big enough to include all the numbers you need to plot.
  ::首先, 绘制数字行。 请确定它足够大, 包括您需要绘制的所有数字 。

  

  [Figure 7]

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  Next, figure out which number is furthest to the left on the number line. This will be your smallest number. -16 is the smallest number.
  ::下一步, 找出数字行左侧最远的数字。 这将是你最小的数字 。 - 16 是最小的数字 。

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  Then, look to see which plotted number is to the right of -16. -12 comes next so it is the second smallest number.
  ::然后,看看哪个绘图数是 -16 的右侧。 - 12 下一个是第二个最小的数字 。

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  Continue by reading the plotted points off the number line from left to right. This will give you the numbers ordered from least to greatest.
  ::继续读取数字线从左到右的绘图点。 这将为您提供从最小到最大排序的数字 。

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  The answer is -16, -12, 1, 4, 5, 6.
  ::答案是 -16 -12 1,4,5,6

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  Example 5
  ::例5

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  Use a number line and write these numbers in order from least to greatest.
  ::使用数字行并写入这些数字,以便从最小到最大顺序。

  −3, −2, −7, −12, −1

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  First, draw a number line. Make sure it is big enough to include all the numbers you need to plot.
  ::首先, 绘制数字行。 请确定它足够大, 包括您需要绘制的所有数字 。

  

  [Figure 8]

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  Next, figure out which number is furthest to the left on the number line. This will be your smallest number. -12 is the smallest number.
  ::下一步, 找出数字行左侧最远的数字。 这将是你最小的数字 。 - 12 是最小的数字 。

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  Then, look to see which plotted number is to the right of -12. -7 comes next so it is the second smallest number.
  ::然后,看看哪个绘图数是 -12 的右侧。 - 7 下一个是第二个最小的数字 。

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  Continue by reading the plotted points off the number line from left to right. This will give you the numbers ordered from least to greatest.
  ::继续读取数字线从左到右的绘图点。 这将为您提供从最小到最大排序的数字 。

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  The answer is -12, -7, -3, -2, -1.
  ::答案是 -12, -7,3,2,1

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  Review
  ::回顾

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  Compare the following integers using inequality symbols.
  ::使用不平等符号比较以下整数。

  1. $\begin{array}{r}-17\underset{\_}{}-19\end{array}$  
  2. $\begin{array}{r}-9\underset{\_}{}-11\end{array}$
  3. $\begin{array}{r}4\underset{\_}{}-3\end{array}$
  4. $\begin{array}{r}5\underset{\_}{}7\end{array}$  
  5. $\begin{array}{r}9\underset{\_}{}-9\end{array}$
  6. $\begin{array}{r}-12\underset{\_}{}-23\end{array}$
  7. $\begin{array}{r}|-9|\underset{\_}{}|8|\end{array}$
  8. $\begin{array}{r}|9|\underset{\_}{}|-9|\end{array}$
  9. $\begin{array}{r}|-2|\underset{\_}{}|-7|\end{array}$  
  10. $\begin{array}{r}|-12|\underset{\_}{}12\end{array}$
  11. $\begin{array}{r}|-22|\underset{\_}{}|22|\end{array}$
  12. $\begin{array}{r}|14|\underset{\_}{}|-7|\end{array}$
  13. $\begin{array}{r}|88|\underset{\_}{}|90|\end{array}$
  14. $\begin{array}{r}|-88|\underset{\_}{}|-92|\end{array}$
  15. $\begin{array}{r}|-27|\underset{\_}{}|27|\end{array}$

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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 。

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  Resources
  ::资源