6.7 绝对值

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  绝对值

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  Absolute Value
  ::绝对值

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  Timmy is trying out his new roller skates. He’s not allowed to cross the street yet, so he skates back and forth in front of his house. If he skates 20 yards east and then 10 yards west, how far is he from where he started? What if he skates 20 yards west and then 10 yards east?
  ::蒂米正在尝试新的溜冰鞋。 他还没有被允许过马路,所以他在家里前后溜冰。 如果他在东20码、西10码滑冰,那么他离起点还有多远? 如果他在西20码、东10码,他又滑了20码、东10码呢?

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  The absolute value of a number is its distance from zero on a number line. There are always two numbers on the number line that are the same distance from zero. For instance, the numbers 4 and -4 are each a distance of 4 units away from zero.
  ::数字的绝对值是数字线上从零到零的距离。数字线上总有两个数字与零一样的距离。例如,数字4和 - 4是距离零的4个单位。

  

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  $\begin{align*}|4|\end{align*}$ represents the distance from 4 to zero, which equals 4.
  ::4 代表4到0的距离,等于4。

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  $\begin{align*}|-4|\end{align*}$ represents the distance from -4 to zero, which also equals 4.
  ::4 表示从 - 4 到零的距离, 也等于 4 。

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  In fact, for any real number $\begin{align*}x\end{align*}$ :
  ::事实上,对于任何实际数字x:

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  $\begin{align*}|x| = x\end{align*}$ if $\begin{align*}x\end{align*}$ is not negative, and $\begin{align*}|x| = -x\end{align*}$ if $\begin{align*}x\end{align*}$ is negative.
  ::x 不是负的, x x x x x x x x x x 。

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  Absolute value has no effect on a positive number, but changes a negative number into its positive inverse .
  ::绝对值对正数没有影响,但将负数变为正反数。

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  Evaluating Absolute Values
  ::评价绝对值

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  Evaluate the following absolute values.
  ::评估以下绝对值。

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  a)  $\begin{align*}|25|\end{align*}$
  :a) ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  $\begin{align*}|25|= 25\end{align*}$ Since 25 is a positive number, the absolute value does not change it.
  ::25+25 因为25是正数 绝对值不会改变

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  b)  $\begin{align*}|-120|\end{align*}$
  :b) 120_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  $\begin{align*}|-120| = 120\end{align*}$ Since -120 is a negative number, the absolute value makes it positive.
  ::120+120 因为-120是负数 绝对值是正数

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  c)  $\begin{align*}|-3|\end{align*}$
  :c) 3________________________________________________________________________________________________________________________________________________________________________________________________

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  $\begin{align*}|-3| = 3\end{align*}$ Since -3 is a negative number, the absolute value makes it positive.
  ::由于 -3是一个负数, 绝对值使它呈正数 。

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  d)  $\begin{align*}|55|\end{align*}$
  :d) _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  $\begin{align*}|55| = 55\end{align*}$ Since 55 is a positive number, the absolute value does not change it.
  ::5555 由于55是正数,绝对值不会改变。

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  e)  $\begin{align*}\left | - \frac{5}{4} \right |\end{align*}$
  :e) ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  $\begin{align*}\left | -\frac{5}{4} \right | =\frac{5}{4}\end{align*}$ Since $\begin{align*}-\frac{5}{4}\end{align*}$ is a negative number, the absolute value makes it positive.
  ::由于54是负数, 绝对值使其呈正数 。

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  Absolute value is very useful in finding the distance between two points on the number line. The distance between any two points $\begin{align*}a\end{align*}$ and $\begin{align*}b\end{align*}$ on the number line is $\begin{align*}|a - b|\end{align*}$ or $\begin{align*}|b - a|\end{align*}$ .
  ::绝对值对于在数字线上找到两个点之间的距离非常有用。数字线上任何两个点a和b之间的距离是 a-b或b-a。

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  For example, the distance from 3 to -1 on the number line is $\begin{align*}|3 - (-1)| = |4| = 4\end{align*}$ .
  ::例如,数字线上从3到-1的距离是%3-(-1)(-1)(4)4。

  

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  We could have also found the distance by subtracting in the opposite order : $\begin{align*}|-1 - 3| = |-4| = 4\end{align*}$ . This makes sense because the distance is the same whether you are going from 3 to -1 or from -1 to 3.
  ::我们也可以通过以相反的顺序去掉来找到距离:+1-344。这是有道理的,因为无论从3到1,还是从-1到3,距离是一样的。

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  Finding the Distance Between Two Points 
  ::寻找两个点之间的距离

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  Find the distance between the following points on the number line.
  ::查找数字线上以下各点之间的距离。

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  Distance is the absolute value of the difference between the two points. 
  ::距离是两个点之间的差的绝对值。

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  a) 6 and 15
  ::a) 6和15

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  $\begin{align*}\text{distance} = |6 - 15| = |-9| = 9\end{align*}$
  ::距離6 - 159999

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  b) -5 and 8
  ::b)-5和8

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  $\begin{align*}\text{distance} = |-5 - 8| = |-13| = 13\end{align*}$
  ::距离5 - 8 13 13 13

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  c) -3 and -12
  :c)-3和-12

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  $\begin{align*}\text{distance} = |-3 - (-12)| = |9| = 9\end{align*}$
  ::距离3 - (- 12)\\\\\\\\\9\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\...\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Remember: When we computed the change in $\begin{align*}x\end{align*}$ and the change in $\begin{align*}y\end{align*}$ as part of the computation, these values were positive or negative, depending on the direction of movement. In this discussion, “distance” means a positive distance only.
  ::记住: 当我们在计算过程中计算 x 和 y 的变化时, 这些值是正的或负的, 取决于运动的方向。 在本次讨论中, “ 距离” 仅指正的距离 。

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  Solve an Absolute Value Equation
  ::解决绝对值等量

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  We now want to solve equations involving absolute values. Consider the following equation :
  ::我们现在要解决包含绝对值的方程式。考虑以下方程式:

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  $$\begin{align*}|x|=8\end{align*}$$
  ::8888888888888888888888888888888888888

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  This means that the distance from the number $\begin{align*}x\end{align*}$ to zero is 8. There are two numbers that satisfy this condition: 8 and -8.
  ::这意味着从数字x到零的距离是8,满足这一条件的两个数字是:8和8。

  

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  When we solve we always consider two possibilities:
  ::当我们解决问题时,我们总是考虑两种可能性:

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  1. The expression inside the absolute value sign is not negative.
     ::绝对值符号内的表达式不是负值。
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  2. The expression inside the absolute value sign is negative.
     ::绝对值符号内的表达式为负值。

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  Then we solve each equation separately.
  ::然后我们分别解决每个方程式

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  Solve the following absolute value equations.
  ::解决以下绝对值方程式。

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  a) There are two possibilities: $\begin{align*}x = 3\end{align*}$ and $\begin{align*}x = -3.\end{align*}$
  :a) 有两种可能性:x=3和x=3。

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  $\begin{align*}|x| = 3\end{align*}$
  ::

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  b) There are two possibilities: $\begin{align*}x = 10\end{align*}$ and $\begin{align*}x = -10.\end{align*}$
  :b) 有两种可能性:x=10和x=10。

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  $\begin{align*}|x| = 10\end{align*}$
  ::

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  Example
  ::示例示例示例示例

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  Example 1
  ::例1

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  Find the distance between the values $\begin{align*}-\frac{1}{3}\end{align*}$ and $\begin{align*}\frac{1}{5}\end{align*}$ on the number line.
  ::在数字行中查找值 -13 和 15 之间的距离。

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  The distance is the absolute value of the difference:
  ::距离是差数的绝对值 :

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  $$\begin{align*}\left|-\frac{1}{3}-\frac{1}{5}\right|&= \quad \text{Set up the absolute value.} \\ \left|-\frac{5}{15}-\frac{3}{15}\right|&= \quad \text{Give the two terms common denominators.}\\ \left|\frac{-5-3}{15}\right|&= \quad \text{Combine the terms.}\\ \left|\frac{-8}{15}\right|&= \quad \text{Simplify.}\\ \frac{8}{15} &= \quad \text{Evaluate.} \end{align*}$$
  ::-=YTET -伊甸园字幕组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=- 翻译组=-

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  The distance between the two points is $\begin{align*}\frac{8}{15}\end{align*}$ .
  ::两点之间的距离是815

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  Review
  ::回顾

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  Evaluate the absolute values.
  ::评估绝对价值。

  1. $\begin{align*}|250|\end{align*}$
  2. $\begin{align*}|-12|\end{align*}$
  3. $\begin{align*}|-0.003|\end{align*}$
  4. $\begin{align*}\left | -\frac{2}{5} \right |\end{align*}$
  5. $\begin{align*}\left | \frac{1}{10} \right |\end{align*}$

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  Find the distance between the points.
  ::查找点之间的距离 。

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  6. 12 and -11
     ::12和11(第12和11条)
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  7. 5 and 22
     ::第5和22条
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  8. -9 and -18
     ::-9和18
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  9. -2 and 3
     ::-2和3
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  10. -0.012 and 1.067
      ::-0.012和1.067
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  11. $\begin{align*}-\frac{2}{3}\end{align*}$ and $\begin{align*}\frac{7}{8}\end{align*}$
      ::- 23和78

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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 。