6.1 不平等表达

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  不平等言论

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  Inequality Expressions
  ::不平等言论

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  Dita has a budget of $350 to spend on a rental car for an upcoming trip, but she wants to spend as little of that money as possible. If the trip will last five days, what range of daily rental rates should she be willing to consider?
  ::Dita有350美元的预算可以花在一辆租车上,用于即将到来的行程,但她想尽量少花这笔钱。 如果旅行持续五天,那么她愿意考虑的每日租金范围是多少?

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  Like equations, inequalities show a relationship between two expressions. We solve and graph inequalities in a similar way to equations—but when we solve an inequality , the answer is usually a set of values instead of just one value.
  ::像等式一样,不平等表现了两个表达式之间的关系。 我们用与等式相似的方式解决和描述不平等 — — 但是当我们解决不平等时,答案通常是一组数值,而不是一个数值。

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  When writing inequalities we use the following symbols:
  ::在写作不平等时,我们使用以下符号:

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  > is greater than
  ::> > 大于

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  $\begin{array}{r}\ge \end{array}$ is greater than or equal to
  ::大于或等于

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  < is less than
  ::< 是小于

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  $\begin{array}{r}\le \end{array}$ is less than or equal to
  ::等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于等于

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  Write and Graph Inequalities in One Variable on a Number Line
  ::在数字行的一个变量中写入和图形不平等

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  Let’s start with the simple inequality $\begin{array}{r}x>3\end{array}$ .
  ::让我们从简单的不平等x>3开始。

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  We read this inequality as “ $\begin{array}{r}x\end{array}$ is greater than 3.” The solution is the set of all real numbers that are greater than three. We often represent the solution set of an inequality with a number line graph.
  ::我们把这种不平等看成是“x大于3 ” 。 解决办法是所有实际数字的一组,超过3个。 我们常常用数字线图代表不平等的一组解决办法。

  

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  Consider another simple inequality: $\begin{array}{r}x\le 4\end{array}$ .
  ::考虑另一个简单的不平等: x4。

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  We read this inequality as “ $\begin{array}{r}x\end{array}$ is less than or equal to 4.” The solution is the set of all real numbers that are equal to four or less than four. We can graph this solution set on the number line.
  ::我们把这种不平等解读为“x小于或等于4”。 解决办法是所有实际数字的一组,等于或小于4或4。 我们可以用数字线来表示这个解决办法。

  

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  Notice that we use an empty circle for the endpoint of a strict inequality (like $\begin{array}{r}x>3\end{array}$ ), and a filled circle for one where the equals sign is included (like $\begin{array}{r}x\le 4\end{array}$ ).
  ::请注意,我们使用空圆来填补严格不平等的终点(如 x>3) , 并使用空圆来填补包含等号的圆( 如 x% 4) 。

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  Graph the following inequalities on the number line.
  ::数字线上的以下不平等图解。

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  a) $\begin{array}{r}x<-3\end{array}$
  ::a) x%3

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  The inequality $\begin{array}{r}x<-3\end{array}$ represents all numbers that are less than -3. The number -3 is not included in the solution, so it is represented by an open circle on the graph.
  ::不平等 x3 代表所有小于 -3. 的数字。 数字 - 3 不包含在解决方案中, 因此在图形上以开放圆表示 。

  

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  b) $\begin{array}{r}x\ge 6\end{array}$
  ::b) x_6

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   The inequality $\begin{array}{r}x\ge 6\end{array}$ represents all numbers that are greater than or equal to 6. The number 6 is included in the solution, so it is represented by a closed circle on the graph.
  ::=============================================================================================================================================== =============================================================================================================================================================================================================================================================================================================================================================================

  

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  c) $\begin{array}{r}x>0\end{array}$
  ::c) x>0

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  The inequality $\begin{array}{r}x>0\end{array}$ represents all numbers that are greater than 0. The number 0 is not included in the solution, so it is represented by an open circle on the graph.
  ::不平等 x>0 代表大于 0 的所有数字。 数字 0 不包含在解决方案中, 因此它由图形上的空圆表示 。

  

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  Write the inequality that is represented by each graph.
  ::写下每个图表所代表的不平等。

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  a)
  ::a) (a)

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  $\begin{array}{r}x\le -12\end{array}$
  ::x 12

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  b)
  :b) b)

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  $\begin{array}{r}x>540\end{array}$
  ::x540x>540

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  c)
  :c) c)

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  $\begin{array}{r}x<6.5\end{array}$
  ::x6.5

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  Inequalities appear everywhere in real life. Here are some simple examples of real-world applications.
  ::不平等现象在现实生活中无处不在,这是现实应用的一些简单例子。

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  Write each statement as an inequality and graph it on the number line.
  ::将每个语句写成不平等,并在数字线上绘制图表。

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  a) You must maintain a balance of at least $2500 in your checking account to get free checking.
  :a) 您必须在支票账户中保持至少2 500美元的余额,以便免费检查。

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  The words “at least” imply that the value of $2500 is included in the solution set, so the inequality is written as $\begin{array}{r}x\ge 2500\end{array}$ .
  ::“至少”两字意味着2 500美元的价值包括在一套解决办法中,因此不平等的字眼是x2500。

  

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  b) You must be at least 48 inches tall to ride the “Thunderbolt” Rollercoaster.
  :b) 你必须至少48英寸高才能骑上“Turnderbolt” Rollercoaster。

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  The words “at least” imply that the value of 48 inches is included in the solution set, so the inequality is written as $\begin{array}{r}x\ge 48\end{array}$ .
  ::“至少”一词的意思是,将48英寸的值包括在一套解决办法中,因此,不平等的字眼是x48。

  

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  c) You must be younger than 3 years old to get free admission at the San Diego Zoo.
  :c) 您必须不满3岁才能免费进入圣地亚哥动物园。

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  The inequality is written as $\begin{array}{r}x<3\end{array}$ .
  ::不平等以 x<3 字写成 。

  

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  Examples
  ::实例

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  Example 1
  ::例1

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  Graph the inequality $\begin{array}{r}x\le 8\end{array}$ on the number line.
  ::在数字行上绘制不平等 x__8 图。

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  The inequality $\begin{array}{r}x\le 8\end{array}$ represents all numbers that are less than or equal to 8. The number 8 is included in the solution, so it is represented by a closed circle on the graph.
  ::不平等 x8 代表所有小于或等于 8. 的数值。 数字 8 包含在解决方案中, 以图形上的封闭圆表示 。

  

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  Example 2
  ::例2

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  Write the inequality that is represented by the graph below.
  ::写下下图所示的不平等。

  

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  $\begin{array}{r}x\ge 85\end{array}$
  ::x85

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  Example 3
  ::例3

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  Write the statement, "the speed limit on the interstate is 65 miles per hour or less" as an inequality .
  ::写下声明:“州际间车速限制为每小时65英里或不到65英里”,

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  Speed limit means the highest allowable speed, so the inequality is written as $\begin{array}{r}x\le 65\end{array}$ .
  ::速度限制是指最高允许速度,所以不平等值是x65。

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  Review
  ::回顾

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  1. Write the inequality represented by the graph.
     ::写下图表所代表的不平等。
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  2. Write the inequality represented by the graph.
     ::写下图表所代表的不平等。
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  3. Write the inequality represented by the graph.
     ::写下图表所代表的不平等。
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  4. Write the inequality represented by the graph.
     ::写下图表所代表的不平等。

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  Graph each inequality on the number line.
  ::在数字线上绘制每个不平等的图表。

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  5. $\begin{array}{r}x<-35\end{array}$
     ::x% 35
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  6. $\begin{array}{r}x>-17\end{array}$
     ::x% 17
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  7. $\begin{array}{r}x\ge 20\end{array}$
     ::x20
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  8. $\begin{array}{r}x\le 3\end{array}$
     ::x% 3
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  9. $\begin{array}{r}x\ge -5\end{array}$
     ::x% 5
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  10. $\begin{array}{r}x>20\end{array}$
      ::x>20 x>20

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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 。