1.9 检查不平等的解决办法

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  检查不平等的解决方案

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  Checking Solutions: Inequalities 
  ::检查解决方案:不平等

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  To check the solution to an inequality , replace the variable in the inequality with the value of the solution. If the solution is correct, the simplified inequality will produce a true statement.
  ::为了检查不平等的解决方案,用解决方案的价值取代不平等的变量。 如果解决方案正确,简化的不平等将产生真实的描述。

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  Checking Solutions to Inequalities
  ::检查不平等的解决方案

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  1. Check that the given number is a solution to the inequality: $\begin{array}{r}a=10;20a\le 250\end{array}$
  ::1. 检查给定数字是否是不平等的解决办法:a=10;20a=250

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  Replace the variable in the inequality with the given value.
  ::用给定值取代不平等变量。

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  $$\begin{array}{rl}20(a)& \le 250\\ 20(10)& \le 250\\ 200& \le 250\end{array}$$

  
  ::20(a)

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  $\begin{array}{r}200\le 250\end{array}$  is a true statement . This means that $\begin{array}{r}a=10\end{array}$ is a solution to the inequality $\begin{array}{r}20a\le 250\end{array}$ .
  ::200250是一个真实的陈述,这意味着a=10是解决20a250不平等的一种解决办法。

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  Note that $\begin{array}{r}a=10\end{array}$ is not the only solution to this inequality. Recall that inequalities may be simplified using essentially the same processes as equations. If we divide both sides of the inequality by 20, we can write it as $\begin{array}{r}a\le 12.5\end{array}$ . This means that actually any number less than or equal to 12.5 is a solution to the inequality.
  ::请注意 a=10 并不是解决这种不平等的唯一办法。 回顾不平等可以用与等式基本相同的程序来简化。 如果我们将不平等的两边除以 20, 我们就可以把它写成 a=12.5。 这意味着任何少于或等于 12.5 的数字实际上都是解决不平等的办法。

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  2. Check that the given number is a solution to the inequality: $\begin{array}{r}b=-0.5;\frac{3-b}{b}>-4\end{array}$
  ::2. 核实给定数字是解决不平等问题的办法:b0.5;3-b4

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  $$\begin{array}{rl}\frac{3-(b)}{(b)}& >-4\\ \frac{3-(-0.5)}{(-0.5)}& >-4\\ \frac{3+0.5}{-0.5}& >-4\\ -\frac{3.5}{0.5}& >-4\\ -7& >-4\end{array}$$

  
  ::3-(b)(b) 43-(-05)(-05)(-05)43+0.5-0.5}4-3.5-0.5}4-74-4

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  This statement is false . This means that $\begin{array}{r}b=-0.5\end{array}$ is not a solution to the inequality $\begin{array}{r}\frac{3-b}{b}>-4\end{array}$ .
  ::这种说法是虚假的,这意味着b0.5不是解决不平等问题3-b4的办法。

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  Real-World Application 
  ::真实世界应用

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  To organize a picnic, Peter needs at least two times as many hamburgers as hot dogs. He has 24 hot dogs. What is the possible number of hamburgers Peter has?
  ::为了组织野餐,彼得至少需要比热狗多两倍的汉堡。他有24只热狗。彼得可能有多少汉堡?

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  Define
  ::定义定义

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  Let $\begin{array}{r}x\end{array}$  represent the number of hamburgers
  ::让 x 代表汉堡数量

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  Translate
  ::翻译

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  Peter needs at least two times as many hamburgers as hot dogs. He has 24 hot dogs.
  ::彼得至少需要比热狗多两倍的汉堡 他有24只热狗

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  This means that twice the number of hot dogs is less than or equal to the number of hamburgers.
  ::这意味着热狗数量的两倍少于或等于汉堡的数量。

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  $$\begin{array}{r}2\times 24\le x,\text{or}48\le x\end{array}$$

  
  ::2x24x, 或 48x

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  Answer
  ::答复答复答复

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  Peter needs at least 48 hamburgers.
  ::彼得至少需要48个汉堡包

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  Check
  ::支票支票支票支票

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  48 hamburgers is twice the number of hot dogs. More than 48 hamburgers is more than twice the number of hot dogs. The answer checks out .
  ::48个汉堡是热狗的两倍,超过48个汉堡是热狗的两倍,答案是肯定的。

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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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  Check that the given number is a solution to the inequality: $\begin{array}{r}x=\frac{3}{4};4x+5\le 8\end{array}$
  ::检查给定数字是否是不平等的解决方案: x=34; 4x+58

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  $$\begin{array}{rl}4(x)+5& \le 8\\ 4\left(\frac{3}{4}\right)+5& \ge 8\\ 3+5& \ge 8\\ 8& \ge 8\end{array}$$

  
  ::4(x)+584(34)+583+588(8)8(8)8

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  The statement $\begin{array}{r}8\ge 8\end{array}$  is true  because this inequality includes an equals sign, and since 8 is equal to itself it is also “ greater than or equal to” itself. This means that $\begin{array}{r}x=\frac{3}{4}\end{array}$ is a solution to the inequality $\begin{array}{r}4x+5\le 8\end{array}$ .
  ::88的表述是真实的,因为这种不平等包括一个等号,而由于8等于本身,它本身也“大于或等于”它本身。 这意味着x=34是解决4x+58不平等的一种解决办法。

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  Review
  ::回顾

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  For 1-4, check whether the given number is a solution to the corresponding inequality.
  ::1-4,检查给定数字是否解决了相应的不平等问题。

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  1. $\begin{array}{r}x=12;2(x+6)\le 8x\end{array}$
     ::x=12; 2(x+6)8x
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  2. $\begin{array}{r}z=-9;1.4z+5.2>0.4z\end{array}$
     ::z9; 1.4z+5.2>0.4z
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  3. $\begin{array}{r}y=40;-\frac{5}{2}y+\frac{1}{2}<-18\end{array}$
     ::y=40; - 52y+1218
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  4. $\begin{array}{r}t=0.4;80\ge 10(3t+2)\end{array}$
     ::t=0.4; 8010( 3t+2)
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  5. On your new job you can be paid in one of two ways. You can either be paid $1000 per month plus 6% commission of total sales or be paid $1200 per month plus you start earning a 5% commission once your sales have exceeded $2000. For what amount of sales is the first option better than the second option? Assume there are always sales over $2000.
     ::在您的新工作上,您可以以两种方式之一获得工资。您可以要么每月1 000美元加总销售额的6%佣金,要么每月1 200美元,一旦您的销售额超过2 000美元,您就可以开始赚取5%的佣金。第一种选择的销售额比第二种选择的多多少?假设总销售额超过2 000美元。

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  For 6-14, suppose a phone company offers a choice of three text-messaging plans. Plan A gives you unlimited text messages for $10 a month; Plan B gives you 60 text messages for $5 a month and then charges you $0.05 for each additional message; and Plan C has no monthly fee but charges you $0.10 per message.
  ::对于 6-14, 假设一个电话公司提供三种短信计划。 A 计划给你无限的短信, 每月10美元; B 计划给你60条短信, 每月5美元, 然后每条附加信息收费0. 5美元; C 计划没有每月收费,但每条信息收费0. 10美元。

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  6. If $\begin{array}{r}m\end{array}$ is the number of messages you send per month, write an expression for the monthly cost of each of the three plans.
     ::如果 m 是您每月发送的信息数量, 请写下三个计划每个计划每月成本的表达式 。
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  7. For what values of $\begin{array}{r}m\end{array}$ is Plan A cheaper than Plan B?
     ::A计划比B计划便宜多少?
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  8. For what values of $\begin{array}{r}m\end{array}$ is Plan A cheaper than Plan C?
     ::A计划比C计划便宜多少?
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  9. For what values of $\begin{array}{r}m\end{array}$ is Plan B cheaper than Plan C?
     ::B计划比C计划便宜多少?
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  10. For what values of $\begin{array}{r}m\end{array}$ is Plan A the cheapest of all? (Hint: for what values is A both cheaper than B and cheaper than C?)
      ::" 计划A " 是什么价值最便宜? (提示:什么价值A比B便宜,C便宜? )
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  11. For what values of $\begin{array}{r}m\end{array}$ is Plan B the cheapest of all? (Careful—for what values is B cheaper than A?)
      ::B计划是什么价值? B计划是最便宜的? (注意B比A更便宜的价值是什么? )
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  12. For what values of $\begin{array}{r}m\end{array}$ is Plan C the cheapest of all?
      ::C计划最便宜的是什么价值?
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  13. If you send 30 messages per month, which plan is cheapest?
      ::如果你每月发送30条信息 哪个计划最便宜?
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  14. What is the cost of each of the three plans if you send 30 messages per month?
      ::如果你每月发送30条信息,这三项计划每个计划的费用是多少?

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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 。