2.7 解决多级不平等

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• 选择节解决多重不平等

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  解决多重不平等

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  You are going to purchase a gym membership. One gym has no activation fee to join and costs $20 per month, but requires you to do a series of five one-on-one sessions with a personal trainer for $30 each. Another gym has a $100 activation fee for your membership and costs $25 per month. Which one of these plans is better for a short- term membership? Which one is better in the long-term? 
  ::你将购买体育馆会员资格。一个体育馆没有活动费可以加入,每个月费用为20美元,但需要你与每个个人教练进行5次一对一的系列培训,每次30美元。另一个体育馆的会员资格为100美元,每月费用为25美元。其中哪一个计划对短期会员更好?哪一个计划在长期更好?

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  In this section, we will discuss how to compare  quantities by solving inequalities with multiple operations like this one.
  ::在本节中,我们将讨论如何通过解决不平等和多种行动(如这一项行动)来比较数量。

  

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  Solving Multi-Step Inequalities
  ::解决多重不平等

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  Like , multi-step inequalities involve combining like terms , the , or, possibly, both of these techniques. We will solve them using the exact same techniques we used for solving multi-step equations, except when multiplying or dividing by a negative number. In that case, we switch the direction of the inequality . 
  ::类似地,多阶段的不平等涉及将类似术语、这些技术或可能这两种技术结合起来。 我们将使用我们用来解决多步方程的完全相同的技术来解决这些问题,除非乘以负数或除以负数。 在这种情况下,我们将改变不平等的方向。

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  by CK-12 demonstrates how to solve multi-step inequalities.
  ::CK-12展示了如何解决多阶段不平等。

   

   

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  Combining Like Terms
  ::将类似术语合并

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  Example 1
  ::例1

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  Solve and graph $\begin{array}{r}8x-5-4x\ge 37-2x\end{array}$ .
  ::解决和图8x-5-4x-37-2x。

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  Solution: Just as we did with equations, we first combine the like terms on the left side. Then, we combine the like terms on opposite sides.
  ::解答:就像我们用方程式一样,我们首先将左侧的类似条件结合起来。然后,我们把对面的类似条件结合起来。

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  $$\begin{array}{rl}8x-5-4x& \ge 37-2x\\ 4x-5& \ge 37-2x& & \text{combine like terms on one side}\\ 4x-5& \ge 37-\cancel{2x}\\ \underset{\_}{\phantom{\cancel{2x}}+2x}& \underset{\_}{+\cancel{2x}}& & \text{combine like terms on both sides}\\ 6x-\cancel{5}& \ge 37\\ \underset{\_}{+\cancel{5}}& \underset{\_}{\phantom{\cancel{5}}+5}\\ \frac{\cancel{6}x}{\cancel{6}}& \ge \frac{42}{6}\\ x& \ge 7\end{array}$$

  
  ::8 - 5 - 4x3-37 - 2x4x4x- 5- 37 - 2x2x+2x2x *2xx_ combine 单面4x-5-37 - 5x5-37+5_ 5+5+5_ 5+5_ 6x6x6_ 426x7等词

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  To check our solution, we substitute the boundary point and a test  point to see if they create true statements in our inequality. Let's start with the boundary point, $\begin{array}{r}x=7\end{array}$ . 
  ::为了检查我们的解决方案, 我们替换了边界点和一个测试点, 看看它们是否在我们的不平等中创造了真实声明。 让我们从边界点 x=7 开始 。

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  $$\begin{array}{rl}8(7)-5-4(7)& =37-2(7)\\ 56-5-28& =37-14\\ 23& =23\end{array}$$

  This is a true statement, so the boundary point is correct. Next, let's check a test point, say  $\begin{array}{r}x=10\end{array}$  .
  ::8(7)-5-4(7)=37-2(7)56-5-28=37-1423=23 这是一个真实的语句, 因此边界点是正确的。 下一步, 让我们检查一个测试点, 比如 x=10 。

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  $$\begin{array}{rl}8(10)-5-4(10)& \ge 37-2(10)\\ 80-5-40& \ge 37-20\\ 35& \ge 17\end{array}$$

   
  This is also a true statement. Therefore , our solution set is  $\begin{array}{r}x\ge 7\end{array}$ or $\begin{array}{r}[7,\mathrm{\infty })\end{array}$ .
  ::8(10)-5-4(10)-37-2(10)-1080-5-4037-203517这也是一个真实的说法,因此,我们的解决方案是x7或[7,]。

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  Graphically, we have:
  ::在图形方面,我们有:

  

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  Example 2
  ::例2

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  You are going to purchase a gym membership.
  ::你要买体育馆的会员资格

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  • One gym is $20 per month and requires five personal training sessions at $30 each. It has no activation fee.
    ::一个健身房每月20美元,需要5次个人培训,每人30美元,没有启动费。
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  • The other gym has a $100 activation fee and costs $25 per month.
    ::另一个健身房有100美元的激活费,每月25美元。

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  Which one of these plans is better for a short-term membership? Which one is better in the long-term?
  ::其中哪一种计划对短期成员更有利?哪一种计划对长期成员更有利?

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  Solution: If we say $\begin{array}{r}m\end{array}$ is the number of months of membership, then we can express the total cost of the first gym membership as $\begin{array}{r}20m+5(30)\end{array}$ and the second gym membership as $\begin{array}{r}100+25m\end{array}$ . Let's consider when the cost of the first membership is less than the second membership.
  ::解决方案:如果我们说M是会员月数,那么我们可以表示第一个健身馆会员的总费用是20米+5(30),第二个健身馆会员的总费用是100+25m。让我们考虑当第一个健身馆会员的费用低于第二个会员时。

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  $$\begin{array}{rl}20m+5(30)& \le 100+25m\\ \cancel{20m}+150& \le 100+25m\\ \underset{\_}{-\cancel{20m}}& \underset{\_}{\phantom{-\cancel{20m}}-20m}& & \text{combine like terms on both sides}\\ 150& \le \cancel{100}+5m\\ \underset{\_}{\phantom{-\cancel{100}}-100}& \underset{\_}{-\cancel{100}}\\ \frac{50}{5}& \le \frac{\cancel{5}m}{\cancel{5}}\\ 10& \le m\end{array}$$

  
  ::20m+5(30)-100+25m20m+150-100+25m20m+25m-20m_20m_20m_20m_20m_combine 双方的类似条件

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  Within the first 10 months or over a short term, the first membership is less expensive. After 10 months or over a longer term, the second membership is less expensive.
  ::在前10个月或短期内,第一个成员费用较低,第二个成员在10个月或更长时间后费用较低。

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  by CK-12 demonstrates how to solve problems involving  applications with  inequalities.
  ::CK-12展示了如何解决涉及不平等申请的问题。

   

   

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  The Distributive Property
  ::分配财产

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  by CK-12 demonstrates how to solve and graph multi-step inequalities that include the distributive property.   
  ::CK-12展示了如何解决和描述包括分配财产在内的多步不平等。

   

   

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  Example 3
  ::例3

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  Solve and graph $\begin{array}{r}2(3x-5)\le x+10.\end{array}$
  ::解决和图2(3x-5)x+10。

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  Solution: Just as we did for equations, we first distribute the 2 on the left side of the inequality.
  ::解答:正如我们对于等式所做的那样,我们首先在不平等的左边分配2。

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  $$\begin{array}{rl}2(3x-5)& \le x+10\\ 6x-10& \le x+10\text{distributive property}\end{array}$$

  
  ::2(3x--5)x+106x-10xx+10分配性财产

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  Now, we can proceed by combining like terms on opposite sides of the inequality . 
  ::现在,我们可以把不平等的对立面的类似条件结合起来。

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  $$\begin{array}{rl}& 6x-10\le \cancel{x}+10\\ & \underset{\_}{-x-\cancel{x}}\\ & 5x-\cancel{10}\le 10\\ & \underset{\_}{+\cancel{10}+10}\\ & \frac{\cancel{5}x}{\cancel{5}}\le \frac{20}{5}\\ & x\le 4\end{array}$$

  
  ::6-10xx+10-10-x-x-x-5x-10-1010+10+10-10_5x5x5}205×4

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  We have a solution set that we need to check. Let's first test the boundary point, $\begin{array}{r}x=4\end{array}$ .
  ::我们有一套解决方案,我们需要检查一下。让我们首先测试边界点 x=4。

  $$\begin{array}{rl}2(3(4)-5)& =4+10\\ 2(7)& =14\\ 14& =14\end{array}$$

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  Since 14 =14 is a true statement, the boundary point checks.
  ::由于14=14是一个真实的声明, 边界点检查。

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  Test another  value in the solution set , like  $\begin{array}{r}x=0.\end{array}$
  ::在解决方案集中测试另一个值, 如 x=0 。

  $$\begin{array}{r}2(3(0)-5)\le 0+10\\ 2(-5)\le 10\\ -10\le 10\end{array}$$

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  This is also a true statement. Therefore, our solution set is  $\begin{array}{r}x\le 4\end{array}$ or $\begin{array}{r}(-\mathrm{\infty },4]\end{array}$ .
  ::这也是一个真实的说法。 因此,我们的解决方案是 x4 或 (4 ) 。

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  The graph for this solution set is :
  ::此解决方案集的图表如下:

  

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  Both Combining Like Terms and the Distributive Property
  ::将 " 类似术语 " 和 " 分配财产 " 合并

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  Example 4
  ::例4

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  Solve and graph $\begin{array}{r}-3(x-10)+18>x-25\end{array}$ .
  ::溶解和图-3(x-10)+18>x-25。

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  Solution: Just as we did for multi-step equations, we will distribute first and then combine like terms.
  ::解决方案:正如我们对于多步方程所做的那样,我们将首先分配,然后合并类似术语。

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  $$\begin{array}{rl}-3(x-10)+18& >x-25\\ -3x+30+18& >x-25& & \text{distributive property}\\ -\cancel{3x}+48& >x-25\\ \underset{\_}{+\cancel{3x}}& \underset{\_}{\phantom{\cancel{3x}}+3x}& & \text{combine like terms}\\ 48& >4x-\cancel{25}\\ \underset{\_}{\phantom{\cancel{25}}+25}& \underset{\_}{+\cancel{25}}\\ \frac{73}{4}& >\frac{\cancel{4}x}{\cancel{4}}\\ 18\frac{1}{4}=\frac{73}{4}& >x\end{array}$$

  To check, first, let's try our boundary point.
  ::-3(x-10)+18x-25-3x+30+18>x-25分配属性-3x+48>x-25+3x_3x+3x_combine,如术语48>4x-2525+25_25_734>4x41814=734>x

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  $$\begin{array}{rl}-3(\frac{73}{4}-10)+18& =\frac{73}{4}-25\\ -3(\frac{73}{4}-\frac{40}{4})+18& =\frac{73}{4}-\frac{100}{4}\\ -3(\frac{33}{4})+18& =-\frac{27}{4}\\ -\frac{99}{4}+18& =-\frac{27}{4}\\ -\frac{99}{4}+\frac{72}{4}& =-\frac{27}{4}\\ -\frac{27}{4}& =-\frac{27}{4}\end{array}$$

  The boundary point is correct. Now, let's check a test point, say $\begin{array}{r}x=0\end{array}$ .
  ::- 3(734-10)+18=734-25-3(734-404)+18=734-1004-3(334)+18+18_274-994+18_274-994+724+274-274-274+274)该边界点是正确的。现在,让我们检查一个测试点,比如 x=0。

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  $$\begin{array}{rl}-3(0-10)+18& >0-35\\ -3(-10)+18& >-35\\ 30+18& >-35\\ 48& >-35\end{array}$$

  This is true, so $\begin{array}{r}\frac{73}{4}>x\end{array}$  or $\begin{array}{r}(-\mathrm{\infty },\frac{73}{4})\end{array}$  is correct.
  ::- 3(0-10)+18+18(0-10)+18(0-35)-3(10)+18(18)+3530+183548}35这是对的,因此734>x或(734)是正确的。

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  For the graph, it is probably easier to think of $\begin{array}{r}\frac{73}{4}\end{array}$  as a mixed number or $\begin{array}{r}18\frac{1}{4}\end{array}$ . We need to graph the interval $\begin{array}{r}(-\mathrm{\infty },18\frac{1}{4})\end{array}$ . We can approximate the location of $\begin{array}{r}18\frac{1}{4}\end{array}$  on the graph. Our graph looks like:
  ::对于图表来说,可能更容易将734视为混合数字或1814年。我们需要将间隔(,1814年)图形化。我们可以将1814年的位置大致放在图中。我们的图表看起来如下:

  

   

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  Example 5
  ::例5

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  Solve and graph $\begin{array}{r}-(y+16)-3y<8.\end{array}$
  ::解析和图形 - (y+16) - 3y< 8 。

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  Solution:
  ::解决方案 :

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  $$\begin{array}{rl}-(y+16)-3y& <8\\ -y-16-3y& <8\text{distributive property}\\ -4y-\cancel{16}& <8\text{combine like terms}\\ \underset{\_}{+\cancel{16}}& \underset{\_}{\phantom{\cancel{16}}+16}\\ \frac{\cancel{-4}y}{\cancel{-4}}& <\frac{24}{-4}\\ y& >-6\end{array}$$

  
  ::- (y+16)- 3y < 8-y- 16- 3y < 8分配属性-4y - 16 < 8combine like items +16_ 16+16_ 4- 4 < 24- 4y_ 6

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  Let's test the boundary point,  $\begin{array}{r}y=-6\end{array}$  .

  $$\begin{array}{rl}-(-6+16)-3(-6)& =8\\ -10+18& =8\\ 8& =8\end{array}$$

  
  ::让我们测试边界点, y6 - (-6+16) - 3 (-6)=8 - 10+18=88=8

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  This statement is true. Let's also test a number in the solution, say $\begin{array}{r}y=0\end{array}$ .
  ::这个语句是真实的。让我们在解答中测试一个数字, 例如 y=0 。

  $$\begin{array}{rl}-(0+16)-3(0)& <8\\ -16+0& <8\\ -16& <8\end{array}$$

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  The  solution set is $\begin{array}{r}y>-6\end{array}$ or $\begin{array}{r}(-6,\mathrm{\infty })\end{array}$ .
  ::解决方案集为 y6 或 (- 6 ) 。

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  Graphically we have:
  ::在图形方面,我们有:

  

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  Summary
  ::摘要

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  • Solve multi-step inequalities just as you would multi-step equations. The one exception is when you multiply or divide by a negative number, you need to switch the direction of the inequality.
    ::解决多步不平等就像解决多步方程一样。 唯一的例外是当您乘以负数或除以负数时, 您需要改变不平等的方向 。

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  Review
  ::回顾

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  Solve and graph the following inequalities.
  ::解决以下不平等问题并绘制图表。

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  1. $\begin{array}{r}5x+16+2x\le -19\end{array}$  
     ::5x+16+2x19
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  2. $\begin{array}{r}15x-23>6x-17\end{array}$  
     ::15-23>6x-17
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  3. $\begin{array}{r}4.2p-3.15>17.36-p\end{array}$  
     ::4.2p-3.15>17.36-p
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  4. $\begin{array}{r}2(u-5)\ge 16\end{array}$  
     ::2(u-55)16
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  5. $\begin{array}{r}-4(3s+7)<20\end{array}$  
     ::- 4(3s+7) <20
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  6. $\begin{array}{r}\frac{5}{2}-5q>4(3-q)\end{array}$  
     ::52-5q>4(3-q)
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  7. $\begin{array}{r}4(2v-1)\ge 3(2v+1)\end{array}$  
     ::4 (2v- 1) 3 (2v+1)
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  8. $\begin{array}{r}11r-\frac{17}{3}-2r\le -(r-\frac{23}{3})\end{array}$  
     ::11-173-2r(r-233)
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  9. $\begin{array}{r}-7(2d-5)+12>-4d-13\end{array}$  
     ::- 7(2d-5)+124d-13
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  10. $\begin{array}{r}-(x-1)+10<-3(x-3)\end{array}$  
      ::- (x-1-1)+103(x-3)

   

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  Explore More
  ::探索更多

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  1. On spring break, you are going to volunteer and paint houses. Each plastic container holds 5 gallons of paint. Your group gets two donations of 10 containers and 5 containers of paint. You will purchase the remaining containers of paint needed to paint 21 houses. For each house you will need 8 gallons of paint . What is the minimum number of containers you need to buy?
  ::1. 春间休息时,你们将自愿参观和油漆房屋,每个塑料容器装有5加仑油漆,你们集团得到2个10个集装箱和5个油漆容器的捐赠,你们将购买21座房屋油漆所需的其余油漆容器,每座房屋需要8加仑油漆。你们需要购买的集装箱最少数量是多少?

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  2. There are 3 sections of algebra. There are twice as many students enrolled in section B as in section A. The number of students enrolled in section C is 16 more than the number of students in section A. The number of students enrolled in section C is greater than or equal to the number of students enrolled in section A plus B. If there are 24 students in section C, how many students could be in section A?
  ::2. 代数分为三个部分:B节的注册学生人数是A节的两倍。 C节的注册学生人数比A节的学生人数多16人。 C节的注册学生人数多于或等于A节加B节的注册学生人数。 如果C节有24名学生,那么A节有多少学生?

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  3. The art department is planning a trip to a museum. The bus costs $100 plus $7 per student. A professor donated $50 to defray the costs. The school is planning on charging students $10 each. How many students need to go on  the trip to not lose money?
  ::3. 艺术系正计划参观博物馆,每名学生公车费用100美元加7美元,一名教授捐赠50美元以支付费用,学校计划向每名学生收费10美元,需要多少学生去旅行才不会亏损?

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  4. You are selling flowers for a fundraiser. You pay $2 for each flower plus $10 for delivery. You are selling the flowers for $4. How many do you need to sell to make a profit?
  ::4. 你卖花是为了募捐,每朵花要付2美元,送10美元,卖花要4美元,要卖多少才能赚钱?

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  5. Does the side of the inequality matter? Try:
  ::5. 不平等的一方重要吗?

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  • Solve $\begin{array}{r}5x+4\le -2(x+3)\end{array}$ by adding the $\begin{array}{r}2x\end{array}$ term on the right to the left-hand side.
    ::将右侧的2x字词添加到左侧,从而解决5x+42(x+3) 2(x+3) 。
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  • Solve $\begin{array}{r}5x+4\le -2(x+3)\end{array}$ by subtracting the $\begin{array}{r}5x\end{array}$ term on the left to the right-hand side.
    ::将左侧的5x条件减到右侧,从而解决5x+42(x+3) 2(x+3) 。

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  Compare your answers. What do you notice?
  ::比较你的答案,你注意到什么了?

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  6. Solve $\begin{array}{r}3x-7>3(x+3)\end{array}$ . What happens? Why?
  ::6. 解决 3x-7>3(x+3) 3(x+3) 。 发生什么了? 为什么?

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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: