节: 应用不平等的应用程序 | 6.4 不平等申请

章节大纲

Suppose you solved an inequality and came up with the solution $\begin{aligned} x > - 3 \end{aligned}$ ? How else could you express this solution?
::假设你解决了不平等问题并提出了解决方案 x3?

Inequalities
::不平等

Ms. Jerome wants to buy identical boxes of art supplies for her 25 students. If she can spend no more than $375 on art supplies, what inequality describes the price can she afford for each individual box of supplies?
::Jerome女士想为25名学生购买一箱相同的艺术用品,如果她花在艺术用品上的费用不超过375美元,那么,她能为每盒用品支付多少价格?

Expressing Solutions of an Inequality
::表达不平等的解决办法

The solution of an inequality can be expressed in four different ways:
::解决不平等的办法可以用四种不同的方式表达:

Inequality notation The answer is simply expressed as $\begin{aligned} x < 15 \end{aligned}$ .
::答案仅以 x<15 表示。

Set notation The answer is expressed as a set: $\begin{aligned} {x | x < 15} \end{aligned}$ . The brackets indicate a set and the vertical line means “such that,” so we read this expression as “the set of all values of $\begin{aligned} x \end{aligned}$ such that $\begin{aligned} x \end{aligned}$ is a real number less than 15”.
::答案以一组表示 {xx < 15} 。括号表示一组,而垂直线则表示“这样,”所以我们将这一表达式解读为“所有 x 值的一组,使 x 实际数字小于 15 ” 。

Interval notation uses brackets to indicate the range of values in the solution. For example, the answer to our problem would be expressed as $\begin{aligned} (- \infty, 15) \end{aligned}$ , meaning “the interval containing all the numbers from $\begin{aligned} - \infty \end{aligned}$ to 15 but not actually including $\begin{aligned} - \infty \end{aligned}$ or 15”.

Square or closed brackets “[” and “]” indicate that the number next to the bracket is included in the solution set .
::方括号或封闭括号“[”和“]”表示括号旁边的数字包括在一套解决办法中。

Round or open brackets “(” and “)” indicate that the number next to the bracket is not included in the solution set. When using infinity and negative infinity ( $\begin{aligned} \infty \end{aligned}$ and $\begin{aligned} - \infty \end{aligned}$ ), we always use open brackets, because infinity isn’t an actual number and so it can’t ever really be included in an interval.
::圆括号或括号( " 和 " ) " 表示括号旁边的数字没有包含在解决方案中。当使用无限性和负无限性(和)时,我们总是使用开放括号,因为无限性不是一个实际数字,因此它不可能真正包含在一个间隔内。

::中间括号使用括号来表示解决方案中的数值范围。例如,我们问题的答案将以(, 15)表示,意思是 " 包含%至15之间所有数字的间隔,但实际上不包括%或15 " 。方括号或封闭括号 " " " 和 " . " 表示括号旁边的数字包含在成套解决方案中。圆括号或开括号 " 和 " . " 表示括号旁边的数字没有包含在设定的解决方案中。在使用无限性和负无限性(和)时,我们总是使用开放括号,因为无限性不是一个实际数字,因此它不可能真正包含在间隔中。

Solution graph shows the solution on the real number line. A closed circle on a number indicates that the number is included in the solution set, while an open circle indicates that the number is not included in the set. For our example, the solution graph is:
::解决方案图显示实际数字行的解决方案。一个数字上的封闭圆表示数字包含在解决方案集中,而一个开放圆表示数字不包括在数据集中。例如,解决方案图是:

[-4, 6] means that the solution is all numbers between -4 and 6 including -4 and 6.
::[4、6] 意指解决办法在-4和6之间,包括-4和6之间的所有数字。

(8, 24) means that the solution is all numbers between 8 and 24 not including the numbers 8 and 24.
:8, 24) 意指解决方案是8到24之间的所有数字,不包括8和24。

[3, 12) means that the solution is all numbers between 3 and 12, including 3 but not including 12.
::[3, 12] 是指解决办法在3到12之间,包括3个,但不包括12个。

$\begin{aligned} (- 10, \infty) \end{aligned}$ means that the solution is all numbers greater than -10, not including -10.
:- 10, ) 意指解决方案的数量均大于 - 10, 不包括 - 10 。

$\begin{aligned} (- \infty, \infty) \end{aligned}$ means that the solution is all real numbers.
:,) 意思是解决方案都是真实数字。

Identify the Number of Solutions of an Inequality
::确定不平等解决方案的数量

Inequalities can have:
::不平等可以有:

A set that has an infinite number of solutions.
::一组有无限数的解决方案。

A set that has a discrete number of solutions.
::一组有不同数种解决方案的集。

No solutions.
::无解决办法。

The inequalities we have solved so far all have an infinite number of solutions, at least in theory. For example, the inequality $\begin{aligned} \frac{5 x - 1}{4} > - 2 (x + 5) \end{aligned}$ has the solution $\begin{aligned} x > - 3 \end{aligned}$ . This solution says that all real numbers greater than -3 make this inequality true, and there are infinitely many such numbers.
::到目前为止,我们解决的不平等问题都有无限数量的解决办法,至少在理论上是如此。比如,5x-14+2(x+5)的不平等问题就有了解决方案 x+*3。这个解决方案指出,所有实际数字都大于 - 3, 使这种不平等成为现实,而且此类数字也无穷无尽。

However, in real life, sometimes we are trying to solve a problem that can only have positive integer answers, because the answers describe numbers of discrete objects.
::然而,在现实生活中,有时我们试图解决一个只能有正整数答案的问题,因为答案描述了离散物体的数量。

For example, suppose you are trying to figure out how many $8 CDs you can buy if you want to spend less than $50. An inequality to describe this situation would be $\begin{aligned} 8 x < 50 \end{aligned}$ , and if you solved that inequality you would get $\begin{aligned} x < \frac{50}{8} \end{aligned}$ , or $\begin{aligned} x < 6.25 \end{aligned}$ .
::例如,假设您正在试图弄清楚如果想要花费不到50美元,您可以买多少8张CD。描述这种情况的不平等程度为8x<50,如果解决了不平等,您将得到 x<508 或 x<6.25。

But could you really buy any number of CDs as long as it’s less than 6.25? No; you couldn’t really buy 6.1 CDs, or -5 CDs, or any other fractional or negative number of CDs. So if we wanted to express our solution in set notation, we couldn’t express it as the set of all numbers less than 6.25, or $\begin{aligned} {x | x < 6.25} \end{aligned}$ . Instead, the solution is just the set containing all the nonnegative whole numbers less than 6.25, or {0, 1, 2, 3, 4, 5, 6}. When we’re solving a real-world problem dealing with discrete objects like CDs, our solution set will often be a finite set of numbers instead of an infinite interval.
::但是,只要不超过6.25,你真的能买到任何数量的CD吗? 不;你真的不能买到6.1个CD,或者 -5个CD,或者其他任何零分数或负数的CD。因此,如果我们想用设定符号表达我们的解决方案,我们就不能把它表达成所有数字少于6.25或{xx<6.25}的一组。相反,解决方案只是包含所有非负数不到6.25的整数,或者{0,1,2,3,3,4,5,6}的一组。当我们解决有关像CD这样的离散物体的现实世界问题时,我们设定的解决方案往往是一组有限的数字,而不是无限的间隔。

An inequality can also have no solutions at all. For example, consider the inequality $\begin{aligned} x - 5 > x + 6 \end{aligned}$ . When we subtract $\begin{aligned} x \end{aligned}$ from both sides, we end up with $\begin{aligned} - 5 > 6 \end{aligned}$ , which is not true for any value of $\begin{aligned} x \end{aligned}$ . We say that this inequality has no solution.
::不平等也不可能有任何解决办法。例如,考虑不平等 x-5>x+6 。当我们从两边减去 x 时,我们最终会得到 - 5>6 。这对x 的任何价值来说都是不正确的。我们说,这种不平等是没有解决办法的。

The opposite can also be true. If we flip the inequality sign in the above inequality, we get $\begin{aligned} x - 5 < x + 6 \end{aligned}$ , which simplifies to $\begin{aligned} - 5 < 6 \end{aligned}$ . That’s always true no matter what $\begin{aligned} x \end{aligned}$ is, so the solution to that inequality would be all real numbers, or $\begin{aligned} (- \infty, \infty) \end{aligned}$ .
::反之亦然。如果我们翻转上述不平等中的不平等标志,我们就会得到x-5<x+6,它简化为~5<6。这始终是事实,不管x是什么,因此解决不平等的办法都是真实数字,或者(,,)。

Solve Real-World Problems Using Inequalities
::利用不平等解决现实世界问题

Solving real-world problems that involve inequalities is very much like solving problems that involve equations.
::解决现实世界中涉及不平等的问题与解决涉及等式的问题非常相似。

Solve the Real-World Problem
::解决现实世界问题

In order to get a bonus this month, Leon must sell at least 120 newspaper subscriptions. He sold 85 subscriptions in the first three weeks of the month. How many subscriptions must Leon sell in the last week of the month?
::为了获得本月的奖金,Leon必须出售至少120份报纸订阅费。他在这个月的前三个星期出售了85份订阅费。 Leon必须在这个月的最后一个星期出售多少订阅费?

Let $\begin{aligned} x = \end{aligned}$ the number of subscriptions Leon sells in the last week of the month. The total number of subscriptions for the month must be greater than 120, so we write $\begin{aligned} 85 + x \geq 120 \end{aligned}$ . We solve the inequality by subtracting 85 from both sides: $\begin{aligned} x \geq 35 \end{aligned}$ .
::Leon在当月最后一周售出订阅次数的xx=Leon。该月的订阅总数必须超过120个,所以我们写85+xx120。我们通过从双方减去85个,解决不平等问题:x}35。

Leon must sell 35 or more subscriptions in the last week to get his bonus.
::里昂上周要卖35个或更多订户才能拿到奖金

To check the answer, we see that $\begin{aligned} 85 + 35 = 120 \end{aligned}$ . If he sells 35 or more subscriptions, the total number of subscriptions he sells that month will be 120 or more. The answer checks out .
::检查答案,我们可以看到85+35=120。如果他出售35个或更多的订阅,那么他这个月销售的订阅总数将是120个或以上。答案可以查清楚。

Solve the Real-World Problem
::解决现实世界问题

Virena’s Scout troop is trying to raise at least $650 this spring. How many boxes of cookies must they sell at $4.50 per box in order to reach their goal?
::维雷纳的童子军今年春天试图募集至少650美元。为了达到目标,他们必须卖出多少盒饼干,每盒4.50美元?

Let $\begin{aligned} x = \end{aligned}$ number of boxes sold. Then the inequality describing this problem is $\begin{aligned} 4.50 x \geq 650 \end{aligned}$ .
::Let x= 售出箱数。然后描述这一问题的不平等度是 4. 50x650 。

We solve the inequality by dividing both sides by 4.50: $\begin{aligned} x \geq 144.44 \end{aligned}$ .
::我们通过将双方除以4.50:x144.44来解决不平等问题。

We round up the answer to 145 since only whole boxes can be sold.
::我们找回145的答案因为只有整个箱子可以卖掉

Virena’s troop must sell at least 145 boxes.
::Virena的部队必须卖掉至少145箱。

If we multiply 145 by $4.50 we obtain $652.50, so if Virena’s troop sells more than 145 boxes they will raise more than $650. But if they sell 144 boxes, they will only raise $648, which is not enough. So they must indeed sell at least 145 boxes. The answer checks out .
::如果我们把145乘以4.5乘以4.50,我们就能获得65250美元,那么如果维雷纳的部队出售了超过145个箱子,他们就能筹集到650多美元。但如果他们卖掉144个箱子,他们只能筹集到648美元,这是不够的。因此他们必须至少卖出145个箱子。答案可以查证。

Example
::示例示例示例示例

The width of a rectangle is 20 inches. What must the length be if the perimeter is at least 180 inches?
::矩形宽度为 20 英寸。如果周边至少 180 英寸, 长度必须是什么 ?

Let $\begin{aligned} x = \end{aligned}$ length of the rectangle. The formula for perimeter is
::x= 矩形的长度。圆周的公式是

$\begin{aligned} Perimeter = 2 \times length + 2 \times width \end{aligned}$

::半径=2x长+2x宽

Since the perimeter must be at least 180 inches, we have $\begin{aligned} 2 x + 2 (20) \geq 180 \end{aligned}$ .
::由于周界必须至少180英寸,我们有2x+2(20)-180。

Simplify: $\begin{aligned} 2 x + 40 \geq 180 \end{aligned}$
::简化: 2x+40-180

Subtract 40 from both sides: $\begin{aligned} 2 x \geq 140 \end{aligned}$
::双方减40:2x140

Divide both sides by 2: $\begin{aligned} x \geq 70 \end{aligned}$
::将两边除以 2 : x / 70

The length must be at least 70 inches.
::长度必须至少70英寸

If the length is at least 70 inches and the width is 20 inches, then the perimeter is at least $\begin{aligned} 2 (70) + 2 (20) = 180 i n c h e s \end{aligned}$ . The answer checks out .
::如果长度至少为70英寸,宽度为20英寸,那么周界至少为2(70)+2(20)=180英寸。答案会查清楚。

Review
::回顾

Solve each inequality. Give the solution in inequality notation and interval notation.
::解决每一种不平等,提供不平等的标记和间隙标记的解决方案。

$\begin{aligned} x + 15 < 12 \end{aligned}$
::x+15 < 12

$\begin{aligned} x - 4 \geq 13 \end{aligned}$
::x-413

$\begin{aligned} 9 x > - \frac{3}{4} \end{aligned}$
::9x34

$\begin{aligned} - \frac{x}{15} \leq 5 \end{aligned}$
::- 15=5 - 15=5

$\begin{aligned} 620 x > 2400 \end{aligned}$
::620x>2400

$\begin{aligned} \frac{x}{20} \geq - \frac{7}{40} \end{aligned}$
::x20_740

$\begin{aligned} \frac{3 x}{5} > \frac{3}{5} \end{aligned}$
::3x5>35

$\begin{aligned} x + 3 > x - 2 \end{aligned}$
::x+3>x-2

Solve each inequality. Give the solution in inequality notation and set notation.
::解决每一种不平等,给出不平等的标记和设置标记的解决方法。

$\begin{aligned} x + 17 < 3 \end{aligned}$
::x+17 <3

$\begin{aligned} x - 12 \geq 80 \end{aligned}$
::x- 12- 1980

$\begin{aligned} - 0.5 x \leq 7.5 \end{aligned}$
::- 0.5x7.5

$\begin{aligned} 75 x \geq 125 \end{aligned}$
::75x=125

$\begin{aligned} \frac{x}{- 3} > - \frac{10}{9} \end{aligned}$
::x-3109

$\begin{aligned} \frac{x}{- 15} < 8 \end{aligned}$
::x-15 < 8

$\begin{aligned} \frac{x}{4} > \frac{5}{4} \end{aligned}$
::x4>54

$\begin{aligned} 3 x - 7 \geq 3 (x - 7) \end{aligned}$
::3x-73(x-7)

Solve the following inequalities, give the solution in set notation, and show the solution graph.
::解决以下不平等问题,在设定符号时给出解决方案,并显示解决方案图。

$\begin{aligned} 4 x + 3 < - 1 \end{aligned}$
::4x+31

$\begin{aligned} 2 x < 7 x - 36 \end{aligned}$
::2x < 7x-36

$\begin{aligned} 5 x > 8 x + 27 \end{aligned}$
::5x8x+27

$\begin{aligned} 5 - x < 9 + x \end{aligned}$
::5-x<9+x

$\begin{aligned} 4 - 6 x \leq 2 (2 x + 3) \end{aligned}$
::4 - 6x%2( 2x+3)

$\begin{aligned} 5 (4 x + 3) \geq 9 (x - 2) - x \end{aligned}$
::5(4x+3)9(x-2)-x

$\begin{aligned} 2 (2 x - 1) + 3 < 5 (x + 3) - 2 x \end{aligned}$
::2(2x-1)+3 <5(x+3)-2x

$\begin{aligned} 8 x - 5 (4 x + 1) \geq - 1 + 2 (4 x - 3) \end{aligned}$
::8x-5( 4x+1) 1+2( 4x-3)

$\begin{aligned} 2 (7 x - 2) - 3 (x + 2) < 4 x - (3 x + 4) \end{aligned}$
::2(7x-2-2)-3(x+2) <4x-(3x+4)

$\begin{aligned} \frac{2}{3} x - \frac{1}{2} (4 x - 1) \geq x + 2 (x - 3) \end{aligned}$
::23-12(4x-1-1)x+2(2x-3)

At the San Diego Zoo you can either pay $22.75 for the entrance fee or $71 for the yearly pass that entitles you to unlimited admission.

At most, how many times can you enter the zoo for the $22.75 entrance fee before spending more than the cost of a yearly membership?
::最多你有多少次能进入动物园收取22.75美元的入场费才能花比每年会员费还多的钱?

Are there a finite or infinite number of solutions to this inequality?
::对这种不平等是否有有限或无限数量的解决办法?

::在圣地亚哥动物园,您既可以支付22.75美元的入场费,也可以支付71美元的年度通行证,这些通行证使您可以不受限制地进入。最多,您在花费超过年度会员费用之前,可以进入动物园多少次22.75美元的入场费? 对这种不平等,您是否有有限或无限数量的解决方案?

Proteek’s scores for four tests were 82, 95, 86, and 88. What will he need to score on his fifth and last test to average at least 90 for the term?
::Proteek的四次考试得分是82、95、86和88,

Review (Answers)
::回顾(答复)

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。

章节大纲

Inequalities ::不平等

Solve the Real-World Problem ::解决现实世界问题

Solve the Real-World Problem ::解决现实世界问题

Example ::示例示例示例示例

Review ::回顾

Review (Answers) ::回顾(答复)

Inequalities
::不平等

Solve the Real-World Problem
::解决现实世界问题

Solve the Real-World Problem
::解决现实世界问题

Example
::示例示例示例示例

Review
::回顾

Review (Answers)
::回顾(答复)