Course: 7.13 数字线上的不平等

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数字行的不平等
A scientific center has been conducting research about bird populations that are being harmed by human development. Scientists are carefully gathering information on shrinking habitats and gathering data about the biology of the birds affected. A group of student scientists went to visit the center and learned about the beak length of various birds. They discovered that the beak length was always 7 inches or less. Can you represent the range of possible beak lengths on a number line?
::一个科学中心一直在研究正在受到人类发展伤害的鸟类种群。科学家们正在仔细收集关于不断缩小的栖息地的信息,并收集关于受影响鸟类生物学的数据。一群学生科学家到中心了解了各种鸟类的嘴部长度。他们发现,嘴部长度总是7英寸或更少。您能否在数字线上代表可能达到的嘴部长度范围 ?

In this concept, you will learn to graph inequalities on a number line.
::在此概念中,您将学会用数字线来描述不平等。

Graphing Inequalities on a Number Line
::数字行的不平等图

An inequality is a mathematical statement that uses one of the following symbols: $\begin{aligned} >, <, \geq, \leq, \end{aligned}$ instead of an equals sign.
::不平等是一种数学说明,它使用以下符号之一: >, <, <,_,_,_,_,_,_,不使用等号。

Here is what those symbols mean:
::这些符号的含义如下:

$\begin{aligned} > \end{aligned}$ greater than
::> 大于

< less than
::<小于

$\begin{aligned} \geq \end{aligned}$ greater than or equal to
::大于或等于

$\begin{aligned} \leq \end{aligned}$ less than or equal to
::少于或等于

Let’s look at an example to see how to interpret these symbols.
::让我们举个例子看看如何解释这些符号。

If you have the statement $\begin{aligned} x > 2 \end{aligned}$ , what does it mean?
::如果您有声明 x>2, 这意味着什么 ?

The inequality $\begin{aligned} x > 2 \end{aligned}$ has a variable , $\begin{aligned} x, \end{aligned}$ that represents an infinite set of numbers. Although $\begin{aligned} x \end{aligned}$ cannot be equal to 2, it can be any real number greater than 2. It is impossible to write out every single number that $\begin{aligned} x \end{aligned}$ can be. For this reason, you will see that a number line is a useful tool to represent the solution. For now, let’s write out some of the numbers that are in the solution of $\begin{aligned} x > 2. \end{aligned}$
::不平等 x> 2 有一个变量 x, 代表无限数组。虽然 x 不能等于 2 。虽然 x 不能等于 2 , 但任何实际数字都可能大于 2 。因此, 您将看到数字行是代表解决方案的有用工具。现在, 让我们写出在 x> 2 解决方案中的一些数字。

To make this statement true, $\begin{aligned} x \end{aligned}$ can be any of the following numbers.
::要使这一说法成为事实,x可以是下列数字中的任何数字。

$\begin{aligned} {2.01, \frac{19}{9}, 2.988, 3, 4, 4.0000001, \frac{20}{3}, 1000, 1001.11111 \dots} \end{aligned}$

You can start to see the range of numbers that make the statement $\begin{aligned} x > 2 \end{aligned}$ true. Though it is helpful to see some of the numbers in the solution for $\begin{aligned} x > 2, \end{aligned}$ it is best to represent the solution on a number line.
::您可以开始看到使语句 x>2 成为真实的数字范围。虽然在 x>2 的解决方案中看到某些数字很有帮助, 但最好在数字行上代表解决方案。

Let’s look at an example where you graph an inequality.
::让我们来举一个例子来说明不平等。

Graph the solution set for the inequality $\begin{aligned} x > 3 \end{aligned}$ on a number line.
::在数字行为不平等 x>3 设定的解决方案图。

To help complete this task, first draw a number line from -5 to 5, marking off ticks at integer intervals.
::为帮助完成此任务, 首先从 - 5 至 5 绘制一个数字行, 按整数间隔标记。

The inequality $\begin{aligned} x > 3 \end{aligned}$ is read as “ $\begin{aligned} x \end{aligned}$ is greater than 3.” So the solution of this inequality includes all numbers greater than 3. It does not , however, actually include 3. To show that the answer does not include 3, you use an open circle.
::不平等 x>3 被解读为“x大于3”。因此,这种不平等的解决办法包括超过3的所有数字。但是,它实际上不包括3。要显示答案不包括3, 您使用一个开放圆圈。

Next, draw an arrow showing all numbers greater than 3. The arrow should point towards the right because greater numbers are to the right on a number line.
::接下来,绘制一个箭头,显示所有数字大于3。箭头应该指向右边, 因为数字线上的右边数字更多。

The answer is
::答案是

Here are some tips for graphing inequalities on a number line.
::以下是用数字线绘制不平等图的一些提示。

Use an open circle to show that a value is not a solution for the inequality. You will use open circles to graph inequalities that include the symbols $\begin{aligned} > or < . \end{aligned}$
::使用开放圆来显示值不是不平等的解决方案。您将使用打开圆来显示包含符号 > 或 < 的不平等。

Use a closed circle to show that a value is a solution for the inequality. You will use closed circles to graph inequalities that include the symbols $\begin{aligned} \geq or \leq . \end{aligned}$
::使用封闭圆来显示一个值是不平等的解决方案。您将使用封闭圆来显示包含或符号的不平等。

Here is another example.
::下面是另一个例子。

Graph the solution for $\begin{aligned} x < - 1 \end{aligned}$ on a number line.
::在数字行上绘制 x<-1 的解决方案。

First, draw a number line from -5 to 5.
::首先,将数字线从 - 5 划为 5 。

The inequality $\begin{aligned} x < - 1 \end{aligned}$ is read as “ $\begin{aligned} x \end{aligned}$ is less than -1.” The solutions of this inequality include all numbers less than -1 , but not -1 itself, so draw an open circle at -1 to show that it is not a solution for this inequality. Then draw an arrow showing all numbers less than -1. The arrow should point towards the left because smaller numbers are to the left on a number line.
::不平等 x < 1 被解读为“ x 低于 - 1 ” 。这种不平等的解决方案包括所有少于 - 1 的数字, 但不包括 - 1 本身, 所以在 - 1 上绘制一个开放的圆, 以显示它不是解决这种不平等的办法。然后绘制一个箭头, 显示所有数字少于 - 1 。箭头应该指向左边, 因为数字线左侧的数字较小。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the students going to the scientific center to study birds and their shrinking environment.
::早些时候,你被问及学生们去科学中心研究鸟类及其萎缩的环境的问题。

These students are collecting data on beak length and found that all the birds they looked at had beaks 7 inches or less. They need to represent the range of beak lengths on a number line.
::这些学生正在收集嘴长度的数据,发现他们所看到的鸟儿都有7英寸或更少的嘴。他们需要在数字线上代表嘴长度的范围。

Since there were beak lengths that were 7 inches you use a solid circle on 7 inches. They also found beaks less than 7 inches. So you draw an arrow to the left.
::既然有7英寸长的嘴线, 你就可以用7英寸长的圆圆。它们也发现了小于7英寸的嘴线。所以, 你向左画一支箭。

One possible solution is the following.
::一种可能的解决办法如下。

This is a fine answer, but can you improve it?
::这是一个很好的答复,但你能改进它吗?

A beak length cannot be less than zero, so a better solution would be a closed circle at 7, with a line going to the left, but then stopping at an open circle at 0. This would show that beak lengths were also always greater than 0.
::圆嘴长度不能小于零,所以更好的解决办法是7时的封闭圆圈,其直线向左,然后在0时的开放圆圈停止。这将显示,圆嘴长度也总是大于0。

Example 2
::例2

Graph the solution of $\begin{aligned} x \geq 0 \end{aligned}$ on a number line.
::数字线上的 x0 的图解。

First, draw a number line from -5 to 5.
::首先,将数字线从 - 5 划为 5 。

The inequality $\begin{aligned} x \geq 0 \end{aligned}$ is read as “ $\begin{aligned} x \end{aligned}$ is greater than or equal to 0.” So, the answer includes zero and all numbers that are greater than 0.
::不平等 x_0 被解读为“ x 大于或等于 0 ” 。因此,答案包括零和大于 0 的所有数字。

Draw a closed circle at 0 to show that 0 is in the solution for this inequality. Then draw an arrow pointing to the right, showing all numbers greater than 0 are also part of the solution set.
::0 绘制一个封闭圆以显示 0 是解决此不平等的办法。然后绘制指向右边的箭头, 显示所有大于 0 的数字也是设定的解决方案的一部分。

The answer is the graph below which shows the solution for the inequality $\begin{aligned} x \geq 0 \end{aligned}$ .
::答案是下图显示不平等的解决方案 x% 0 。

Example 3
::例3

True or false: An open circle on a number line means that the circled number is not included in the solution set.
::真实的或假的:数字行上的开放圆是指圆形数没有包括在所设定的解决方案中。

Open circles are a way to indicate that the circled number is not part of the solution.
::打开圆圈可以表明圆形数不是解决方案的一部分。

The answer is true.
::答案是真实的。

Example 4
::例4

True or false: An inequality does not include the number referenced in the solution set.
::真实的或虚假的:不平等不包括成套解决方案中引用的数字。

Inequalities include greater than or equal to $\begin{aligned} (\geq) \end{aligned}$ and less than or equal to $\begin{aligned} (\leq) \end{aligned}$ . When these symbols are used the number referenced is included in the solution set.
::不平等包括大于或等于(____)和小于或等于(____)的不平等。当使用这些符号时,所引用的数字包括在成套解决办法中。

The answer is false.
::答案是虚假的。

Example 5
::例5

True or false: A closed circle on a graph means that the number is included in the solution set.
::真实的或假的:图形上的封闭圆是指数字包含在成套解决方案中。

The answer is true.
::答案是真实的。

Review
::回顾

Graph the solution for each inequality on the given number line.
::在给定数字行上绘制每种不平等的解决方案。

$\begin{aligned} x < - 3 \end{aligned}$
::x% 3

$\begin{aligned} x > - 5 \end{aligned}$
::x% 5

$\begin{aligned} n \leq 2 \end{aligned}$
::n2

$\begin{aligned} 1 \leq n \end{aligned}$
::1n

$\begin{aligned} x > 6 \end{aligned}$
::x>6 x6

$\begin{aligned} n < - 5 \end{aligned}$
::n 5 个

$\begin{aligned} n \leq - 6 \end{aligned}$
::n6

$\begin{aligned} x \geq - 9 \end{aligned}$
::x% 9

$\begin{aligned} x \leq - 2 \end{aligned}$
::x% 2

$\begin{aligned} x > - 5 \end{aligned}$
::x% 5

$\begin{aligned} x \leq - 8 \end{aligned}$
::x% 8

$\begin{aligned} x > - 10 \end{aligned}$
::x 10

$\begin{aligned} x < - 6 \end{aligned}$
::x 6

$\begin{aligned} x > - 7 \end{aligned}$
::x% 7

$\begin{aligned} x \leq - 8 \end{aligned}$
::x% 8

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。

Section outline

General

数字行的不平等

Graphing Inequalities on a Number Line ::数字行的不平等图

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Graphing Inequalities on a Number Line
::数字行的不平等图

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)