Course: 6.11 一个变量中的不平等图

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一个变量中的不平等图
Graphs of Inequalities in One Variable
::一个变量中的不平等图

A linear inequality in two variables takes the form $\begin{aligned} y > m x + b \end{aligned}$ or $\begin{aligned} y < m x + b \end{aligned}$ . Linear inequalities are closely related to graphs of straight lines; recall that a straight line has the equation $\begin{aligned} y = m x + b \end{aligned}$ .
::两个变量的线性不平等以 y>mx+b 或 y <mx+b 的形式出现。线性不平等与直线的图表密切相关; 提醒注意, 直线有 y=mx+b 等式 y=mx+b 。

When we graph a line in the coordinate plane , we can see that it divides the plane in half:
::当我们在坐标平面上绘制一条线时,我们可以看到它将平面除以一半:

The solution to a linear inequality includes all the points in one half of the plane. We can tell which half by looking at the inequality sign:
::线性不平等的解决方案包括一半平面上的所有点。通过观察不平等标志,我们可以分辨出哪一半:

> The solution set is the half plane above the line.
::> 设定的解决方案是线上半平面。

$\begin{aligned} \geq \end{aligned}$ The solution set is the half plane above the line and also all the points on the line.
::确定的解决办法是线上半平面和线上所有点。

< The solution set is the half plane below the line.
::< 设定的解决方案是线下半平面。

$\begin{aligned} \leq \end{aligned}$ The solution set is the half plane below the line and also all the points on the line.
::确定的解决办法是线下半平面和线上所有点。

For a strict inequality, we draw a dashed line to show that the points in the line are not part of the solution. For an inequality that includes the equals sign, we draw a solid line to show that the points on the line are part of the solution.
::对于严格的不平等,我们划出一条破折的线以表明线中各点不是解决方案的一部分。对于包含平等标志的不平等,我们划出一条坚实的线以表明线上各点是解决方案的一部分。

Solution Sets
::解决方案集

This is a graph of $\begin{aligned} y \geq m x + b \end{aligned}$ ; the solution set is the line and the half plane above the line.
::这是 ymx+b 的图形; 解决方案设置为线和线上半平面。

This is a graph of $\begin{aligned} y < m x + b \end{aligned}$ ; the solution set is the half plane above the line, not including the line itself.
::这是 y < mx+b 的图形; 解决方案是线上半平面, 不包括线本身。

Graph Linear Inequalities in One Variable in the Coordinate Plane
::坐标平面中一个变量的线性不平等

In the last few sections we graphed inequalities in one variable on the number line. We can also graph inequalities in one variable on the coordinate plane. We just need to remember that when we graph an equation of the type $\begin{aligned} x = a \end{aligned}$ we get a vertical line, and when we graph an equation of the type $\begin{aligned} y = b \end{aligned}$ we get a horizontal line.
::在最后几节中,我们用数字线上的一个变量绘制不平等的图解。我们也可以用坐标平面上的一个变量绘制不平等的图解。我们只需要记住,当我们用X=a型方程式绘制时,我们得到一条垂直线,当我们用 y=b 型方程式绘制时,我们得到一条水平线。

Graphing Inequalities
::不平等图表

1. Graph the inequality $\begin{aligned} x > 4 \end{aligned}$ on the coordinate plane.
::1. 在坐标平面上绘制不平等 x>4图。

First let’s remember what the solution to $\begin{aligned} x > 4 \end{aligned}$ looks like on the number line.
::首先让我们记住 x>4 的解决方案在数字线上是什么样子。

The solution to this inequality is the set of all real numbers $\begin{aligned} x \end{aligned}$ that are bigger than 4, not including 4. The solution is represented by a line.
::解决这种不平等的办法,是所有实际数字x大于4的一组数字x,不包括4。解决办法由一条线代表。

In two dimensions , the solution still consists of all the points to the right of $\begin{aligned} x = 4 \end{aligned}$ , but for all possible $\begin{aligned} y - \end{aligned}$ values as well. This solution is represented by the half plane to the right of $\begin{aligned} x = 4 \end{aligned}$ . (You can think of it as being like the solution graphed on the number line, only stretched out vertically .)
::在两个维度中, 解决方案仍然包含 x=4 右侧的所有点, 但对于所有可能的 y- 值也包含。此解决方案由 x=4 右侧的半平面表示。 ( 您可以把它看作是数字线上绘制的解决方案, 仅垂直延伸。 )

The line $\begin{aligned} x = 4 \end{aligned}$ is dashed because the equals sign is not included in the inequality, meaning that points on the line are not included in the solution.
::x=4 行被冲破,因为等号不包含在不平等中,这意味着线上的点没有包含在解决方案中。

2. Graph the inequality $\begin{aligned} | y | < 5 \end{aligned}$
::2. 分析不平等状况 5

The absolute value inequality $\begin{aligned} | y | < 5 \end{aligned}$ can be re-written as $\begin{aligned} - 5 < y < 5 \end{aligned}$ . This is a compound inequality which can be expressed as
::绝对值不平等 y 5 可以改写为 - 5 < y < 5 。这是一种复杂的不平等,可以表述为

$\begin{aligned} y > - 5 and y < 5 \end{aligned}$

::y 5andy < 5

In other words, the solution is all the coordinate points for which the value of $\begin{aligned} y \end{aligned}$ is larger than -5 and smaller than 5. The solution is represented by the plane between the horizontal lines $\begin{aligned} y = - 5 \end{aligned}$ and $\begin{aligned} y = 5 \end{aligned}$ .
::换句话说,解决方案就是所有坐标点,其y值大于-5,小于-5。解决方案由水平线y5和y=5之间的平面代表。

Both horizontal lines are dashed because points on the lines are not included in the solution.
::这两条水平线均被冲破,因为两条水平线上的点没有包括在解决方案中。

Example
::示例示例示例示例

Example 1
::例1

Graph the inequality $\begin{aligned} | x | \geq 2 \end{aligned}$ .
::不平等图 x2。

The absolute value inequality $\begin{aligned} | x | \geq 2 \end{aligned}$ can be re-written as a compound inequality:
::绝对价值不平等 x2 可以重写为复合不平等:

$\begin{aligned} x \leq - 2 or x \geq 2 \end{aligned}$

::x2orx%2

In other words, the solution is all the coordinate points for which the value of $\begin{aligned} x \end{aligned}$ is smaller than or equal to -2 or greater than or equal to 2. The solution is represented by the plane to the left of the vertical line $\begin{aligned} x = - 2 \end{aligned}$ and the plane to the right of line $\begin{aligned} x = 2 \end{aligned}$ .
::换句话说,解决方案是所有坐标点,其x值小于或等于-2或大于或等于2。解决方案由垂直线x2左侧的平面和X=2线右侧的平面代表。

Both vertical lines are solid because points on the lines are included in the solution.
::两条垂直线都是固态的,因为两条线上的点包括在溶液中。

Review
::回顾

Graph the following inequalities on the coordinate plane.
::图1 坐标平面上以下的不平等。

$\begin{aligned} x < 20 \end{aligned}$
::x <20

$\begin{aligned} y \geq - 5 \end{aligned}$
::y 5

$\begin{aligned} x > 0.5 \end{aligned}$
::x>0.5

$\begin{aligned} x \leq \frac{1}{2} \end{aligned}$
::x12

$\begin{aligned} y > - \frac{2}{3} \end{aligned}$
::y y 23

$\begin{aligned} y < - 0.2 \end{aligned}$
::y0.2

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

$\begin{aligned} | y | \leq 7 \end{aligned}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

$\begin{aligned} | y | < \frac{1}{3} \end{aligned}$
::y13

$\begin{aligned} | x | \geq - 10 \end{aligned}$
::

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。

Texas Instruments Resources
::得克萨斯州工具资源

In the CK-12 Texas Instruments Algebra I FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See .
::在CK-12得克萨斯州仪器代数I FlexBook资源中,有图表计算活动,旨在补充本章某些经验教训的目标。

Section outline

General

一个变量中的不平等图

Graphs of Inequalities in One Variable ::一个变量中的不平等图

Solution Sets ::解决方案集

Graph Linear Inequalities in One Variable in the Coordinate Plane ::坐标平面中一个变量的线性不平等

Graphing Inequalities ::不平等图表

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Texas Instruments Resources ::得克萨斯州工具资源

Graphs of Inequalities in One Variable
::一个变量中的不平等图

Solution Sets
::解决方案集

Graph Linear Inequalities in One Variable in the Coordinate Plane
::坐标平面中一个变量的线性不平等

Graphing Inequalities
::不平等图表

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)

Texas Instruments Resources
::得克萨斯州工具资源