课程： 2.11 两个变量中的不平等图示

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以两个变量绘制不平等图
Xpress Taxi Service charges $1.50 per minute traveled minus $0.25 per minute spent in stopped traffic. You only have $10 in your wallet, so that is the maximum amount you can spend on your ride. In which quadrant(s) would the graph represented by this situation fall?
::Xpress 出租车服务公司每分钟收费1.50美元,从每分钟0.25美元到每分钟0.25美元。您钱包里只有10美元,所以这是您在旅途上可以花费的最大金额。以这种情况为代表的图表会跌落到哪个区块?

Graphing Inequalities
::不平等图表

Graphing linear inequalities is very similar to graphing lines. First, you need to change the inequality into slope-intercept form . At this point, we will have a couple of differences. If the inequality is in the form $\begin{aligned} y < m x + b \end{aligned}$ or $\begin{aligned} y > m x + b \end{aligned}$ , the line will be dotted or dashed because it is not a part of the solution. If the line is in the form $\begin{aligned} y \leq m x + b \end{aligned}$ or $\begin{aligned} y \geq m x + b \end{aligned}$ , the line will be solid to indicate that it is included in the solution.
::绘制线性不平等图与图形线非常相似。首先, 您需要将不平等改变为斜坡界面。此时, 我们将会有一些差异。如果不平等以 y < mx+b 或 y> mx+b 的形式出现, 则该线将因它不是解决方案的一部分而被点化或破折。如果该线系以 y=mx+b 或 y=mx+b 的形式出现, 则该线条将非常可靠, 以表明它是否包含在解决方案中。

The second difference is the shading. Because these are inequalities, not just the line is the solution. Depending on the sign, there will be shading above or below the line. If the inequality is in the form $\begin{aligned} y < m x + b \end{aligned}$ or $\begin{aligned} y \leq m x + b \end{aligned}$ , the shading will be below the line, in reference to the $\begin{aligned} y - \end{aligned}$ axis.
::第二个差别是阴影。由于这些是不平等,不只是线就是解决办法。根据符号,在线上或线下会有阴影。如果不平等的形式为 y ' mx+b 或 y'mx+b,那么阴影将在线下,参照 y - 轴值。

If the inequality is in the form $\begin{aligned} y > m x + b \end{aligned}$ or $\begin{aligned} y \geq m x + b \end{aligned}$ , the shading will be above the line.
::如果不平等以 y>mx+b 或 y'mx+b 的形式存在,则阴影将位于线条之上。

Let's graph the following inequalities.
::让我们用图表来说明以下的不平等。

$\begin{aligned} 4 x - 2 y < 10 \end{aligned}$
::4 - 2y < 10

First, change the inequality into slope-intercept form. Remember, that if you have to divide or multiply by a negative number, you must flip the inequality sign.
::首先,将不平等改变为斜坡拦截形式。记住, 如果您需要除以负数或乘以负数, 您必须翻转不平等符号。

Now, graph the inequality as if it was a line. Plot $\begin{aligned} y = 2 x - 5 \end{aligned}$ . However, the line will be dashed because of the “greater than” sign.
::现在, 请将不平等状况描述为一行。绘制 y= 2x-5 。但是, 这条线会因为“ 大于” 符号而破折。

Now, we need to determine the shading. You can use one of two methods to do this. The first way is to use the graphs and forms from above. The equation , in slope-intercept form, matches up with the purple dashed line and shading. Therefore , we should shade above the dashed blue line.
::现在,我们需要确定阴影。您可以使用两种方法中的一个方法来做到这一点。第一个方法是使用上面的图表和表单。方程式, 以斜坡界面的形式, 与紫色破折线和阴影相匹配。因此, 我们应该在破折的蓝线上方阴影。

The alternate method would be to test a couple of points to see if they work. If a point is true, then the shading is over that side of the line. If we pick (-5, 0), the inequality yields $\begin{aligned} - 20 < 10 \end{aligned}$ , which tells us that our shading is correct.
::另一种方法是测试几个点以检验它们是否有效。如果一个点是真实的, 那么阴影就会在线的另一边。如果我们选择( 5, 0) , 不平等就会产生 - 20 < 10, 这告诉我们我们的阴影是正确的。

$\begin{aligned} y \leq - \frac{2}{3} x + 6 \end{aligned}$
::y 23x+6

This inequality is already in slope-intercept form. So, graph the line, which will be solid, and then determine the shading. Looking at the example graphs above, this inequality should look like the red inequality, so shade below the line.
::这种不平等已经以斜坡界面的形式出现。因此, 请绘制线条图, 线条会坚固, 然后决定阴影。看一下上面的示例图, 这种不平等应该看起来像红色的不平等, 因此在线下阴影。

Test a point to make sure our shading is correct. An easy point in the shaded region is (0, 0). Plugging this into the inequality, we get $\begin{aligned} 0 \leq 6 \end{aligned}$ , which is true.
::测试点可以确保我们的阴影是正确的。阴影区域的一个简单点是( 0, 0 ) 。将它融入不平等, 我们得到的就是0, 6, 这是正确的。

Let's determine the linear inequality that is graphed below.
::我们来确定线性不平等,如下图所示。

Find the equation of the line portion just like you've done before. The given points on the line are (0, 8) and (6, 2) (from the points drawn on the graph). This means that the $\begin{aligned} y - \end{aligned}$ intercept is (0, 8). Then, using triangles we fall 6 and run 6 to get to (6, 2). This means the slope is $\begin{aligned} \frac{- 6}{6} \end{aligned}$ or -1. Because we have a dotted line and the shading is above, our sign will be the > sign. Putting it all together, the equation of our linear inequality is $\begin{aligned} y > - x + 8 \end{aligned}$ .
::查找线段部分的方程式, 就象您以前所做的一样。线段上的给定点是 (0, 8) 和 (6, 2) (来自图中绘制的点) 。这意味着 y - 截取是 (0, 8) 。这意味着 y - 截取是 (0, 8) 。然后, 使用三角形, 我们掉6 , 跑6 来到达 (6, 2) 。这意味着斜坡是 ~ 66 或 - 1 。因为我们有一个点线条, 阴影在上方, 我们的标志就是 > 符号。把它们加在一起, 我们线性不平等的方程式是 y x+ 8 。

$\begin{aligned} ^{*} \end{aligned}$ When finding the equation of an inequality, like above, it is easiest to find the equation in slope-intercept form. To determine which inequality sign to use, look at the shading along the $\begin{aligned} y - \end{aligned}$ axis. If the shaded $\begin{aligned} y - \end{aligned}$ values get larger, the line will be in the form or $\begin{aligned} y > m x + b \end{aligned}$ or $\begin{aligned} y \geq m x + b \end{aligned}$ . If they get smaller, the line will be in the form $\begin{aligned} y < m x + b \end{aligned}$ or $\begin{aligned} y \leq m x + b \end{aligned}$ .
::∗ 如上所示, 当找到不平等的方程式时, 最容易找到斜度界面中的方程式。要确定要使用哪种不平等符号, 请查看 y - 轴周围的阴影。如果阴暗 y - 值变大, 该线将以 y ' mx+b 或 y ' mx+b 或 y ' mx+b 的形式出现。如果线条变小, 则以 y < mx+b 或 y ' x+b 的形式出现。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find which quadrant(s) into which the graph represented by the situation would fall.
::早些时候,有人要求你找到以情况表示的图表会跌落到哪个象限。

To solve this taxi cab problem, we must first set up an inequality to represent the situation.
::为了解决出租车问题,我们必须首先建立不平等,以代表情况。

$\begin{aligned} 1.5 x - 0.25 y \leq 10 \end{aligned}$
::1.5x-0.25y10

You can't travel a negative number of miles or sit in traffic for a negative number of minutes. Therefore both x and y must have zero or positive values. When both x and y are positive, the graph occurs in the first quadrant only. Graph the function to check this answer.
::您不能旅行负数的英里数, 也不能在流量中停留负数的分钟数。因此, x 和 y 的数值必须为零或正值。如果 x 和 y 均为正数, 则图只出现在第一个象限中。绘制功能以检查此答案。

Example 2
::例2

Graph $\begin{aligned} 3 x - 4 y > 20 \end{aligned}$ .
::图3x-4y>20。

First, change the inequality into slope-intercept form.
::首先,将不平等改变为斜坡干扰形式。

$\begin{aligned} 3 x - 4 y & > 20 \\ - 4 y & > - 3 x + 20 \\ y & < \frac{3}{4} x - 5 \end{aligned}$

::3x-4y>20-4y3x+20y<34x-5

Now, we need to determine the type of line and shading. Because the sign is “<,” the line will be dashed and we will shade below.
::现在,我们需要确定线条和阴影的类型。因为标志是“ < ” , 线条会被冲破, 而我们会在下面阴影。

Test a point in the shaded region to make sure we are correct. If we test (6, -6) in the original inequality, we get $\begin{aligned} 42 > 20 \end{aligned}$ , which is true.
::在阴影区域测试一个点以确保我们是正确的。如果我们在原始不平等中测试( 6, - 6 ) , 我们就会得到42> 20, 这是正确的。

Example 3
::例3

Graph $\begin{aligned} x < - 1 \end{aligned}$ .
::图 x% 1 。

To graph this line on the $\begin{aligned} x - y \end{aligned}$ plane, recall that all vertical lines have the form $\begin{aligned} x = a \end{aligned}$ . Therefore, we will have a vertical dashed line at -1. Then, the shading will be to the left of the dashed line because that is where $\begin{aligned} x \end{aligned}$ will be less than the value of the line.
::要在 x - y 平面上绘制这条线图, 请记住所有垂直线都有表单 x=a。因此, 我们将会在 - 1 处设置一条垂直破折线。然后, 阴影将位于破折线的左侧, 因为这里的 x 将低于线值。

Example 4
::例4

What is the equation of the linear inequality?
::线性不平等的等式是什么?

Looking at the line, the $\begin{aligned} y - \end{aligned}$ intercept is (0, 1). Using a slope triangle, to count down to the next point, we would fall 4, and run backward 1. This means that the slope is $\begin{aligned} \frac{- 4}{- 1} = 4 \end{aligned}$ . The line is solid and the shading is above, so we will use the $\begin{aligned} \geq \end{aligned}$ sign. Our inequality is $\begin{aligned} y \geq 4 x + 1 \end{aligned}$ .
::从线上看, y- interview is (0, 1) y- interview 是 (0, 1) 。使用一个斜度三角形, 向下倒数到下一点, 我们会跌到 4 , 然后向后退 1 。这意味着斜度是 4 - 1= 4 。线是固的, 阴影是上面的, 所以我们将使用 + 符号。我们的不平等是 y 4x+ 1 。

Review
::回顾

Graph the following inequalities.
::如下图所示的不平等情况。

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

$\begin{aligned} 3 x - 2 y \geq 4 \end{aligned}$
::3x-2-2y4

$\begin{aligned} y < - 3 x + 8 \end{aligned}$
::y3x+8 y3x+8

$\begin{aligned} x + 4 y \leq 16 \end{aligned}$
::x+4y16

$\begin{aligned} y < - 2 \end{aligned}$
::y2

$\begin{aligned} y < - \frac{1}{2} x - 3 \end{aligned}$
::y12x-3

$\begin{aligned} x \geq 6 \end{aligned}$
::x_ 6

$\begin{aligned} 8 x + 4 y \geq - 20 \end{aligned}$
::8+4y20

$\begin{aligned} - 4 x + y \leq 7 \end{aligned}$
::- 4x+y=7 - 4x+y=7

$\begin{aligned} 5 x - 3 y \geq - 24 \end{aligned}$
::5-3y24

$\begin{aligned} y > 5 x \end{aligned}$
::y>5x y>5x

$\begin{aligned} y \leq 0 \end{aligned}$
::y0

Determine the equation of each linear inequality below.
::确定以下每一线性不平等的方程式。

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

常规

以两个变量绘制不平等图

Graphing Inequalities ::不平等图表

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Graphing Inequalities
::不平等图表

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)