7.10 线性不平等制度

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  线性不平等制度

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  Systems of Linear Inequalities 
  ::线性不平等制度

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  T o graph a linear inequality in two variables , you graph  the equation of the straight line on the coordinate plane . The line is  solid for $\begin{align*}\le\end{align*}$ or $\begin{align*}\ge\end{align*}$ signs (where the equals sign is included), and the line is  dashed for < or > signs (where the equals sign is not included). Then , shade above the line (if the inequality begins  with $\begin{align*}y > \end{align*}$ or $\begin{align*}y \ge\end{align*}$ ) or below the line (if it begins  with $\begin{align*}y < \end{align*}$ or $\begin{align*}y \le\end{align*}$ ).
  ::要在两个变量中绘制线性不平等图,请绘制坐标平面上直线线的方程式。直线对于 {或 {( 包含等号的) 符号是固态的,而线条对于 < 或 > 标志是破折的( 不包括等号的) 。然后,线条上方的阴影( 如果不平等以 y> 或 y}开始) 或线下方的阴影( 如果以 y < 或 y}开始) 。

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  Here you will  see how to graph two or more linear inequalities on the same coordinate plane. The inequalities are graphed separately on the same graph, and the solution for the system is the common shaded region between all the inequalities in the system. One linear inequality in two variables divides the plane into two half-planes . A system of two or more linear inequalities can divide the plane into more complex shapes.
  ::这里您将看到如何在同一坐标平面上绘制两个或更多线性不平等图。 不平等图是在同一平面上单独绘制的, 而这个系统的解决方案是系统中所有不平等之间的共同阴影区域。 两个变量中的一条线性不平等将平面分为两个半平面。 两个或更多线性不平等的系统可以将平面分割成更复杂的形状。

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  Let’s start by solving a system of two inequalities.
  ::让我们首先解决两种不平等的制度。

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  Graph a System of Two Linear Inequalities
  ::a 双线不平等体系图

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  Solve the following system:
  ::解决以下系统:

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  $$\begin{align*}\begin{cases}2x + 3y \le 18\\ x - 4y \le 12\end{cases}\end{align*}$$
  ::{2x+3y18x-4y12

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  Solving systems of linear inequalities means graphing and finding the intersections. So we graph each inequality, and then find the intersection regions of the solution.
  ::线性不平等的解决系统意味着图形化和找到交叉点。所以我们用图表来绘制每个不平等点,然后找到解决方案的交叉区域。

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  First, let’s rewrite each equation in slope-intercept form . (Remember that this form makes it easier to tell which region of the coordinate plane to shade.) Our system becomes
  ::首先,让我们以斜坡界面的形式重写每一个方程式。 (记住,这个方程式更容易辨别坐标平面的哪个区域。 )我们的系统变成一个系统。

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  $$\begin{align*}3y \le -2x + 18 \quad & \Rightarrow \quad y \le - \frac{2}{3}x + 6\\ \\ -4y \le -x + 12 \quad & \Rightarrow \quad y \ge \frac{x}{4} - 3\end{align*}$$
  ::32x+182x23x+6-444x+124x4-3

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  Notice that the inequality sign in the second equation changed because we divided by a negative number!
  ::注意第二个方程式中的不平等标志已经改变 因为我们除以负数!

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  For this first example, we’ll graph each inequality separately and then combine the results.
  ::首先,我们将分别列出每个不平等情况,然后将结果合并在一起。

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  Here’s the graph of the first inequality:
  ::以下是第一个不平等的图示:

  

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  The line is solid because the equals sign is included in the inequality. Since the inequality is less than or equal to, we shade below the line.
  ::这条线是牢固的,因为平等标志包含在不平等中。 由于不平等程度小于或等于,我们向线下阴影。

  

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  And here’s the graph of the second inequality:
  ::以下是第二个不平等的图示:

  

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  The line is solid again because the equals sign is included in the inequality. We now shade above the line because $\begin{align*}y\end{align*}$ is greater than or equal to.
  ::线线再次坚固, 因为等号包含在不平等中 。 我们现在在线上阴影, 因为 y 大于或等于 。

  

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  When we combine the graphs, we see that the blue and red shaded regions overlap. The area where they overlap is the area where both inequalities are true. Thus that area (shown below in purple) is the solution of the system.
  ::当我们合并图表时,我们看到蓝色和红色阴影区域重叠。它们重叠的区域是两种不平等都真实存在的区域。因此,这个区域(以紫色显示在下面)是系统的解决方案。

  

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  The kind of solution displayed in this example is called unbounded , because it continues forever in at least one direction (in this case, forever upward and to the left).
  ::这个例子所展示的解决方案被称为无约束的, 因为它至少会永远持续一个方向(在此情况下, 永远向上和向左 ) 。

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  Systems with No Solution 
  ::无解决方案的系统

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  There are also situations where a system of inequalities has no solution. For example, let’s solve this system.
  ::不平等制度也存在无法解决问题的情况。 比如,让我们解决这个制度。

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  $$\begin{align*}\begin{cases}y \le 2x - 4\\ y > 2x + 6\end{cases}\end{align*}$$
  ::{y2x- 4y>2x+6

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  We start by graphing the first line. The line will be solid because the equals sign is included in the inequality. We must shade downwards because $\begin{align*}y\end{align*}$ is less than .
  ::我们从绘制第一行图开始。 这条线将坚固, 因为等号包含在不平等中。 我们必须向下阴影, 因为 y 小于 。

  

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  Next we graph the second line on the same coordinate axis. This line will be dashed because the equals sign is not included in the inequality. We must shade upward because $\begin{align*}y\end{align*}$ is greater than .
  ::接下来,我们将在同一坐标轴上绘制第二行的图。这条线会被冲破,因为等号没有包括在不平等中。我们必须向上阴影,因为 y 大于 。

  

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  It doesn’t look like the two shaded regions overlap at all. The two lines have the same , so we know they are parallel; that means that the regions indeed won’t ever overlap since the lines won’t ever cross. So this system of inequalities has no solution.
  ::似乎这两个阴暗区域根本不是重叠的。 这两条线是相同的,因此我们知道它们是平行的;这意味着这两个区域的确不会重叠,因为两条线永远不会交叉。 因此,这种不平等体系是无法解决的。

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  But a system of inequalities can sometimes have a solution even if the lines are parallel. For example, what happens if we swap the directions of the inequality signs in the system we just graphed?
  ::但是,即使线条是平行的,不平等体系有时也会有解决办法。 比如,如果我们在刚刚绘制的体系中交换不平等迹象的方向,结果会如何?

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  G raph the system
  ::系统图图

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  $$\begin{align*}\begin{cases}y \ge 2x - 4\!\\ y < 2x + 6\end{cases}\end{align*}$$
  ::{y% 2x- 4y < 2x+6

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  D raw the same lines we drew for the previous system, but we shade upward for the first inequality and downward for the second inequality. Here is the result:
  ::绘制我们为前一个系统绘制的相同线条,但我们向上偏向第一个不平等线,向下偏向第二个不平等线。结果如下:

  

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  You can see that this time the shaded regions overlap. The area between the two lines is the solution to the system.
  ::您可以看到, 这次阴影区域重叠。 两行之间的区域是系统解决方案 。

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  Graph a System of More Than Two Linear Inequalities
  ::a 大于两个线性不平等的系统图

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  When we solve a system of just two linear inequalities, the solution is always an unbounded region (one that continues infinitely in at least one direction). I f we put together a system of more than two inequalities, sometimes we can get a solution that is bounded - a finite region with three or more sides.
  ::当我们解决了两个线性不平等的体系时,解决方案总是一个没有界限的区域(一个在至少一个方向上无限延续下去 ) 。 如果我们把一个超过两个不平等的体系结合在一起,有时我们可以找到一个有界限的解决方案 — — 一个有三面或更多面的有限区域。

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  Let’s look at a simple example.
  ::让我们看看一个简单的例子。

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  Find the solution to the following system of inequalities.
  ::找到解决以下不平等制度的办法。

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  $$\begin{align*}\begin{cases}3x - y < 4\\ 4y + 9x < 8\\ x \ge 0\\ y \ge 0\end{cases}\end{align*}$$
  ::{3x-y <44y+9x <8x_0y_0}0

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  Let’s start by writing our inequalities in slope-intercept form.
  ::让我们首先写下我们在斜坡接触形式的不平等。

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  $$\begin{align*}y & > 3x - 4\\ y & < - \frac{9}{4}x + 2\\ x & \ge 0\\ y & \ge 0\end{align*}$$
  ::y>3x- 4y\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Now we can graph each line and shade appropriately. First we graph $\begin{align*}y > 3x - 4\end{align*}$ :
  ::现在我们可以适当绘制每一行和阴影的图表。我们首先绘制 y>3x- 4 :

  

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  Next we graph $\begin{align*}y < - \frac{9}{4}x + 2\end{align*}$ :
  ::下图 y94x+2 :

  

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  Finally we graph $\begin{align*}x \ge 0\end{align*}$ and $\begin{align*}y \ge 0\end{align*}$ , and we’re left with the region below; this is where all four inequalities overlap.
  ::最后,我们用图表x0和y°0,我们只剩下以下区域;这是所有四种不平等重叠的地方。

  

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  The solution is bounded because there are lines on all sides of the solution region. In other words, the solution region is a bounded geometric figure, in this case a triangle.
  ::解决方案的界限在于解决方案区域所有两边都有线条,换句话说,解决方案区域是一个界限的几何数字,在这种情况下是一个三角。

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  Notice, too, that only three of the lines we graphed actually form the boundaries of the region. Sometimes when we graph multiple inequalities, it turns out that some of them don’t affect the overall solution; in this case, the solution would be the same even if we’d left out the inequality $\begin{align*}y > 3x - 4\end{align*}$ . That’s because the solution region of the system formed by the other three inequalities is completely contained within the solution region of that fourth inequality; in other words, any solution to the other three inequalities is automatically a solution to that one too, so adding that inequality doesn’t narrow down the solution set at all.
  ::也注意到我们绘制的线条中只有三条实际上构成了该地区的边界。 有时,当我们绘制多重不平等图时,结果发现其中一些并不影响整体解决方案;在这种情况下,即使我们忽略了不平等 y>3x-4,解决方案也是一样的。 这是因为由其他三种不平等构成的体系的解决方案区域完全包含在第四种不平等的解决方案区域中;换句话说,其他三种不平等的任何解决方案也自动成为这一解决方案的解决方案,因此,不平等不会缩小设定的解决方案范围。

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  T hat wasn’t obvious until we actually drew the graph!
  ::直到我们真的绘制了图表,

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  Example 
  ::示例示例示例示例

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  Example 1
  ::例1

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  Write the system of inequalities shown below.
  ::将不平等制度写成如下:

  

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  There are two boundary lines, so there are two inequalities. Write each one in slope-intercept form.
  ::有两条边界线, 所以有两条不平等。 以斜坡界面写一个 。

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  $$\begin{align*}y & \le \frac{1}{4} x + 7\\ y & \ge -\frac{5}{2} x - 5\end{align*}$$
  ::y}14x+7y\\\\\\\\\\\\\\\\\\\\\\\\\5\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Review 
  ::回顾

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  1. Consider the system $\begin{align*}\begin{cases}y < 3x - 5\!\\ y > 3x - 5\end{cases}\end{align*}$ Is it consistent or inconsistent? Why?
     ::考虑系统 {y<3x- 5y>3x- 5} 是否一致或不一致? 为什么?
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  2. Consider the system $\begin{align*}\begin{cases}y \le 2x + 3\!\\ y \ge 2x + 3\end{cases}\end{align*}$ Is it consistent or inconsistent? Why?
     ::考虑一下系统 {y% 2x+3y% 2x+3 是否一致或不一致? 为什么?
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  3. Consider the system $\begin{align*}\begin{cases}y \le -x + 1\!\\ y > -x + 1\end{cases}\end{align*}$ Is it consistent or inconsistent? Why?
     ::考虑系统 {yx+1yx+1} x+1 是否一致或不一致? 为什么?
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  4. In example 3 in this lesson, we solved a system of four inequalities and saw that one of the inequalities, $\begin{align*}y > 3x - 4\end{align*}$ , didn’t affect the solution set of the system.
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     1. What would happen if we changed that inequality to $\begin{align*}y < 3x - 4\end{align*}$ ?
        ::如果我们把不平等变成你们3x4会怎么样?
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     2. What’s another inequality that we could add to the original system without changing it? Show how by sketching a graph of that inequality along with the rest of the system.
        ::我们可以在不改变原制度的情况下给原制度增加另一个不平等又是什么? 显示如何通过绘制不平等与制度其他部分的图表来说明不平等。
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     3. What’s another inequality that we could add to the original system to make it inconsistent? Show how by sketching a graph of that inequality along with the rest of the system.
        ::另一种不平等又是什么,我们可以在原有体系中增加哪些不平等,使它变得不一致? 显示如何绘制不平等与体系其他部分的图表。
     
     ::在举例3中,我们解决了四种不平等的体系,发现其中一种不平等,即y>3x-4,并没有影响这一体系的解决方案。如果我们将不平等变为y<3x-4,那会怎样?我们可以在不改变原体系的情况下增加另一种不平等,而不会改变它吗?显示如何通过绘制不平等与体系其他部分的图表来显示另一种不平等?我们可以在原体系中增加另一种不平等,使其不一致?展示如何通过绘制不平等与体系其他部分的图表来显示不平等。
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  5. Recall the compound inequalities in one variable that we worked with back in chapter 6. Compound inequalities with “and” are simply systems like the ones we are working with here, except with one variable instead of two.
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     1. Graph the inequality $\begin{align*}x > 3\end{align*}$ in two dimensions. What’s another inequality that could be combined with it to make an inconsistent system?
        ::将不平等 x> 3 分为两个维度来图解。 另一种不平等又可以与它结合到一起,从而形成一种不一致的制度吗?
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     2. Graph the inequality $\begin{align*}x \le 4\end{align*}$ on a number line. What two-dimensional system would have a graph that looks just like this one?
        ::在数字行上图解不平等 x4。什么二维系统会有一个看上去和这个相似的图表?
     
     ::回顾我们在第6章中合作过的一个变量中的复合不平等。 “和”的复合不平等与我们在这里合作过的系统一样,只是简单的系统,只有一个变量而不是两个变量除外。用两个维度来分析不平等 x>3 。另一个不平等可以与它结合起来,形成一个不一致的系统吗?用数字线来分析不平等 x4 。哪个二维系统可以有一个和这个相似的图表?

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  Find the solution region of the following systems of inequalities.
  ::找到以下不平等制度的解决办法区域。

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  6. $\begin{align*}\begin{cases}x - y < -6\!\\ 2y \ge 3x + 17\end{cases}\end{align*}$
     ::{x-y 62y3x+17}
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  7. $\begin{align*}\ \\ \begin{cases}4y - 5x < 8\!\\ -5x \ge 16 - 8y\end{cases}\\ \ \\\end{align*}$
     ::{4y- 5x < 8- 5x_ 16- 8y
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  8. $\begin{align*}\begin{cases}5x - y \ge 5\!\\ 2y - x \ge -10\end{cases}\end{align*}$
     ::{5x-y52y-x10}
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  9. $\begin{align*}\ \\ \begin{cases}5x + 2y \ge -25\!\\ 3x - 2y \le 17\!\\ x - 6y \ge 27 \end{cases} \\ \ \\\end{align*}$
     ::{5x+2y} 253x-2y17x-6y}27
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  10. $\begin{align*}\begin{cases}2x - 3y \le 21\!\\ x + 4y \le 6\!\\ 3x + y \ge -4\end{cases}\end{align*}$
      ::{2x--3y21x+4y}63x+y4
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  11. $\begin{align*}\ \\ \begin{cases}12x - 7y < 120\!\\ 7x - 8y \ge 36\!\\ 5x + y \ge 12\end{cases}\\ \ \\\end{align*}$
      ::{12x-7y < 1207x-8y_365x+y12

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  Review (Answers)
  ::回顾(答复)

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  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 。

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

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  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。