Course: 3.10 线性不平等体系解决方案核对

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检查线性不平等体系的解决方案
Three times Aubrey's age minus twice Dakar's age is less than or equal to 25. Four times Aubrey's age plus three times Dakar's age is greater than or equal to 90. Which could be their ages?
::Aubrey的3岁减去达喀尔的2岁,低于或等于25岁,Aubrey的4倍加达喀尔的3倍,大于或等于90岁,他们哪个年龄?

A. Aubrey is 10 and Dakar is 15.
::A. Aubrey为10岁,达喀尔为15岁。

B. Aubrey is 16 and Dakar is 8.
::Bubrey为16岁,达喀尔为8岁。

C. Aubrey is 17 and Dakar is 13.
::C. Aubrey为17岁,达喀尔为13岁。

Solutions to a System of Linear Inequalities
::解决线性不平等制度的解决办法

A linear system of inequalities has an infinite number of solutions. Recall that when graphing a linear inequality the solution is a shaded region of the graph which contains all the possible solutions to the inequality. In a system, there are two linear inequalities. The solution to the system is all the points that satisfy both inequalities or the region in which the shading overlaps.
::线性不平等体系有无限数量的解决方案。回顾当绘制线性不平等图时,解决方案是图中的阴影区域,包含所有可能的不平等解决方案。在一个体系中,存在两种线性不平等。对体系的解决方案是满足不平等的所有要点或阴影重叠的区域。

Given the system of linear inequalities shown in the graph, let's determine which points are solutions to the system.
::鉴于图中显示的线性不平等制度,让我们确定哪些点是这个制度的解决办法。

(0, -1)

The point (0, -1) is not a solution to the system of linear inequalities. It is a solution to $\begin{align*}y \le \frac{2}{3}x+3\end{align*}$ (graphed in $\begin{align*}{\color{blue}\mathbf{blue}}\end{align*}$ ), but it lies on the line $\begin{align*}y=-\frac{4}{5}x-1\end{align*}$ which is not included in the solution to $\begin{align*}y>-\frac{4}{5}x-1\end{align*}$ (shown in $\begin{align*}{\color{red}\mathbf{red}}\end{align*}$ ). The point must satisfy both inequalities to be a solution to the system.
::点( 0, - 1) 不是线性不平等体系的解决方案, 是 y 23x+3 ( 以蓝色标出) 的解决方案, 但它在 y 45x - 1 的 y 445x - 1 线上, 不包括在 y 45x - 1 ( 以红色标出) 的解决方案中。这一点必须满足两种不平等, 才能解决这个体系。

(2, 3)

The point (2, 3) lies in the overlapping shaded region and therefore is a solution to the system.
::点(2,3)在于重叠的阴影区域,因此是系统的解决办法。

(-2, -1)

The point (-2, -1) lies outside the overlapping shaded region and therefore is not a solution the system.
::点(2-2-1)位于重叠的阴影区域之外,因此不是系统的解决办法。

(3, 5)

The point (3, 5) lies on the line $\begin{align*}y=\frac{2}{3}x+3\end{align*}$ , which is included in the solution to $\begin{align*}y \le \frac{2}{3}x+3\end{align*}$ . Since this part of the line is included in the solution to $\begin{align*}y>-\frac{4}{5}x-1\end{align*}$ , it is a solution to the system.
::点(3,5)位于y=23x+3号线上,该线已包括在y23x+3的解决方案中。由于该线的这一部分已包括在y45x-1号的解决方案中,因此它是系统的一个解决方案。

Let's determine whether the following points are solutions to the system of linear inequalities:
::让我们确定以下各点是否是线性不平等制度的解决办法:

$\begin{aligned} 3 x + 2 y & \geq 4 \\ x + 5 y & < 11 \end{aligned}$
$\begin{align*}3x+2y & \ge 4\\ x+5y &< 11\end{align*}$
::3x+2y4x+5y < 11

This time we do not have a graph with which to work. Instead, we will plug the points into the equations to determine whether or not they satisfy the linear inequalities. A point must satisfy both linear inequalities to be a solution to the system.
::这一次,我们没有可以工作的图表。相反,我们将在方程式中插入点数,以确定它们是否满足线性不平等。一个点必须满足线性不平等,才能解决这一体系。

(3, 1)

Yes, $\begin{align*}3(3)+2(1) \ge 4 \end{align*}$ and $\begin{align*}(3)+5(1)<11\end{align*}$ are both valid inequalities. Therefore, (3, 1) is a solution to the system.
::是,3(3)+2(1)(4)和(3)+5(1)<11都是有效的不平等,因此,(3,1)是解决该制度的办法。

(1, 2)

No, $\begin{align*}3(1)+2(2) \ge 4 \end{align*}$ is true, but $\begin{align*}(1)+5(2)=11\end{align*}$ , so the point fails the second inequality.
::否, 3(1)+2(2)4是真实的, 但是 (1)+5(2)=11, 所以点不能消除第二个不平等。

(5, 2)

No, $\begin{align*}3(5)+2(2) \ge 4 \end{align*}$ is true, but $\begin{align*}(5)+5(2)>11\end{align*}$ , so the point fails the second inequality.
::否, 3(5)+2(2)4是真实的, 但(5)+5(2)>11, 所以点忽略了第二个不平等。

(-3, 1)

No, $\begin{align*}3(-3)+2(1)=-9+2=-7<4\end{align*}$ , so the point fails the first inequality. There is no need to check the point in the second inequality since it must satisfy both to be a solution.
::否, (3(-3) +2(1) 9+27 < 4) , 因此点不能达到第一个不平等点。没有必要检查第二个不平等点, 因为它必须满足这两个点才能解决问题。

Now, let's answer the following question.
::现在,让我们回答下面的问题。

Is (-9, 0) a solution to the system below?
:9,0)是以下系统的解决办法吗?

$\begin{aligned} y & > 3 x - 11 \\ x + 2 y \leq 4 \end{aligned}$
$\begin{align*}y &> 3x -11 \\ x + 2y \le 4\end{align*}$
::y>3x- 11x+2y4

Substitute the point into each equation and see if the inequalities hold true.
::在每个方程式中以点替代点,看看不平等是否真实。

$\begin{aligned} 0 & > 3 (- 9) - 11 \\ 0 & > 38 \end{aligned}$
$\begin{align*} 0 &> 3(-9) -11 \\ 0 &> 38 \end{align*}$

The first inequality is true, let's test the second.
::第一个不平等是真实的,让我们测试第二个。

$\begin{aligned} - 9 + 2 (0) \leq 4 \\ - 9 \leq 4 \end{aligned}$
$\begin{align*}-9 + 2(0) \le 4 \\ -9 \le 4 \end{align*}$

This inequality is also true, therefore (-9, 0) is a solution.
::这种不平等也是真实的,因此(9,0)是一个解决办法。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find Aubrey's and Dakar's ages.
::早些时候,你被要求找到奥布里和达喀尔的年纪

We don't have a graph to work with so we must plug the ages into the system of inequalities to determine whether or not they satisfy both. The system of linear inequalities represented by this situation is:
::我们没有图表可以用来工作,因此我们必须将年龄纳入不平等体系,以确定它们是否满足这两个条件。

$\begin{aligned} 3 A - 2 D \leq 25 \\ 4 A + 3 D \geq 90 \end{aligned}$
$\begin{align*}3A - 2D \le 25\\ 4A + 3D \ge 90\end{align*}$
::3A-2D254A+3D90

Now we test each of the possibilities.
::现在,我们测试每一种可能性。

For A = 10 and D = 15: $\begin{align*}3(10) - 2(15) \le 25\end{align*}$ is fulfilled. $\begin{align*}4(10) + 3(15) \ge 90\end{align*}$ is NOT fulfilled.
::对于 A = 10 和 D = 15: 3(10)-2(15)-25 满足。 4(10)+3(15)-90 不满足。

For B = 16 and D = 8: $\begin{align*}3(16) - 2(8) \le 25\end{align*}$ is NOT fulfilled and we can stop here.
::对于B=16和D=8:3(16)-2(8)-25没有满足,我们可以停在这里。

For B = 17 and D = 13: $\begin{align*}3(17) - 2(13) \le 25\end{align*}$ is fulfilled. $\begin{align*}4(17) + 3(13) \ge 90\end{align*}$ is fulfilled.
::B=17,D=13:3(17)-2(13)-25已满足。 4(17)+3(13)-90已满足。

Therefore, of the answer choices given only C could be Aubrey and Dakar's ages.
::因此,在只给C的答案选择中,只有Aubrey和Dakar的年纪。

Example 2
::例2

Determine whether the given points are solutions to the systems shown in the graph:
::确定给定点是否是图中显示的系统的解决办法:

(-3, 3)

(-3, 3) is a solution to the system because it lies in the overlapping shaded region.
:3,3)是该系统的一个解决办法,因为它位于重叠的阴影区域。

(4, 2)

(4, 2) is not a solution to the system. It is a solution to the red inequality only.
:4,2)不是解决这个制度的办法,它只是解决红色不平等问题的办法。

(3, 2)

(3, 2) is not a solution to the system because it lies on the dashed blue line and therefore does not satisfy that inequality.
::该系统(3,2)不是解决这一系统的办法,因为它依靠破碎的蓝线,因此不能消除这种不平等。

(-4, 4)

(-4, 4) is a solution to the system since it lies on the solid red line that borders the overlapping shaded region.
:4-4)是该系统的一个解决办法,因为它位于与重叠的阴影区域相邻的坚固的红线上。

Example 3
::例3

Determine whether the following points are solutions to the system:
::确定以下各点是否是系统的解决办法:

$\begin{aligned} y & < 11 x - 5 \\ 7 x - 4 y & \geq 1 \end{aligned}$
$\begin{align*}y &< 11x-5\\ 7x-4y & \ge 1\end{align*}$
::y < 11x- 57x- 4y1

(4, 0)

Yes, $\begin{align*}0<11(4)-5 \end{align*}$ , and $\begin{align*}7(4)-4(0) \ge 1 \end{align*}$ .
::是, 0<11(4)-5, 和 7(4)-4(0)1。

(0, -5)

No, $\begin{align*}-5=11(0)-5\end{align*}$ so the first inequality is not satisfied.
::否,-5=11(0)-5,因此第一种不平等不能得到满足。

(7, 12)

Yes, $\begin{align*}12<11(7)-5 \end{align*}$ , and $\begin{align*}7(7)-4(12) \ge 1\end{align*}$ .
::是,12<11(7)-5和7(7)-4(12)1。

(-1, -3)

No, $\begin{align*}-3>11(-1)-5\end{align*}$ so the first inequality is not satisfied.
::否,-3>11(-1)-5,因此第一种不平等不能得到满足。

Review
::回顾

Given the four linear systems graphed below, match the point with the system(s) for which it is a solution.
::鉴于以下四条线性系统图示,该点与作为解决办法的系统相匹配。

A.

::A. A.

B.

::B. B. .

C.

::C. 中国家

D.

::D. 专计 D.

(0, 3)

(2, -1)

(4, 3)

(-2, 1)

(-3, 0)

(4, -1)

(-1, -2)

(2, 1)

(2, 5)

(0, 0)

Given the four linear systems below, match the point with the system(s) for which it is a solution.
::鉴于以下四个线性系统,将点与作为解决办法的系统相匹配。

A.
$\begin{aligned} 5 x + 2 y & \leq 10 \\ 3 x - 4 y & > - 12 \end{aligned}$
$\begin{align*}5x+2y & \le 10\\ 3x-4y &> -12\end{align*}$
::A. A. 5x+2+2y103x-4y12

B.
$\begin{aligned} 5 x + 2 y & < 10 \\ 3 x - 4 y & \leq - 12 \end{aligned}$
$\begin{align*}5x+2y &< 10\\ 3x-4y & \le -12\end{align*}$
::B. B. 5x+2y < 103x-4y12

C.
$\begin{aligned} 5 x + 2 y & > 10 \\ 3 x - 4 y & < - 12 \end{aligned}$
$\begin{align*}5x+2y &> 10\\ 3x-4y &< -12\end{align*}$
::C. 5x+2y>103x-4y12

D.
$\begin{aligned} 5 x + 2 y & \geq 10 \\ 3 x - 4 y & \geq - 12 \end{aligned}$
$\begin{align*}5x+2y & \ge 10\\ 3x-4y & \ge -12\end{align*}$
::D. 5x+2y%103x-4y12

(0, 0)

(4, 6)

(0, 5)

(-3, 4)

(4, 3)

(0, 3)

(-8, -3)

(1, 6)

(4, -5)

(4, -2)

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

检查线性不平等体系的解决方案

Solutions to a System of Linear Inequalities ::解决线性不平等制度的解决办法

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Solutions to a System of Linear Inequalities
::解决线性不平等制度的解决办法

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)