课程： 2.10 两个变量中线性不平等的测试解决方案

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选择节两个变量中线性不平等的测试解决方案

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两个变量中线性不平等的测试解决方案
A taxi cab charges $2 per mile plus $0.20 per stopped minute in traffic. If your cab bill totals less than $10 but more than $5, which of the following could have occurred during your ride?
::如果你的计程车费总额不到10美元,但超过5美元,那么在您乘车时会发生以下哪一种情况?

A. You travelled 5 miles and sat in traffic for 3 minutes.
::A. 你走了5英里,在交通中坐了3分钟。

B. You travelled 2 miles and sat in traffic for 2 minutes.
::B. 你走了2英里,在交通中坐了2分钟。

C. You travelled 4 miles and sat in traffic for 6 minutes.
::C. 你走了4英里,在交通中坐了6分钟。

Solution for Linear Inequality
::解决线性不平等问题

A linear inequality is very similar to the equation of a line, but with an inequality sign. They can be written in one of the following ways:
::线性不平等与线性的等式非常相似,但带有不平等标志。

$\begin{aligned} A x + B y < C & A x + B y > C & A x + B y \leq C & A x + B y \geq C \end{aligned}$
$\begin{align*}Ax + By < C && Ax + By > C && Ax + By \le C && Ax + By \ge C\end{align*}$
::Ax+By <CAx+By>CAx+ByCAx+ByCAx+ByCAx+ByCAx+ByCAx+ByCAx+ByCAx+ByCAx+ByByC

Notice that these inequalities are very similar to the standard form of a line . We can also write a linear inequality in slope-intercept form .
::请注意,这些不平等与一条线的标准形式非常相似。我们也可以以斜坡间接形式写出线性不平等。

$\begin{aligned} y < m x + b & y > m x + b & y \leq m x + b & y \geq m x + b \end{aligned}$
$\begin{align*}y < mx + b && y > mx + b && y \le mx + b && y \ge mx + b\end{align*}$
::y < mx+bby>mx+by%mx+b > y <mx+b > > y <mx+b > >

In all of these general forms, the $\begin{align*}A, B, C, m\end{align*}$ , and $\begin{align*}b\end{align*}$ represent the exact same thing they did with lines.
::在所有这些一般形式中,A、B、C、m和b代表着它们用线所做的完全相同的事情。

An ordered pair , or point, is a solution to a linear inequality if it makes the inequality true when the values are substituted in for $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$ .
::定购一对或一对点是线性不平等的解决办法,如果它使不平等成为真实的话,如果数值被x和y所取代的话。

Let's determine the solutions to the following inequalities.
::让我们决定解决以下不平等的解决办法。

Which ordered pair is a solution to $\begin{align*}4x - y > -12\end{align*}$ ?
::哪种配对是4x -y12的解决方案?

Plug in each point to see if they make the inequality true.
::在每个点上加插,看它们是否使不平等成为事实。

a) (6, -5)
:a) (6,5)

$\begin{aligned} 4 (6) - (- 5) & > - 12 \\ 24 + 5 & > - 12 \\ 29 & > - 12 \end{aligned}$
$\begin{align*}4(6) - (-5) & >-12\\ 24 + 5 & > -12\\ 29 & > -12\end{align*}$

b) (-3, 0)
:b) (3,0)

$\begin{aligned} 4 (- 3) - 0 & > - 12 \\ - 12 & > - 12 \end{aligned}$
$\begin{align*}4(-3)-0 & >-12\\ -12 & \ \cancel{>} - 12\end{align*}$

c) (-5, 4)
:c) (-5,4)

$\begin{aligned} 4 (- 5) - 4 & > - 12 \\ - 20 - 4 & > - 12 \\ - 24 & > - 12 \end{aligned}$
$\begin{align*}4(-5) - 4 & > -12\\ -20 -4 & > -12\\ -24 & \ \cancel{>} -12\end{align*}$

Of the three points, a) is the only one where the inequality holds true. b) is not true because the inequality sign is only “ greater than ,” not “ greater than or equal to .”
::在三点中,(a)是不平等唯一真实的。 (b) 是不真实的,因为不平等标志只是“大于”,而不是“大于或等于”。

Is the point (-9, 1) a solution for $\begin{align*} y < 5x + 1 \end{align*}$ ?
::点 (- 9, 1) 是 y < 5x+1 的解决方案吗 ?

Substitute in the point values for x and y and see if the inequality holds true.
::代替x和y的点值,看看不平等是否真实。

$\begin{aligned} 1 < 5 \cdot - 9 + 1 \\ 1 < - 45 + 1 \\ 1 < - 44 \end{aligned}$
$\begin{align*}1<5 \cdot -9 +1 \\ 1<-45+1 \\ 1<-44 \end{align*}$

This is false. Therefore , (-9, 1) is not a solution.
::这是虚假的,因此(9,1)不是解决办法。

Determine 3 solutions to the inequality $\begin{align*}2x-7y > -12\end{align*}$ .
::确定3个解决不平等问题的办法 2x-7y12。

Select values for x and y that would make the inequality true. If $\begin{align*}x=2\end{align*}$ and $\begin{align*}y=-2\end{align*}$ , the inequality is true, $\begin{align*}4+14 > -12\end{align*}$ . Another easy point would be the origin. Testing it, we have $\begin{align*}0>-12\end{align*}$ . Lastly, we could select a point where y is zero and the x value is positive. For example, the points (1, 0), (2, 0), (3, 0), etc... would all work. There are infinitely many solutions.
::选择 x 和 y 的值, 使不平等变为真实。如果 x=2 和 y , 不平等是真实的, 4+14\\\ 12。另一个简单点是源。测试它, 我们有 0\\ 12。最后, 我们可以选择一个 y 是零, x 值是正数的点。例如, 点 ( 1, 0, 2, 0, 0, 3, 0) 等都会有效。有无限多的解决方案。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find which scenario could have occurred during your taxi ride.
::早些时候,你被要求寻找在乘出租车途中可能发生的情况。

To solve th is problem, we must first set up an inequality to represent the scenario.
::为了解决这一问题,我们必须首先建立不平等,以代表这种情景。

$\begin{align*} 5 < 2x + 0.2y < 10\end{align*}$ , where x equals the miles traveled and y equals the number of minutes in stopped traffic.
::5<2x+0.2y<10,其中x等于行驶的里程,y等于停止交通的分钟数。

Now let's test each of the possibilities to see if they fit the inequality.
::现在让我们来测试每一种可能性看看它们是否适合不平等。

A: $\begin{align*}2(5) + 0.2(3) = 10 + 0.6 = 10.6 > 10\end{align*}$ so this possibility could not have occurred. B: $\begin{align*}2(2) + 0.2(2) = 4 + 0.4 = 4.4 <5\end{align*}$ so this possibility could not have occurred. C: $\begin{align*}2(4) + 0.2(6) = 8 + 1.2 = 9.2\end{align*}$ ; $\begin{align*}5 < 9.2 < 10\end{align*}$ so this possibility could have occurred.
::A:2(5)+0.2(3)=10+0.6=10.6>10,因此不可能发生这种可能性。B:2(2)+0.2(2)=4+0.4=4.4=5,不可能发生这种可能性。C:2(4)+0.2(6)=8+1.2=9.2;5 < 9.2 < 10,因此可能发生这种可能性。

Example 2
::例2

Which inequality is (-7, 1) a solution for?
::哪种不平等(-7,1)是解决办法?

Plug (-7, 1) in to each equation.
::每个方程式的插件( 7, 1) 。

$\begin{align*}y < 2x - 1\end{align*}$
::y < 2x- 1

$\begin{aligned} 1 & < 2 (- 7) - 1 \\ 1 & < - 15 \end{aligned}$
$\begin{align*}1 &< 2(-7) -1\\ 1 & \ \bcancel{<} - 15\end{align*}$

$\begin{align*}4x -3y \ge 9\end{align*}$
::4x-33y9

$\begin{aligned} 4 (- 7) - 3 (1) & \geq 9 \\ - 28 - 3 & \geq 9 \\ - 31 & \geq 9 \end{aligned}$
$\begin{align*}4(-7) -3(1) &\ge 9\\ -28 -3 &\ge 9\\ -31 & \ \cancel{\ge} \ 9\end{align*}$

$\begin{align*}y > -4\end{align*}$
::y 4

$\begin{aligned} 1 > - 4 \end{aligned}$
$\begin{align*}1 > -4\end{align*}$

(-7, 1) is only a solution to #3, $\begin{align*}y > -4\end{align*}$ .
:7,1)只是解决3,y4的方法。

E xample 3
::例3

List three possible solutions for $\begin{align*}5x - y \le 3\end{align*}$ .
::为 5x-y3 列出三种可能的解决办法。

To find possible solutions, plug in values to the inequality. There are infinitely many solutions. Here are three: (-1, 0), (-4, 3), and (1, 6).
::为了找到可能的解决方案, 将价值加到不平等中。有很多解决方案。这里有三个( 1, 0, (4, 3) 和 (1, 6) 。

$\begin{aligned} 5 (- 1) - 0 & \leq 3 & 5 (- 4) - 3 \leq 3 & 5 (1) - 6 \leq 3 \\ - 5 & \leq 3 & - 23 \leq 3 & - 1 \leq 3 \end{aligned}$
$\begin{align*}5(-1) -0 & \le 3 && 5(-4) -3 \le 3 && 5(1) -6 \le 3\\ -5 & \le 3 && \qquad -23 \le 3 && \quad \ \ -1 \le 3\end{align*}$

Review
::回顾

Using the four inequalities below, determine which point is a solution for each one. There may be more than one correct answer. If the answer is none, write none of these .
::使用下面的四种不平等, 确定哪个点是每个点的解决方案。答案可能不止一个。如果答案是否定的, 请不要写这些。

A) $\begin{align*}y \le \frac{2}{3}x - 5\end{align*}$
:A)y23x-5

B) $\begin{align*}5x +4y > 20\end{align*}$
::B) 5x+4y>20

C) $\begin{align*}x - y \ge -5\end{align*}$
::C) x-y5

D) $\begin{align*}y > -4x + 1\end{align*}$
::D) y4x+1

(9, -1)

(0, 0)

(-1, 6)

(-3, -10)

Determine which inequality each point is a solution for. There may be more than one correct answer. If the answer is none, write none of these .
::确定每个点的不平等是哪个点的解决方法。答案可能不止一个。如果答案是零, 请不要写这些。

A) (-5, 1)
:A) (-5,1)

B) (4, 2)
::B) (4,4,2)

C) (-12, -7)
::C) (12,-7)

D) (8, -9)
::D) (8,-9)

$\begin{align*}2x -3y > 8\end{align*}$
::2x-3y>8

$\begin{align*}y \le -x -4\end{align*}$
::yx - 4

$\begin{align*}y \ge 6x + 7\end{align*}$
::y=6x+7 y=6x+7

$\begin{align*}8x +3y < -3\end{align*}$
::8+3y+33

Is (-6, -8) a solution to $\begin{align*}y < \frac{1}{2}x -6\end{align*}$ ?
:6, -8)是解决你12x -6的办法吗?

Is (10, 1) a solution to $\begin{align*}y \ge -7x + 1\end{align*}$ ?
:10,1)是解决y7x+1的办法吗?

For problems 11-15, find three solutions for each inequality.
::对于问题11-15,每种不平等都有三种解决办法。

$\begin{align*}5x -y >12\end{align*}$
::5 - y>12

$\begin{align*}y \le -2x + 9\end{align*}$
::y2x+9

$\begin{align*}y \ge -4\end{align*}$
::y 4

$\begin{align*}3x + 4y < -5\end{align*}$
::3x+4y5

$\begin{align*}x \le 7\end{align*}$
::x_ 7x7

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

两个变量中线性不平等的测试解决方案

Solution for Linear Inequality ::解决线性不平等问题

Examples ::实例

Example 1 ::例1

Example 2 ::例2

E xample 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Solution for Linear Inequality
::解决线性不平等问题

Examples
::实例

Example 1
::例1

Example 2
::例2

E xample 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)