Course: 9.16 解决线性不平等

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解决线性不平等
You are going on a trip in a jeep out to the rainforest. You need to have enough fuel to get there and back. It is 180 miles one way. The jeep you are driving requires two types of fuel - gasoline and ethanol because it is designed to run on a combination of both fuels. Gasoline has a gas mileage of 21 mpg while ethanol has a mileage of 17 mpg. The fuel tank of the jeep will hold a maximum of 20 gallons of fuel. How can you figure out how many gallons of each fuel you will need for the trip to and from the rainforest?

In this concept, you will learn to solve linear inequalities.
::在这个概念中,你会学会解决线性不平等。

Linear Inequalities
::线性不平等

When solving a linear equation with one variable the goal is to determine a value for the variable that makes the equation true. When solving a linear equation with two variables the goal is to determine an ordered pair ( $\begin{align*}x, y\end{align*}$ ) that makes the equation true. When solving a system of two linear equations with two variables the goal is to determine an ordered pair ( $\begin{align*}x, y\end{align*}$ ) that makes both equations true. The goals are the same for solving an inequality with one variable, two variables or a system of linear inequalities.
::当用一个变量解决线性方程式时,目标是确定使方程式成为真实的变量的值。当用两个变量解决线性方程式时,目标是确定使方程式成为真实的定单对(x,y) 。当用两个变量解决两个线性方程式的系统时,目标是确定两个方程式的定单对(x,y),使两个方程式都成为真实的。用一个变量、两个变量或线性不平等体系解决不平等的目标是一样的。

Unlike an equation that has a single solution, an inequality has more than one solution which is indicated by the inequality symbol. An equation has an equals sign but an inequality will have one of the following symbols:
::与单一解决办法的等式不同,不平等有不止一种解决办法,不平等符号就表明了这种解决办法。等式具有同等的标志,但不平等将具有以下标志之一:

$\begin{align*}>\end{align*}$ which is read as “greater than”
::> 被读为“大于”

$\begin{align*}\ge\end{align*}$ which is read as “ greater than and equal to”
::改为“大于或等于”

< which is read as “less than”
::< 以“小于”表示

$\begin{align*}\le\end{align*}$ which is read as “ less than and equal to”
::改为“小于或等于”

Let’s look at an example.
::让我们举个例子。

Solve the following inequality:
::解决以下不平等:

$\begin{aligned} 4 x + 2 < 18 \end{aligned}$
$\begin{align*}4x+2 < 18\end{align*}$
::4x+2 < 18

Solve the inequality as if it were a linear equation.
::解决不平等,仿佛是线性等式。

First, subtract 2 from both sides of the inequality and simplify both sides of the equation.
::首先,从不平等的两侧减去2,简化等式的两侧。

$\begin{aligned} \begin{array}{rcl} 4 x + 2 & < & 18 \\ 4 x + 2 - 2 & < & 18 - 2 \\ 4 x & < & 16 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 4x+2 &<& 18\\ 4x+2-2 &<& 18-2\\ 4x &<& 16 \end{array}\end{align*}$
::4x+2 < 184x+2-2 < 18-24x < 16

Next, divide both sides of the inequality by 4 to solve for the variable ‘ $\begin{align*}x\end{align*}$ ’.
::接下来,将不平等的两边除以4, 解决变数 'x ' 。

$\begin{aligned} \begin{array}{rcl} 4 x & < & 16 \\ \frac{\overset{1}{4}}{4} x & < & \frac{\overset{4}{16}}{4} \\ x & < & 4 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 4x &<& 16\\ \frac{\overset{1}{\cancel{4}}}{\cancel{4}}x &<& \frac{\overset{4}{\cancel{16}}}{\cancel{4}}\\ x &<& 4 \end{array}\end{align*}$
::4 < 16414x < 16414x < 16444x < 4

Any value for the variable less than four is a solution for the given inequality. The solution can be expressed as $\begin{align*}x<4\end{align*}$ or as $\begin{align*}\{x|x <4, x \in R \}\end{align*}$ using set notation .
::变量小于 4 的任何值是给定不平等的解决方案。可用设置符号表示为 x < 4 或 {xx < 4,xR} 。

If at any time, when solving an inequality, multiplication or division by a negative quantity is performed then the inequality symbol reverses.
::如果在任何时间,在解决不平等、乘数或负数分开时,不平等的象征就倒转。

Let’s look at an example.
::让我们举个例子。

Solve the following inequality:
::解决以下不平等:

$\begin{aligned} 5 - 3 y \geq 23 \end{aligned}$
$\begin{align*}5-3y \ge 23\end{align*}$
::5-33y23

First, subtract 5 from both sides of the inequality and simplify.
::首先,从不平等的两面减去5,并简化。

$\begin{aligned} \begin{array}{rcl} 5 - 3 y & \geq & 23 \\ 5 - 5 - 3 y & \geq & 23 - 5 \\ - 3 y & \geq & 18 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 5-3y &\ge& 23\\ 5-5-3y &\ge& 23-5\\ -3y &\ge& 18 \end{array}\end{align*}$
::5-3-23-23-5-3-3-18年 5-3-3-2335-5-3-5-3-5-3-3-18

Next, divide both sides of the inequality by -3 to solve for the variable ‘ $\begin{align*}y\end{align*}$ ’.
::接下来,将不平等的两边除以 - 3, 以解决变量“y” 。

$\begin{aligned} \begin{array}{rcl} - 3 y & \geq & 18 \\ \frac{\overset{1}{- 3}}{- 3} y & \geq & \frac{\overset{- 6}{18}}{- 3} \\ y & \leq & - 6 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} -3y &\ge& 18\\ \frac{\overset{1}{\cancel{-3}}}{\cancel{-3}}y &\ge& \frac{\overset{-6}{\cancel{18}}}{\cancel{-3}}\\ y &\le& -6 \end{array}\end{align*}$
::- 3y18 - 31 - 3y18 - 6 - 3y6

Notice the inequality sign has been reversed because both sides of the equation were divided by -6.
::由于等式的两面除以-6,不平等标志被逆转。

The solution is any value for the variable less than and equal to -6. This solution can be expressed as:
::解决方案是变量小于 -6 且等于 -6 的任何值。此解决方案可以表现为 :

$\begin{aligned} y \leq - 6 or {y | y \leq - 6, y \in R} \end{aligned}$
$\begin{align*}y \le -6 \ \text{or} \ \{y|y \le -6, y \in R \}\end{align*}$
::y'6 或yy6,yR}

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the trip to the rainforest. It is 180 miles one way so the total distance you must travel is 360 miles. The jeep will hold a maximum of 20 gallons of fuel which is a mixture of gasoline and ethanol. You need to figure out how many gallons of gasoline and ethanol you need to buy for the return trip to the rainforest.
::早些时候,有人给了你一个去雨林旅行的问题。这是一个180英里的单程路程, 因此您必须旅行的总距离是360英里。吉普车将持有最多20加仑的燃料, 燃料是汽油和乙醇的混合物。您需要弄清楚您需要购买多少加仑的汽油和乙醇才能返回雨林。

Use a system of linear inequalities to calculate the number of gallons of each fuel type is needed.
::需要使用线性不平等系统来计算每种燃料类型的加仑数量。

First, write a system of linear inequalities to model the problem.
::首先,写出一个线性不平等制度,作为这一问题的模型。

$\begin{align*}\text{Let }x = \# \text{ gallons of ethanol}\end{align*}$
::让 x加仑乙醇

$\begin{align*}\text{Let }y = \# \text{ gallons of gasoline}\end{align*}$
::让加仑汽油给你

$\begin{aligned} \begin{array}{rcl} 17 x + 21 y & \geq & 360 \\ x + y & \leq & 20 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 17x+21y &\ge& 360\\ x+y &\le& 20 \end{array}\end{align*}$
::17x+21y=360x+y=20

Next, solve the system of linear inequalities using the substitution method.
::其次,用替代方法解决线性不平等制度。

Next, subtract $\begin{align*}y\end{align*}$ from both sides of the second inequality and simplify.
::其次,从第二个不平等的两侧减去y,并简化。

$\begin{aligned} \begin{array}{rcl} x + y & \leq & 20 \\ x + y - y & \leq & 20 - y \\ x & \leq & 20 - y \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} x+y &\le& 20\\ x+y-y &\le& 20-y\\ x &\le& 20-y \end{array}\end{align*}$
::x+y%20x+y-y%20-yx%20-yy

Next, substitute the expression for the variable ‘ $\begin{align*}x\end{align*}$ ’ into the first inequality.
::其次,将变量 'x ' 改为第一个不平等。

$\begin{aligned} \begin{array}{rcl} 17 x + 21 y & \geq & 360 \\ 17 (20 - y) + 21 y & \geq & 360 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 17x+21y &\ge& 360\\ 17(20-y)+21y &\ge& 360 \end{array}\end{align*}$
::17x+21y%36017(20-y)+21y___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Next, perform the multiplication to clear the parenthesis and simplify.
::下一步,执行乘法以清除括号并简化。

$\begin{aligned} \begin{array}{rcl} 17 (20 - y) + 21 y & \geq & 360 \\ 340 - 17 y + 21 y & \geq & 360 \\ 340 + 4 y & \geq & 360 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 17(20-y)+21y &\ge& 360\\ 340-17y+21y &\ge& 360\\ 340+4y &\ge& 360 \end{array}\end{align*}$
::17(20-y)+21y_360340-17y+21y_360340+4y____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Next, subtract 340 from both sides of the inequality and simplify.
::其次,从不平等的两边减去340,简化。

$\begin{aligned} \begin{array}{rcl} 340 + 4 y & \geq & 360 \\ 340 - 340 + 4 y & \geq & 360 - 340 \\ 4 y & \geq & 20 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 340+4y &\ge& 360\\ 340-340+4y &\ge& 360-340\\ 4y &\ge& 20 \end{array}\end{align*}$
::340 - 340+4y360-3404y20

Next, divide both sides of the inequality by 4 to solve for the variable ‘ $\begin{align*}y\end{align*}$ ’.
::接下来,将不平等的两边除以4,

$\begin{aligned} \begin{array}{rcl} 4 y & \geq & 20 \\ \overset{1}{\frac{4 y}{4}} & \geq & \frac{\overset{5}{20}}{4} \\ y & \geq & 5 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 4y &\ge& 20\\ \overset{1}{\frac{\cancel{4}y}{\cancel{4}}} &\ge& \frac{\overset{5}{\cancel{20}}}{\cancel{4}}\\ y &\ge& 5 \end{array}\end{align*}$
::4y204y412054y5

Then, substitute the value 5 into the expression for the variable ‘ $\begin{align*}x\end{align*}$ ’ and simplify.
::然后将值 5 替换为变量 'x ' 并简化。

$\begin{aligned} \begin{array}{rcl} x & \leq & 20 - y \\ x & \leq & 20 - 5 \\ x & \leq & 15 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} x &\le& 20-y\\ x &\le& 20-5\\ x &\le& 15 \end{array}\end{align*}$
::x% 20 - yx% 20 - 5x% 15

The optimal solution is the ordered pair $\begin{align*}(15, 5)\end{align*}$ which means 15 gallons of ethanol and 5 gallons of gasoline are needed for the return trip to the rainforest.
::最佳解决办法是定购一对(15.5),这意味着返回雨林需要15加仑乙醇和5加仑汽油。

Example 2
::例2

Translate the following statement into an inequality and solve it.
::将以下声明转换为不平等并加以解决。

The difference between a number and seven is greater than 12.
::数字与7之间的差大于12。

$\begin{aligned} x - 7 > 12 \end{aligned}$
$\begin{align*}x-7>12\end{align*}$
::x-7 > 12

First, add 7 to both sides of the inequality to solve for the variable ‘ $\begin{align*}x\end{align*}$ ’.
::首先,在不平等的两面加7,以解决变数 'x ' 。

$\begin{aligned} \begin{array}{rcl} x - 7 & > & 12 \\ x - 7 + 7 & > & 12 + 7 \\ x & > & 19 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} x-7 &>& 12\\ x-7+7 &>& 12+7\\ x &>& 19 \end{array}\end{align*}$
::x-7>12x-7+7>12+7x>19

Any value for the variable greater than 19 is a solution for the inequality. The solution can be expressed as:
::大于19的变量的任何价值都是不平等的解决办法。

$\begin{aligned} x > 19 or {x | x > 19, x \in R} \end{aligned}$
$\begin{align*}x>19 \ \text{or} \ \{x|x >19, x \in R \}\end{align*}$

::x19 或 {xx>19,xR}

Example 3
::例3

Translate the following statement into an inequality and solve it.
::将以下声明转换为不平等并加以解决。

Two times a number increased by six is less than twenty.
::两次乘以六是20以下。

$\begin{aligned} 2 x + 6 < 20 \end{aligned}$
$\begin{align*}2x+6<20\end{align*}$
::2x6+6 < 20

First, subtract 6 from both sides of the inequality and simplify.
::首先,从不平等的两侧减去6,并简化。

$\begin{aligned} \begin{array}{rcl} 2 x + 6 & < & 20 \\ 2 x + 6 - 6 & < & 20 - 6 \\ 2 x & < & 14 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 2x+6 &<& 20\\ 2x+6-6 &<& 20-6\\ 2x &<& 14 \end{array}\end{align*}$
::2x+6 <202x+6-6<202-62x<14

Next, divide both sides of the inequality by 2 to solve for the variable ‘ $\begin{align*}x\end{align*}$ ’.
::接下来,将不平等的两边除以2, 解决变数 'x ' 。

$\begin{aligned} \begin{array}{rcl} 2 x & < & 14 \\ \overset{1}{\frac{2}{2}} x & < & \frac{\overset{7}{14}}{2} \\ x & < & 7 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 2x &<& 14\\ \overset{1}{\frac{\cancel{2}}{\cancel{2}}} \ x &<& \frac{\overset{7}{\cancel{14}}}{{\cancel{2}}}\\ x &<& 7 \end{array}\end{align*}$
::2x < 14221 x < 1472x < 7

Any value for the variable less than 7 is a solution for the inequality. The solution can be expressed as:
::变量小于7的任何值是不平等的解决办法。

$\begin{aligned} x < 7 or {x | x < 7, x \in R} \end{aligned}$
$\begin{align*}x<7 \ \text{or} \ \{x|x <7, x \in R \}\end{align*}$
::x7 或 {xx< 7,xR}

Example 4
::例4

Translate the following statement into an inequality and solve it.
::将以下声明转换为不平等并加以解决。

Three less than four times a number is greater than and equal to 41.
::数字比41大于或等于41乘以3至4倍。

$\begin{aligned} 4 x - 3 \geq 41 \end{aligned}$
$\begin{align*}4x-3 \ge 41\end{align*}$
::4x-341

First, add three to both sides of the inequality and simplify.
::首先,在不平等的两面加上三点,并简化。

$\begin{aligned} \begin{array}{rcl} 4 x - 3 & \geq & 41 \\ 4 x - 3 + 3 & \geq & 41 + 3 \\ 4 x & \geq & 44 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 4x-3 &\ge& 41\\ 4x-3+3 &\ge& 41+3\\ 4x &\ge& 44 \end{array}\end{align*}$
::4 - 3414x-3+341+34×44

Next, divide both sides of the inequality by 4 to solve for the variable ‘ $\begin{align*}x\end{align*}$ ’.
::接下来,将不平等的两边除以4, 解决变数 'x ' 。

$\begin{aligned} \begin{array}{rcl} 4 x & \geq & 44 \\ \frac{\overset{1}{4}}{4} x & \geq & \frac{\overset{11}{44}}{4} \\ x & \geq & 11 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 4x &\ge& 44\\ \frac{\overset{1}{\cancel{4}}}{\cancel{4}}x &\ge& \frac{\overset{11}{\cancel{44}}}{{\cancel{4}}} \\ x &\ge& 11 \end{array}\end{align*}$
::4x44414x4141414x1414x11

Any value for the variable greater than and equal to 11 is a solution for the inequality. The solution can be expressed as:
::大于和等于11的变量的任何价值是不平等的解决办法。

$\begin{aligned} x \geq 11 or {x | x \geq 11, x \in R} \end{aligned}$
$\begin{align*}x \ge 11 \ \text{or} \ \{x|x \ge 11, x \in R \}\end{align*}$

::x11 或 {xx11,xR}

Example 5
::例5

The difference between 20 and a number is greater than 12.
::20和数之差大于12。

$\begin{aligned} 20 - x > 12 \end{aligned}$
$\begin{align*}20-x > 12\end{align*}$
::20 - x > 12

First, subtract 20 from both sides of the inequality and simplify.
::首先,从不平等的两边减去20个,并简化。

$\begin{aligned} \begin{array}{rcl} 20 - x & > & 12 \\ 20 - 20 - x & > & 12 - 20 \\ - x & > & - 8 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 20-x &>& 12\\ 20-20-x &>& 12-20\\ -x &>& -8 \end{array}\end{align*}$
::20 - 20 - 1220 - 20 - 20 - x> 12 - 20 - x% 8

Next, divide both sides of the inequality by -1 to solve for the variable ‘ $\begin{align*}x\end{align*}$ ’.
::接下来,将不平等的两边除以-1, 以解决变量“ x” 。

$\begin{aligned} \begin{array}{rcl} - x & > & - 8 \\ \frac{\overset{1}{- 1} x}{- 1} & > & \frac{\overset{8}{- 8}}{- 1} \\ x & < & 8 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} -x &>& -8\\ \frac{\overset{1}{\cancel{-1}}x}{\cancel{-1}} &>& \frac{\overset{8}{\cancel{-8}}}{\cancel{-1}}\\ x &<& 8 \end{array}\end{align*}$
::- 8-11x-188-1x<8

Notice the inequality sign has been reversed because both sides of the equation were divided by -1.
::由于等式的两面除以-1,不平等标志被逆转。

The solution is any value for the variable less than and equal to 8. This solution can be expressed as:
::解决方案是变量小于或等于 8 的任何值。此解决方案可以表述为 :

$\begin{aligned} y < 8 or {y | y < 8, y \in R} \end{aligned}$
$\begin{align*}y<8 \ \text{or} \ \{y|y <8, y \in R \}\end{align*}$
::y < 8 或{yy < 8 yR}

Review
::回顾

Translate each statement into an inequality.
::将每个说法都转换成不平等。

1. The difference between two numbers is greater than 8.
::1. 两个数字之间的差额大于8。

2. Half of one number is at least 3 times another number.
::2. 一个数字的一半至少是另一个数字的三倍。

3. A quarter the sum of two numbers is less than 15.
::3. 四分之一,两个数字的总和不到15。

4. Seven times one number plus 3 less than another is not more than -16.
::4. 比其他数字少7倍于1+3,不超过 -16。

5. Six times a number is greater than negative thirty.
::5. 数字的六倍大于负30。

6. Five times a number and six is less than or equal to 39.
::6. 数字的五倍和数字的六小于或等于39。

7. Twelve divided by a number is less than seven.
::7. 十二除以一个数字不到七。

8. Six times a number and two less than another number is less than or equal to -12.
::8. 数字小于或等于-12,是数字的六倍,比数字少二倍。

Which ordered pairs make the inequalities true?
::哪个定购的对子能让不平等成为事实呢?

$\begin{aligned} 2 x - 5 y \geq 20 \end{aligned}$
$\begin{align*}2x-5y \ge 20\end{align*}$
::2 - 5y20

9. $\begin{align*}(10, 5)\end{align*}$

10. $\begin{align*}(10, 4)\end{align*}$

11. $\begin{align*}(-10, -10)\end{align*}$

12. $\begin{align*}(0, 0)\end{align*}$

$\begin{aligned} y > - 4 x - 2 \end{aligned}$
$\begin{align*}y >-4x-2\end{align*}$
::y4x-2

13. $\begin{align*}(1, 1)\end{align*}$

14. $\begin{align*}(0, -6)\end{align*}$

15. $\begin{align*}(2, -9)\end{align*}$

16. $\begin{align*}(-1, 0)\end{align*}$

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

解决线性不平等

Linear Inequalities ::线性不平等

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Linear Inequalities
::线性不平等

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)