Course: 1.11 解决多重不平等

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解决多重不平等
Adding antifreeze to a car's cooling system can lower the freezing point of the water-based liquids that enable the vehicle to run . (It also raises the boiling point of these liquids.) You add enough antifreeze to lower the freezing point of the liquids to at most $\begin{aligned} - 35^{\circ} C \end{aligned}$ (in Celsius). Write the described inequality in Fahrenheit.
::将防冻剂添加到汽车冷却系统中可以降低使车辆能够运行的水基液体的冷藏点。 (它也会提高这些液体的沸点。 ) 您添加足够的防冻剂, 将液体的冷冻点降低到最多至-35°C( 摄氏度) 。写下所描述的法赫伦海特的不平等。

Multi-Step Inequalities
::多重不平等

Like , multi-step inequalities can involve having variables on both sides, the , and combining like terms . Again, the only difference when solving inequalities is the sign must be flipped when multiplying or dividing by a negative number.
::类似地,多重不平等可能涉及双方的变数,如条件等。同样,解决不平等的唯一区别是当负数乘以或除以负数时必须翻转信号。

Let's check whether $\begin{aligned} x = - 3 \end{aligned}$ is a solution to $\begin{aligned} 2 (3 x - 5) \leq x + 10 \end{aligned}$ .
::让我们检查 x3 是否是 2( 2 3x- 5) x+10 的解决方案。

Plug in -3 for $\begin{aligned} x \end{aligned}$ and see if the inequality is true.
::x 插插到 - 3, 看看不平等是否真实。

$\begin{aligned} 2 (3 (- 3) - 5) & \leq (- 3) + 10 \\ 2 (- 9 - 5) & \leq 7 \\ 2 \cdot - 14 & \leq 7 \\ - 28 & \leq 7 \end{aligned}$

This is a true inequality statement. -3 is a solution.
::这是一个真正的不平等声明。 -3是一个解决办法。

Now, let's solve the following multi-step inequalities.
::现在,让我们解决以下多重不平等。

Solve and graph the inequality $\begin{aligned} 2 (3 x - 5) \leq x + 10. \end{aligned}$
::解决不平等问题并绘制图2(2,3x-5,5)x+10。

First, distribute the 2 on the left side of the inequality.
::首先,在不平等的左边分配两个。

$\begin{aligned} 2 (3 x - 5) & \leq x + 10 \\ 6 x - 10 & \leq x + 10 \end{aligned}$

::2(3x-5)x+106x-10x+10

Now, subtract the $\begin{aligned} x \end{aligned}$ on the right side to move it to the left side of the inequality. You can also add the 10’s together and solve.
::现在,将右侧的 x 减去以将其移到不平等的左侧。您也可以将10 相加并解决。

$\begin{aligned} 6 x - 10 \geq x + 10 \\ \underline{- x + 10 - x + 10} \\ \frac{5 x}{5} \leq \frac{20}{5} \\ x \leq 4 \end{aligned}$

::6-10xx+10-10x+10 - x10+10 - 10+10_5x5x5=205x%4

Test a solution, $\begin{aligned} x = 0 : 2 (3 (0) - 5) \leq 0 + 10 ✓ \end{aligned}$
::测试溶液, x=0:2:3(0)-5:0+10

$\begin{aligned} - 10 \leq 10 \end{aligned}$

The graph looks like:
::图表看起来像:

Solve $\begin{aligned} 8 x - 5 - 4 x \geq 37 - 2 x \end{aligned}$ .
::解决8x-5-4x-37-2x。

First, combine like terms on the left side. Then, solve for $\begin{aligned} x \end{aligned}$ .
::首先,将左侧的术语合并,然后解决 x 。

$\begin{aligned} 8 x - 5 - 4 x \geq 37 - 2 x \\ 4 x - 5 \geq 37 - 2 x \\ \underline{+ 2 x + 5 + 5 + 2 x} \\ \frac{6 x}{6} \geq \frac{42}{6} \\ x \geq 7 \end{aligned}$

::8 - 5 - 4x__ 4x__ 37 - 2x4x - 5- 37 - 2x+2x+5+5+5+2x - 6x66__ 426 x 7

Test a solution, $\begin{aligned} x = 10 : 8 (10) - 5 - 4 (10) \geq 37 - 2 (10) \end{aligned}$
::测试溶液, x=10:8(10)-5-4(10)-237-2(10)

$\begin{aligned} 80 - 5 - 40 \geq 37 - 20 \end{aligned}$

$\begin{aligned} 35 \geq 17 ✓ \end{aligned}$

Examples
::实例

Example 1
::例1

Earlier, you were asked to write the inequality regarding antifreeze in Fahrenheit.
::更早之前,你被要求写关于法氏冰冷的不平等问题的文章。

Recall that the conversion formula for Celsius to Fahrenheit is $\begin{aligned} C = \frac{5}{9} (F - 32) \end{aligned}$ . The temperature can be equal to or greater than $\begin{aligned} - 35^{\circ} C \end{aligned}$ .
::回顾摄氏至华氏的转换公式为C=59(F-32),温度可等于或大于-35°C。

$\begin{aligned} \frac{5}{9} (F - 32) \geq - 35 \\ F - 32 \geq - 35 \cdot 95 \\ F - 32 \geq - 63 \\ F \geq - 31 \end{aligned}$

::59(F-32)_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

So, the temperature can be equal to or higher than $\begin{aligned} - 31^{\circ} F \end{aligned}$ .
::因此,温度可以等于或高于-31F。

Example 2
::例2

Is $\begin{aligned} x = 12 \end{aligned}$ a solution to $\begin{aligned} - 3 (x - 10) + 18 \geq x - 25 \end{aligned}$ ?
::x=12 是- 3(x- 10) +18x- 25的解决方案吗 ?

Plug in 12 for $\begin{aligned} x \end{aligned}$ and simplify.
::x 和简化, 12 插件 12 。

$\begin{aligned} - 3 (12 - 10) + 18 & \geq 12 - 35 \\ - 3 \cdot 2 + 18 & \geq - 13 \\ - 6 + 18 & \geq - 13 \end{aligned}$

This is true because $\begin{aligned} 12 \geq - 13 \end{aligned}$ , so 12 is a solution.
::这是正确的,因为12+13,所以12是一个解决办法。

Example 3
::例3

Solve and graph $\begin{aligned} - (x + 16) + 3 x > 8 \end{aligned}$ .
::解析和图形 -(x+16)+3x>8。

Distribute the negative sign on the left side and combine like terms.
::将左侧的负符号分布在左侧, 并结合类似条件。

$\begin{aligned} - (x + 16) + 3 x > 8 \\ - x - 16 + 3 x > 8 \\ 2 x - 16 > 8 \\ \underline{+ 16 + 16} \\ \frac{2 x}{2} > \frac{24}{2} \\ x > 12 \end{aligned}$

::- (x+16)+3x8-x16+3x >8 2x-16>8+16+16_2x2>242 x12

Test a solution, $\begin{aligned} x = 15 : \end{aligned}$
::测试溶液, x=15:

$\begin{aligned} - (15 + 16) + 3 (15) & > 8 \\ - 31 + 45 & > 8 \\ 14 & > 8 \end{aligned}$

Example 4
::例4

Solve and graph $\begin{aligned} 24 - 9 x < 6 x - 21 \end{aligned}$ .
::解决和图24-9x<6x-21。

First, add $\begin{aligned} 9 x \end{aligned}$ to both sides and add 21 to both sides.
::首先,在双方增加9x,在双方增加21。

$\begin{aligned} 24 - 9 x < 6 x - 21 \\ \underline{+ 9 x + 9 x} \\ 24 < 15 x - 21 \\ \underline{+ 21 + 21} \\ \frac{45}{15} < \frac{15 x}{15} \\ 3 < x \end{aligned}$

::24-9x < 6x-21+9x+9x+9x_ 24 < 15x-21+21+21_4515 < 15x153 <x

Test a solution, $\begin{aligned} x = 10 : \end{aligned}$
::测试溶液, x=10:

$\begin{aligned} 24 - 9 (10) & < 6 (10) - 21 \\ 24 - 90 & < 60 - 21 \\ - 66 & < 39 \end{aligned}$

Review
::回顾

Determine if the following numbers are solutions to $\begin{aligned} - 7 (2 x - 5) + 12 > - 4 x - 13 \end{aligned}$ .
::确定以下数字是否为 -7( 2x- 5) +12+ 4x- 13 的解决方案。

$\begin{aligned} x = 4 \end{aligned}$
::x=4x=4

$\begin{aligned} x = 10 \end{aligned}$
::x=10x=10

$\begin{aligned} x = 6 \end{aligned}$
::x=6x=6

Solve and graph the following inequalities.
::解决以下不平等问题并绘制图表。

$\begin{aligned} 2 (x - 5) \geq 16 \end{aligned}$
::2(x-5-5)16

$\begin{aligned} - 4 (3 x + 7) < 20 \end{aligned}$
::- 4(3x+7) <20

$\begin{aligned} 15 x - 23 > 6 x - 17 \end{aligned}$
::15-23>6x-17

$\begin{aligned} 5 x + 16 + 2 x \leq - 19 \end{aligned}$
::5x+16+2x19

$\begin{aligned} 4 (2 x - 1) \geq 3 (2 x + 1) \end{aligned}$
::4( 2x- 1)3(2x+1)

$\begin{aligned} 11 x - 17 - 2 x \leq - (x - 23) \end{aligned}$
::11-17-2x(x-23)

Solve the following inequalities.
::解决以下不平等。

$\begin{aligned} 5 - 5 x > 4 (3 - x) \end{aligned}$
::5-5x>4(3-x)

$\begin{aligned} - (x - 1) + 10 < - 3 (x - 3) \end{aligned}$
::- (x-1-1)+103(x-3)

Solve $\begin{aligned} 5 x + 4 \leq - 2 (x + 3) \end{aligned}$ by adding the $\begin{aligned} 2 x \end{aligned}$ term on the right to the left-hand side.
::将右侧的2x字词添加到左侧,从而解决5x+42(x+3) 2(x+3) 。

Solve $\begin{aligned} 5 x + 4 \leq - 2 (x + 3) \end{aligned}$ by subtracting the $\begin{aligned} 5 x \end{aligned}$ term on the left to the right-hand side.
::将左侧的5x条件减到右侧,从而解决5x+42(x+3) 2(x+3) 。

Compare your answers from 12 and 13. What do you notice?
::比较12和13的答案你注意到什么了?

Challenge Solve $\begin{aligned} 3 x - 7 > 3 (x + 3) \end{aligned}$ . What happens?
::挑战解决 3x-7>3(x+3) 时如何?

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

解决多重不平等

Multi-Step Inequalities ::多重不平等

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)

Multi-Step Inequalities
::多重不平等

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)