Course: 4.8 通过绝对值求整数的差

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通过绝对值求整数的差
This summer Logan is spending all of July driving across the United States. At the start of the trip Logan had $500. Over the course of the trip, Logan spent $792 traveling and charged it all to his credit card. How can Logan figure out how much money he has now?
::这个夏天,洛根正在全七月份开着车横跨美国。在洛根旅行开始时,他有500美元。在旅行过程中,洛根花了792美元旅行,并将所有钱都记在信用卡上。洛根怎么知道他现在有多少钱?

In this concept, you will learn how to find the difference of integers using opposites.
::在这个概念中,你会学会如何用反向来发现整数的差别。

Subtracting Integers Using Absolute Value
::使用绝对值进行减法整数

The opposite of any integer, $\begin{aligned} b \end{aligned}$ , is $\begin{aligned} - b \end{aligned}$ . To find the opposite of an integer, change its sign.
::任何整数的反面是- b。要找到整数的反面,请修改整数的符号。

Here are some examples of integers and their opposites.
::以下是整数及其反面的一些例子。

The opposite of 2 is -2.
::2的对面是-2。

The opposite of -4 is 4.
::与-4相反的是-4。

The opposite of -100 is 100.
::与-100相反的是100。

The opposite of 17 is -17.
::与17相反的是 -17。

There are many different strategies for subtracting integers. One strategy for subtracting involves opposites. To use this strategy you will be changing your subtraction problem into an addition problem.
::减整数有许多不同的策略。减整数的一个策略是反向的。要使用这个策略, 您将会将减整问题改变为附加问题。

To find $\begin{aligned} a - b \end{aligned}$ , add $\begin{aligned} a + (- b) \end{aligned}$ .
::要找到 - b, 增加 + (- b) 。

In other words, to subtract two integers:
::换句话说,减去两个整数:

Take the opposite of the integer being subtracted (the second integer).
::取正对的整数减去的整数(第二个整数)。

Add that opposite to the first integer using your strategies for .
::使用您的策略添加第一个整数的对面。

Let's look at an example.
::让我们举个例子。

Find the difference of $\begin{aligned} 5 - (- 8) \end{aligned}$ .
::找出5-(-8)的差别。

First, you need to take the opposite of your second integer, -8. The opposite of -8 is 8.
::首先,你需要拿第二个整数的对面,8,8,8的对面是8。

Now you can rewrite your subtraction problem as an addition problem.
::现在您可以将减法问题重写为添加问题。

$\begin{aligned} 5 - (- 8) = 5 + 8 \end{aligned}$

Finally, you can solve your addition problem.
::最后,你可以解决你的添加问题。

$\begin{aligned} 5 + 8 = 13 \end{aligned}$

The answer is 13. $\begin{aligned} 5 - (- 8) = 13 \end{aligned}$ .
::答案是13,5-(-8)=13。

Let's look at another example.
::让我们再举一个例子。

Find the difference of $\begin{aligned} - 12 - 2 \end{aligned}$ .
::找出 - 12 - 2的差别。

First, you need to take the opposite of your second integer, 2. The opposite of 2 is -2.
::首先,你需要取第二个整数的反面 2, 2的反面是-2。

Now you can rewrite your subtraction problem as an addition problem.
::现在您可以将减法问题重写为添加问题。

$\begin{aligned} - 12 - 2 = - 12 + (- 2) \end{aligned}$

Next, you can solve your addition problem. Here you have two integers with the same sign. To add, find the absolute value of each integer and add. $\begin{aligned} | - 12 | = 12 \end{aligned}$ and $\begin{aligned} | - 2 | = 2 \end{aligned}$ so you have
::接下来, 您可以解决您的添加问题。您在这里使用相同的符号有两个整数。要添加, 请找到每个整数的绝对值, 并添加。 @ \ - 12 = 12 = 12 = 和 _ 2 = 2 = 2 。所以您已经找到了。 @ info/ plain

$\begin{aligned} 12 + 2 = 14 \end{aligned}$ .

Finally, determine the sign of your final answer by looking at the signs of the original integers. -12 and -2 were both negative so your answer will be negative.
::最后,通过查看原始整数的符号来确定您最后回答的符号。 -12 和 -2 均为负数,所以您的回答将是负数。

$\begin{aligned} - 12 + (- 2) = - 14 \end{aligned}$

The answer is $\begin{aligned} - 14. - 12 - 2 = - 14 \end{aligned}$ .
::答案是 - 14 - 12 - 2 = - 14。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Logan and his trip across the United States.
::早些时候,你得到一个问题关于洛根和他的旅行穿越美国。

He started out with $500 and spent $792 on his credit card over the course of his trip. He wants to know how much money he has now.
::他从500美元起步,在旅途中用信用卡支付了792美元。他想知道他现在有多少钱。

Since Logan spent more money than he has, he is now in debt and essentially has negative money. To figure out how much money he has, you need to subtract:
::既然洛根花的钱比他多,他现在还欠债,基本上有负钱。

$\begin{aligned} 500 - 792 \end{aligned}$

First, you need to take the opposite of your second integer, 792. The opposite of 792 is -792.
::首先,你需要拿第二个整数,792的反面。 792的反面是792的反面是792。

Now you can rewrite your subtraction problem as an addition problem.
::现在您可以将减法问题重写为添加问题。

$\begin{aligned} 500 - 792 = 500 + (- 792) \end{aligned}$

Next, you can solve your addition problem. Here we have two integers with different signs. To add, find the absolute value of each integer and subtract. $\begin{aligned} | 500 | = 500 \end{aligned}$ and $\begin{aligned} | - 792 | = 792 \end{aligned}$ so you have
::接下来, 您可以解决您的添加问题。这里我们有两个有不同符号的整数。要添加, 请找到每个整数的绝对值, 并减去。 @ 500 = 500 = 500 和 ~ ~ 792 = 792 = 792 。

$\begin{aligned} 792 - 500 = 292 \end{aligned}$ .

Finally, determine the sign of your final answer by looking at the sign of the integer with the greater absolute value. -792 had the greater absolute value and it is negative, so your answer will be negative.
::最后,通过以更大的绝对值来查看整数的符号来确定你最后回答的符号。 -792的绝对值更大,是负的,所以你的答案将是负的。

$\begin{aligned} 500 + (- 792) = - 292 \end{aligned}$

The answer is -292. $\begin{aligned} 500 - 792 = - 292 \end{aligned}$ .
::答案是 - 292. 500 - 792 = - 292。

Logan now has -$292.
::罗根现在有292美元

Example 2
::例2

The temperature inside a laboratory freezer was $\begin{aligned} - 10^{\circ} C \end{aligned}$ . A scientist at the lab then lowered the temperature inside the freezer so it was $\begin{aligned} 5^{\circ} C \end{aligned}$ less. What was the new temperature inside the freezer?
::实验室冷冻室的温度是- 10°C. 实验室的一名科学家然后降低了冷冻室的温度,因此是5°C。冷冻室内的新温度是多少?

First, turn the word problem into a subtract problem. The room started out at -10 and then the temperature went down by 5 degrees. The subtraction problem is
::首先,将单词问题转换为减法问题。房间从 - 10 开始, 然后温度下降5 度。减法问题是

$\begin{aligned} - 10 - 5 \end{aligned}$ .

To solve the subtraction problem, you first need to take the opposite of your second integer, 5. The opposite of 5 is -5.
::为了解决减法问题, 您首先需要取第二个整数的对面, 5, 5的对面是 - 5 。

Now you can rewrite your subtraction problem as an addition problem.
::现在您可以将减法问题重写为添加问题。

$\begin{aligned} - 10 - 5 = - 10 + (- 5) \end{aligned}$

Next, you can solve your addition problem. Here we have two integers with the same sign. To add, find the absolute value of each integer and add. $\begin{aligned} | - 10 | = 10 \end{aligned}$ and $\begin{aligned} | - 5 | = 5 \end{aligned}$ so you have
::接下来,您可以解决您的添加问题。这里我们有两个带有相同符号的整数。要添加, 请找到每个整数的绝对值, 并添加。 @ ~ ~ 10 = 10 = 10 和 ~ 5 = 5 = 5 。

$\begin{aligned} 10 + 5 = 15 \end{aligned}$

Finally, determine the sign of your final answer by looking at the signs of the original integers. -10 and -5 were both negative so your answer will be negative.
::最后,通过查看原始整数的符号来确定最后答案的符号。 - 10 和 - 5 均为负数,所以你的答案将是负数。

$\begin{aligned} - 10 + (- 5) = - 15 \end{aligned}$

The answer is -15. $\begin{aligned} - 10 - 5 = - 15 \end{aligned}$
::答案是 -15。 - 10 - 5 = - 15

The new temperature inside the freezer is $\begin{aligned} - 15^{\circ} C \end{aligned}$ .
::冰柜内的新温度是15°C

Example 3
::例3

Find the difference of $\begin{aligned} - 5 - 7 \end{aligned}$ .
::找出5-7的差别。

First, you need to take the opposite of your second integer, 7. The opposite of 7 is -7.
::首先,你需要采取与你的第二整数相反的7,7的对7是 -7。

Now you can rewrite your subtraction problem as an addition problem.
::现在您可以将减法问题重写为添加问题。

$\begin{aligned} - 5 - 7 = - 5 + (- 7) \end{aligned}$

Next, you can solve your addition problem. Here you have two integers with the same sign. To add, find the absolute value of each integer and add. $\begin{aligned} | - 5 | = 5 \end{aligned}$ and $\begin{aligned} | - 7 | = 7 \end{aligned}$ so you have
::接下来, 您可以解决您的添加问题。您在这里使用相同的符号有两个整数。要添加, 请找到每个整数的绝对值, 并添加。 @ ~ 5 = 5 = 5 = 7 = 7 = 7 。 @ label

$\begin{aligned} 5 + 7 = 12 \end{aligned}$

Finally, determine the sign of your final answer by looking at the signs of the original integers. -5 and -7 were both negative so your answer will be negative.
::最后,通过查看原始整数的符号来确定你最后回答的符号。 -5和 -7都是负的,所以你的回答将是负的。

$\begin{aligned} - 5 + (- 7) = - 12 \end{aligned}$

The answer is -12. $\begin{aligned} - 5 - 7 = - 12 \end{aligned}$ .
::答案是 -12。 - 5 - 7 = - 12。

Example 4
::例4

Find the difference of $\begin{aligned} 8 - (- 4) \end{aligned}$ .
::发现差值为8 - (-4) 。

First, you need to take the opposite of your second integer, -4. The opposite of -4 is 4.
::首先,你需要取第二个整数的对面, -4。4的对面是4。

Now you can rewrite your subtraction problem as an addition problem.
::现在您可以将减法问题重写为添加问题。

$\begin{aligned} 8 - (- 4) = 8 + 4 \end{aligned}$

Finally, you can solve your addition problem.
::最后,你可以解决你的添加问题。

$\begin{aligned} 8 + 4 = 12 \end{aligned}$

The answer is 12. $\begin{aligned} 8 - (- 4) = 12 \end{aligned}$ .
::答案是12、8-(-4)=12。

Example 5
::例5

Find the difference of $\begin{aligned} - 12 - (- 8) \end{aligned}$ .
::找出 - 12 - (-8)的差别。

First, you need to take the opposite of your second integer, -8. The opposite of -8 is 8.
::首先,你需要拿第二个整数的对面,8,8,8的对面是8。

Now you can rewrite your subtraction problem as an addition problem.
::现在您可以将减法问题重写为添加问题。

$\begin{aligned} - 12 - (- 8) = - 12 + 8 \end{aligned}$

Next, you can solve your addition problem. Here you have two integers with different signs. To add, find the absolute value of each integer and subtract. $\begin{aligned} | - 12 | = 12 \end{aligned}$ and $\begin{aligned} | 8 | = 8 \end{aligned}$ so you have
::接下来, 您可以解决您的添加问题。您在这里有两个有不同符号的整数。要添加, 请找到每个整数的绝对值, 并减去。 @ \ - 12 = 12 = 12 和 8 = 8 = 8 。所以您有

$\begin{aligned} 12 - 8 = 4 \end{aligned}$ .

Finally, determine the sign of your final answer by looking at the sign of the integer with the greater absolute value. -12 had the greater absolute value and it is negative, so your answer will be negative.
::最后,通过以更大的绝对值来查看整数的符号来确定最后答案的符号。 -12的绝对值更大,是负的,所以你的答案将是负的。

$\begin{aligned} - 12 + 8 = - 4 \end{aligned}$

The answer is -4. $\begin{aligned} - 12 - (- 8) = - 4 \end{aligned}$ .
::答案是 -4. - 12 - (-8) = - 4。

Review
::回顾

Find each difference using opposites.
::用对立面来找出每个差异。

$\begin{aligned} 15 - 7 \end{aligned}$

$\begin{aligned} - 7 - 12 \end{aligned}$

$\begin{aligned} 0 - 4 \end{aligned}$

$\begin{aligned} 13 - (- 9) \end{aligned}$

$\begin{aligned} - 21 - 4 \end{aligned}$

$\begin{aligned} 33 - (- 4) \end{aligned}$

$\begin{aligned} - 11 - (- 8) \end{aligned}$

$\begin{aligned} 18 - 28 \end{aligned}$

$\begin{aligned} 28 - (- 8) \end{aligned}$

$\begin{aligned} 13 - 18 \end{aligned}$

$\begin{aligned} - 21 - (- 8) \end{aligned}$

$\begin{aligned} - 9 - (- 38) \end{aligned}$

$\begin{aligned} 22 - 8 \end{aligned}$

$\begin{aligned} 25 - 38 \end{aligned}$

$\begin{aligned} 19 - (- 19) \end{aligned}$

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。

Resources
::资源

Section outline

General

通过绝对值求整数的差

Subtracting Integers Using Absolute Value ::使用绝对值进行减法整数

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)

Resources ::资源

Subtracting Integers Using Absolute Value
::使用绝对值进行减法整数

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)

Resources
::资源