Course: 2.2 增加反面和绝对值

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增加反反和绝对值
Additive Inverses and Absolute Values
::增加反反和绝对值

Every number has an opposite. On the number line, a number and its opposite are, predictably, opposite each other. In other words, they are the same distance from zero, but on opposite sides of the number line. The opposite of zero is defined to be simply zero.
::每个数字都相反。在数字线上,一个数字和相反的数字可以预见地相互对立。换句话说,它们与零是相同的距离,但在数字线的对面。零的对面被定义为零。

The sum of a number and its opposite is always zero, as shown in the following example.
::如下例所示,一个数字的总和与其相反始终为零。

Proving the Sum of a Number and its Opposite Equals Zero
::证明一个数字的总数及其对面等值为零

The numbers 3 and -3 are opposites because: $\begin{aligned} 3 + - 3 = 0 \end{aligned}$
::数字3和3是相反的,因为:3+3=0

The numbers 4.2 and -4.2 are opposites because: $\begin{aligned} 4.2 + - 4.2 = 0 \end{aligned}$
::数字4.2和-4.2相反,因为:4.2*4.2=0

This is because adding 3 and -3 is like moving 3 steps to the right along the number line, and then 3 steps back to the left. The number and its opposite cancel each other out, leaving zero.
::这是因为添加 3 和 - 3 就像在数字行向右移动 3 个步骤, 然后向左移动 3 个步骤。数字和相反的数字互相取消, 留下零。

Another way to think of the opposite of a number is that it is simply the original number multiplied by -1.
::另一种与数字相反的思考方法是,它仅仅是原来的数字乘以-1。

Additive Inverses
::添加反移

The opposite of 4 is $\begin{aligned} 4 \times - 1 \end{aligned}$ or -4, and the opposite of -2.3 is $\begin{aligned} - 2.3 \times - 1 \end{aligned}$ or just 2.3.
::4的反面是 4 1 或 4, 而 -2.3 的反面是-2.3 1 或仅2.3。

Another term for the opposite of a number is the additive inverse .
::与数字相反的另一个术语是添加反比。

Finding the Opposite
::寻找对面

Find the opposite of each of the following:
::发现以下各点的相反之处:

Since we know that opposite numbers are opposite sides of zero, we can simply multiply each expression by -1. This changes the sign of the number to its opposite-- if it's negative, it becomes positive, and vice versa.
::由于我们知道相反的数字是零的对面, 我们可以简单地将每个表达式乘以 - 1。这将数字的符号改变为相反 -- 如果数字是负的, 它会变成正的, 反之亦然。

a) 19.6
:a) 19.6

The opposite of 19.6 is -19.6.
::19.6反比为19.6。

b) $\begin{aligned} - \frac{4}{9} \end{aligned}$
:b)-49

The opposite of $\begin{aligned} - \frac{4}{9} \end{aligned}$ is $\begin{aligned} \frac{4}{9} \end{aligned}$ .
::与49相反的是49。

c) $\begin{aligned} x \end{aligned}$
:c)x

The opposite of $\begin{aligned} x \end{aligned}$ is $\begin{aligned} - x \end{aligned}$ .
::x的反面是-x。

d) $\begin{aligned} x y^{2} \end{aligned}$
::d) xy2

The opposite of $\begin{aligned} x y^{2} \end{aligned}$ is $\begin{aligned} - x y^{2} \end{aligned}$ .
::xy2 的对面是 - xy2。

e) $\begin{aligned} (x - 3) \end{aligned}$
:e) (x-3)

The opposite of $\begin{aligned} (x - 3) \end{aligned}$ is $\begin{aligned} - (x - 3) \end{aligned}$ , or $\begin{aligned} (3 - x) \end{aligned}$ .
::反之(x-3)是-(x-3)或(3-x)。

Note: With the last example you must multiply the entire expression by -1. A common mistake in this example is to assume that the opposite of $\begin{aligned} (x - 3) \end{aligned}$ is $\begin{aligned} (x + 3) \end{aligned}$ . Avoid this mistake!
::注意: 使用最后一个示例, 您必须将整个表达式乘以 - 1. 。此示例中常见的错误是假设( x-3) 与 (x+ 3) 的相反是 (x+ 3) , 避免这个错误 !

Find Absolute Values
::查找绝对值

When we talk about absolute value , we are talking about distances on the number line. For example, the number 7 is 7 units away from zero—and so is the number -7. The absolute value of a number is the distance it is from zero, so the absolute value of 7 and the absolute value of -7 are both 7.
::当我们谈论绝对值时,我们谈论的是数字线上的距离。例如,数字线上的数字7是距离零的7个单位,数字7也是7个单位。数字的绝对值是距离零,7的绝对值和7的绝对值都是7。

We write the absolute value of -7 as $\begin{aligned} | - 7 | \end{aligned}$ . We read the expression $\begin{aligned} | x | \end{aligned}$ as “the absolute value of $\begin{aligned} x \end{aligned}$ .”
::我们把 -7 的绝对值写为 @ @ @ 7 @ 。我们把 {x} 的表达方式读为“ x 的绝对值 ” 。

Treat absolute value expressions like " data-term="Parentheses" role="term" tabindex="0"> parentheses . If there is an operation inside the absolute value symbols, evaluate that operation first.
::处理绝对值表达式, 如括号。如果在绝对值符号内有操作, 请先评估操作。

The absolute value of a number or an expression is always positive or zero. It cannot be negative. With absolute value, we are only interested in how far a number is from zero, and not in which direction.
::数字或表达式的绝对值总是正值或零值。它不能是负值。如果是绝对值,我们只关心数字与零的距离,而不是方向。

Evaluating Absolute Value Expressions
::评价绝对值表达式

Evaluate the following absolute value expressions.
::评估以下绝对值表达式。

(Remember to treat any expressions inside the absolute value sign as if they were inside parentheses and evaluate them first.)
:注意将绝对值符号内的任何表达式当作括号内的任何表达式处理,并先进行评估。 )

a) $\begin{aligned} | 5 + 4 | \end{aligned}$
::a) 5+4+5+4

$\begin{aligned} | 5 + 4 | = | 9 | = 9 \end{aligned}$

b) $\begin{aligned} - | 7 - 22 | \end{aligned}$
:b) 7-22____________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} - | 7 - 22 | = - | - 15 | = - (15) = - 15 \end{aligned}$

Examples
::实例

Example 1
::例1

Find the opposite of each of the following:
::发现以下各点的相反之处:

Since we know that opposite numbers are on opposite sides of zero, we can simply multiply each expression by -1.
::由于我们知道相反数字在零的对面,我们可以简单地将每个表达式乘以-1。

a) $\begin{aligned} \frac{2}{x} \end{aligned}$
::a) 2x

The opposite of $\begin{aligned} \frac{2}{x} \end{aligned}$ is $\begin{aligned} \frac{- 2}{x} \end{aligned}$ .
::2xx对面

b) $\begin{aligned} - 2 y \end{aligned}$
:b)-2yy

The opposite of $\begin{aligned} - 2 y \end{aligned}$ is $\begin{aligned} 2 y \end{aligned}$ .
::——2y的反面是2y。

Example 2
::例2

Evaluate the following absolute value expressions.
::评估以下绝对值表达式。

a) $\begin{aligned} 3 - | 4 - 9 | \end{aligned}$
:a) 3+4-9______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} 3 - | 4 - 9 | = 3 - | - 5 | = 3 - 5 = - 2 \end{aligned}$

b) $\begin{aligned} | - 5 - 11 | \end{aligned}$
:b)5-11_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} | - 5 - 11 | = | - 16 | = 16 \end{aligned}$

Review
::回顾

Find the opposite of each of the following.
::发现以下各点正好相反。

1.001

$\begin{aligned} (5 - 11) \end{aligned}$

$\begin{aligned} (x + y) \end{aligned}$
:x+y)

$\begin{aligned} (x - y) \end{aligned}$
:x-y)

$\begin{aligned} (x + y - 4) \end{aligned}$
:x+y-4) (x+y-4)

$\begin{aligned} (- x + 2 y) \end{aligned}$
:-x+2y)

Simplify the following absolute value expressions.
::简化以下绝对值表达式。

$\begin{aligned} 11 - | - 4 | \end{aligned}$

$\begin{aligned} | 4 - 9 | - | - 5 | \end{aligned}$

$\begin{aligned} | - 5 - 11 | \end{aligned}$

$\begin{aligned} 7 - | 22 - 15 - 19 | \end{aligned}$

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

$\begin{aligned} | - 2 - 88 | - | 88 + 2 | \end{aligned}$

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。

Section outline

General

增加反反和绝对值

Additive Inverses and Absolute Values ::增加反反和绝对值

Proving the Sum of a Number and its Opposite Equals Zero ::证明一个数字的总数及其对面等值为零

Additive Inverses ::添加反移

Finding the Opposite ::寻找对面

Find Absolute Values ::查找绝对值

Evaluating Absolute Value Expressions ::评价绝对值表达式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)

Additive Inverses and Absolute Values
::增加反反和绝对值

Proving the Sum of a Number and its Opposite Equals Zero
::证明一个数字的总数及其对面等值为零

Additive Inverses
::添加反移

Finding the Opposite
::寻找对面

Find Absolute Values
::查找绝对值

Evaluating Absolute Value Expressions
::评价绝对值表达式

Examples
::实例

Example 1
::例1

Example 2
::例2

Review
::回顾

Review (Answers)
::回顾(答复)