1.10 绝对值
章节大纲
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Introduction
::导言What if you were given two points like -8 and 12? How could you find the distance between them on a number line?
::如果给您两个点, 比如 - 8 和 12 点呢? 您如何在数字线上找到它们之间的距离 ?Absolute value is used to find distances, whether the distance of a number from zero, or the distance between numbers, as in this case.
::绝对值用于寻找距离,无论是数字与零的距离,还是数字之间的距离,如本例。Absolute Value
::绝对值The absolute value is the distance between a number and zero on a number line. There are always two numbers on the number line that are the same distance from zero. For instance, the numbers 4 and -4 are each a distance of 4 units away from zero.
::绝对值是数字行中数字和零之间的距离。数字行中总有两个数字与零的距离相同。例如,数字4和 - 4是距离零的4个单位。represents the distance from 4 to zero, which equals 4.
::4 代表4到0的距离,等于4。represents the distance from -4 to zero, which also equals 4.
::4 表示从 - 4 到零的距离, 也等于 4 。Absolute Value
::绝对值For any real number ,
::对于任何真实数字x,
::*% xx 全部 x0 xx (读取反 x) 全部 x <0Absolute value has no effect on a positive number or zero, but changes a negative number into its positive additive inverse .
::绝对值对正数或零没有影响,但将负数改变为正复数。Absolute value situations can also involve unknown variables. For example, suppose the distance from zero is 16. What two points can this represent?
::绝对值情况还可能涉及未知变量。例如,假设与零的距离是16,这代表了哪些两点?Begin by writing an absolute value sentence to represent this situation:
::开始写出绝对值句,以代表这种情况:
::n=缺失值Which two numbers are 16 units from zero?
::哪个数字是零的16个单位?
::n=16 或 n16Absolute value situations can also involve distances from points other than zero. We treat such cases by separating the problem into two independent equations and solving them separately.
::绝对值情况还可能涉及距离零点以外的其他点。 我们通过将问题分为两个独立的方程式分别解决来对待此类情况。Absolute value is very useful in finding the distance between two points on the number line. The distance between any two points and on the number line is or .
::绝对值对于在数字线上找到两个点之间的距离非常有用。数字线上任何两个点a和b之间的距离是 a-b或b-a。For example, the distance from 3 to -1 on the number line is .
::例如,数字线上从3到-1的距离是%3-(-1)(-1)(4)4。We could have also found the distance by subtracting in the opposite order : . This makes sense because the distance is the same whether you are going from 3 to -1 or from -1 to 3.
::我们也可以通过以相反的顺序去掉来找到距离:+1-344。这是有道理的,因为无论从3到1,还是从-1到3,距离是一样的。Note: When we compute the change in and the change in as part of a computation, these values are positive or negative, depending on the direction of movement. In this discussion, “distance” means a positive distance only.
::注:当我们在计算过程中计算 x 和 y 的变化时,这些值是正或负的,视移动方向而定。在本次讨论中,“距离”仅指正距离。The following video gives an overview of finding the distance between two points on a number line, and it also previews the next section on solving basic absolute value equations:
::以下视频概述了在数字线上发现两点之间的距离,并预览了下一节关于解决基本绝对值方程的部分:Solve an Absolute Value Equation
::解决绝对值等量We now want to solve equations involving absolute values. Consider the following equation :
::我们现在要解决包含绝对值的方程式。考虑以下方程式:
::8888888888888888888888888888888888888This means that the distance from the number to zero is 8. There are two numbers that satisfy this condition: 8 and -8.
::这意味着从数字x到零的距离是8,满足这一条件的两个数字是:8和8。When we solve absolute value equations we always consider two possibilities:
::当我们解决绝对价值方程式时 我们总是考虑两种可能性:-
The
expression
inside the absolute value sign is not negative.
::绝对值符号内的表达式不是负值。 -
The expression inside the absolute value sign is negative.
::绝对值符号内的表达式为负值。
Then we solve each equation separately.
::然后我们分别解决每个方程式Examples
::实例Example 1
::例1Evaluate the following absolute values:
::评价以下绝对值:a)
:a) ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Solution:
::解决方案 :Since 25 is a positive number, the absolute value does not change it.
::25+25 因为25是正数 绝对值不会改变b)
:b) 120_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Solution:
::解决方案 :Since -120 is a negative number, the absolute value makes it positive.
::120+120 因为-120是负数 绝对值是正数c)
:c) 3________________________________________________________________________________________________________________________________________________________________________________________________
Solution:
::解决方案 :Since -3 is a negative number, the absolute value makes it positive.
::由于 -3是一个负数, 绝对值使它呈正数 。d)
:d) _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Solution:
::解决方案 :Since 55 is a positive number, the absolute value does not change it.
::5555 由于55是正数,绝对值不会改变。e)
:e) ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Solution:
::解决方案 :Since is a negative number, the absolute value makes it positive.
::由于54是负数, 绝对值使其呈正数 。Example 2
::例2Find the distance between the following points on the number line:
::在数字行找到以下点之间的距离 :a) 6 and 15
::a) 6和15Solution:
::解决方案 :Distance is the absolute value of the difference between the two points.
::距离是两个点之间的差的绝对值。
::距離6 - 159999b) -5 and 8
::b)-5和8Solution:
::解决方案 :
::距离5 - 8 13 13 13c) -3 and -12
:c)-3和-12
Solution:
::解决方案 :
::距离3 - (- 12)\\\\\\\\\9\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\...\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Example 3
::例3Solve the following absolute value equations:
::解决以下绝对值方程式:a)
:a) x3
Solution:
::解决方案 :There are two possibilities: and
::有两种可能性: x=3 和 x3。b)
:b) _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Solution:
::解决方案 :There are two possibilities: and
::有两种可能性:x=10和x=10。Example 4
::例4Solve .
::解决 #% 2x- 7# 6 。Solution:
::解决方案 :Begin by separating this into its separate equations:
::开始将它分离成不同的方程式 :
::2-7=6和2x-7=7=6Solve each equation independently:
::独立解决每个方程式:
::2 - 7=62x-7_ 7_ 62x-7+7=6+7 2x-7+7_ 6+72x=13 2x=1x=132x=12Example 5
::例5A company packs coffee beans in airtight bags. Each bag should weigh 16 ounces, but it is hard to fill each bag to the exact weight. After being filled, each bag is weighed, and if it is more than 0.25 ounces overweight or underweight, it is emptied and repacked. What are the lightest and heaviest acceptable bags?
::公司在密封袋中包装咖啡豆,每个袋子应重16盎司,但很难填满每个袋子的准确重量。 装满后,每个袋子都要称重,如果超过0.25盎司超重或体重不足,则清空并重新包装。 最轻和最重的袋子是什么?Solution:
::解决方案 :The varying quantity is the weight of the bag of coffee beans. Choosing a letter to represent this quantity, and writing an absolute value equation, yields critical values of :
::不同的数量是咖啡豆袋的重量。 选择一个字母 w 来表示这个数量, 并写一个绝对值方程, 得出关键值w :
::-160.25Separate and solve:
::分隔和解析 :
::w-16=0.25w-160.25w=16.25w=15.75w=15.75The lightest bag acceptable is 15.75 ounces, and the heaviest bag acceptable is 16.25 ounces.
::可接受的最轻袋为15.75盎司,可接受的最重袋为16.25盎司。The following video is one more example of an equation with absolute values:
::以下视频是具有绝对值的方程式的又一个例子:Review
::回顾Evaluate the absolute values:
::评估绝对值:Find the distance between the points:
::查找点之间的距离 :-
12 and -11
::12和11(第12和11条) -
5 and 22
::第5和22条 -
-9 and -18
::-9和18 -
-2 and 3
::-2和3 -
-0.012 and 1.067
::-0.012和1.067 -
and
::- 23和78
S olve the absolute value equations and interpret the results by graphing the solutions on a number line:
::解析绝对值方程式, 并用数字线绘制解决方案图解结果 :-
::7u77 -
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ -
::5 -6 9 -
::16+5z5 -
::8x32 -
::m81 -
::X+26 -
::5x-23 -
::511 - 5b -
::8=310y+5 -
::4x-119 -
::8x6+48
Solve the following:
::解决以下问题:-
A company manufactures rulers. Its 12-inch rulers pass quality control if they're within
inch of the ideal length. What is the longest and shortest ruler that can leave the factory?
::一个公司制造统治者。它的12英寸统治者如果在理想长度的132英寸之内,就会通过质量控制。什么是能够离开工厂的最长和最短的统治者? -
The height of a football goal crossbar is 10 feet. The height of the goal post is 30 feet. What is the absolute value of the distance from the crossbar to the top of the goal post?
::足球目标横栏的高度是 10 英尺。 目标柱的高度是 30 英尺。 从 横栏到 目标柱顶部的距离的绝对值是 多少 ? -
The ideal selling price of a car is $17,000. The dealer allows this price to vary $850. What is the lowest price at which this dealer will sell this car?
::汽车的理想销售价格是17,000美元。经销商允许这一价格在850美元之间浮动。 该经销商出售这辆车的最低价格是多少?
Review (Answers)
::回顾(答复)Please see the Appendix.
::请参看附录。 -
The
expression
inside the absolute value sign is not negative.