Course: 9.4 添加和减减有类似分号的有理表达式

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添加和减减有类似引号的理性表达式
A car company constructs x vehicles and adds a new component to each car. If the cost of producing one part of the component is $\begin{aligned} 12 x + 100, \end{aligned}$ and the cost of producing the other part is $\begin{aligned} 14.5 x + 460 \end{aligned}$ , what is the total average cost of producing the new component? Since the number of parts is the same, these average costs will have the same denominator. We discuss how to combine them in this section.
::汽车公司制造x车辆,并为每辆车增加一个新的部件。如果生产一个部件的成本为12x+100,而生产另一个部件的成本为14.5x+460,那么生产新部件的平均成本总额是多少?由于部件的数量相同,这些平均成本将具有相同的分母。我们在本节讨论如何将它们合并。

Adding or Subtracting Rational Expressions With Like Denominators
::以类似分号添加或减缩有理表达式

Recall that when you add or subtract fractions, the denominators must be the same. The same is true of adding and subtracting rational expressions.
::请注意,当您添加或减减分数时,分母必须是相同的。增减理性表达式也是相同的。

Adding or Subtracting Rational Expressions With Like Denominators
::以类似分号添加或减缩有理表达式

For any expressions A, B, and C :
::任何A、B和C等词:

$\begin{aligned} \frac{A}{C} \pm \frac{B}{C} = \frac{A \pm B}{C} . \end{aligned}$

::ACBC=ABC。

Example 1
::例1

Add $\begin{aligned} \frac{x}{x - 6} + \frac{7}{x - 6} \end{aligned}$ .
::添加xx-6+7x-6。

Solution: Since the denominators are the same, add the numerators and simplify if needed.
::解决方案:由于分母是相同的,添加点数,必要时简化。

$\begin{aligned} \frac{x}{x - 6} + \frac{7}{x - 6} = \frac{x + 7}{x - 6} \end{aligned}$

::xx-6+7x-6=x+7x-6

This video by CK-12 demonstrates how to add rational expressions with like denominators.
::CK-12的这段影片展示了如何添加像样分母的理性表达方式。

Example 2
::例2

Subtract $\begin{aligned} \frac{x^{2} - 4}{x - 3} - \frac{2 x - 1}{x - 3} \end{aligned}$ .
::减号 x2 - 4x - 3 - 2x - 1x - 3 。

Solution: You will need to be more careful with subtraction . The entire expression in the 2nd numerator is being subtracted. Distribute $\begin{aligned} - 1 \end{aligned}$ to all of the terms in the 2nd numerator.
::解答 : 您需要更加小心地减法。第二位数中的全部表达式正在被减。将 - 1 分配到第二位数中的所有术语。

$\begin{aligned} \frac{x^{2} - 4}{x - 3} - \frac{2 x - 1}{x - 3} & = \frac{x^{2} - 4 - (2 x - 1)}{x - 3} \\ = \frac{x^{2} - 4 - 2 x + 1}{x - 3} \\ = \frac{x^{2} - 2 x - 3}{x - 3} \end{aligned}$

::x2 - 4x - 3 - 2x - 2x - 1x - 3=x2 - 4 - (2x - 1)x - 3x - 3x2 - 4 - 2x+1x - 3=x2 - 2x - 3x - 3

Now , factor the numerator and simplify if possible.
::现在,计数器,并尽可能简化。

$\begin{aligned} \frac{x^{2} - 2 x - 3}{x - 3} = \frac{(x - 3) (x + 1)}{x - 3} = x + 1 \end{aligned}$

::x2-2x-3x-3=(x-3)(x+1)x-3=x1)

Example 3
::例3

Add $\begin{aligned} \frac{x + 7}{2 x^{2} + 14 x + 20} + \frac{x + 1}{2 x^{2} + 14 x + 20} \end{aligned}$ .
::添加 x+72x2+14x+20+x+12x2+14x+20。

Solution: Add the numerators. Simplify if possible .
::解决方案: 添加分子。如果可能, 简化。

$\begin{aligned} \frac{x + 7}{2 x^{2} + 14 x + 20} + \frac{x + 1}{2 x^{2} + 14 x + 20} & = \frac{2 x + 8}{2 x^{2} + 14 x + 20} \\ = \frac{2 (x + 4)}{2 (x + 5) (x + 2)} \\ = \frac{(x + 4)}{(x + 5) (x + 2)} \end{aligned}$

::x+72x2+14x+20+x+12x2+14x+20=2x82x2+14x+20=2(x+4)2(x+5)(x+2)=(x+4)(x+5)(x+2)=(x+4)(x+5)(x+2)

by CK-12 demonstrates how to add and subtract rational expressions with like denominators.
::以 CK-12 演示如何以类似分母增减理性表达式。

Example 4
::例4

Subtract $\begin{aligned} \frac{x^{2} + 2}{4 x^{2} - 4 x - 3} - \frac{x^{2} - 2 x + 1}{4 x^{2} - 4 x - 3} \end{aligned}$ .
::减号 x2+24x2 - 4x - 3 - x2 - 2x+14x2 - 4x - 3 。

Solution: Distribute $\begin{aligned} - 1 \end{aligned}$ to all the terms in the 2nd denominator, and then add.

$\begin{aligned} \frac{x^{2} + 2}{4 x^{2} - 4 x - 3} - \frac{x^{2} - 2 x + 1}{4 x^{2} - 4 x - 3} & = \frac{x^{2} + 2 - (x^{2} - 2 x + 1)}{4 x^{2} - 4 x - 3} \\ = \frac{x^{2} + 2 - x^{2} + 2 x - 1}{4 x^{2} - 4 x - 3} \\ = \frac{2 x + 1}{4 x^{2} - 4 x - 3} \end{aligned}$

::解析度: 将 -1 分布到 2nd 立方体中的所有术语中, 然后添加 x2+24x2 - 4x2 - 4x3 - x2 - 2x2 - 2x+14x2 - 4x2=x2+2 - (x2 - 2x+1 x1) 4x2 - 4x - 4x3=x2 - 2x2+2+2 - 14x2 - 4x3=2+14x2 - 4x3

At this point, factor the denominator to see if any factors cancel with the numerator.
::在此点,将分母计数,以查看是否有因数与分子取消。

$\begin{aligned} \frac{2 x + 1}{4 x^{2} - 4 x - 3} = \frac{2 x + 1}{(2 x + 1) (2 x - 3)} = \frac{1}{2 x - 3} \end{aligned}$

::2x+14x2-4x-3=2x1(2x+1)(2x-3)=12x-3

Example 5
::例5

A car company constructs $\begin{aligned} x \end{aligned}$ vehicles and adds a new component to each car. If the cost of producing one part of the component is $\begin{aligned} 12 x + 100, \end{aligned}$ and the cost of producing the other part is $\begin{aligned} 14.5 x + 460 \end{aligned}$ , what is the total average cost of producing the new component?
::如果生产一个部件的成本是12x+100,而生产另一个部件的成本是14.5x+460,那么生产新部件的平均总成本是多少?

Solution: The average cost of producing one part is the total cost divided by the number of parts. For the first part, it is $\begin{aligned} \frac{12 x + 100}{x} \end{aligned}$ . For the 2nd part, it is $\begin{aligned} \frac{14.5 x + 460}{x} . \end{aligned}$ The total average cost is their sum.
::解决办法:生产一个部件的平均成本是总成本除以部件数量。第一部分为12x+100x。第二部分为14.5x+460x。总平均成本是总成本。

$\begin{aligned} \frac{12 x + 100}{x} + \frac{14.5 x + 460}{x} = \frac{26.5 x + 560}{x} \end{aligned}$

::12x+100x+14.5x+460x=26.5x+560x

WARNING
::警告

For expressions A , B , and C :
::A、B和C的表述:

$\begin{aligned} \frac{A}{B + C} \neq \frac{A}{B} + \frac{A}{C} \end{aligned}$

::AB+AC(AB+C+AB+AC) (AB+C+AB+AC) (AB+C+AB+AC)

For example,
$\begin{aligned} \frac{16}{8 + 4} & = \frac{4}{3} \\ \neq \frac{16}{8} + \frac{16}{4} = 2 + 4 = 6. \end{aligned}$

::例如,168+4=43168+164=2+4=6。

Feature: Spread It Out!
::传播它!

by Meredith Beaton
::梅雷迪思·贝顿(签名)

Imagine the George Washington Bridge in New York City, the Golden Gate Bridge in San Francisco, the Seven Mile Bridge in Florida, and the Lake Pontchartrain Causeway in Louisiana during a bumper-to-bumper traffic jam. There must be tens of thousands of pounds of weight on these bridges, and yet none of them collapse.
::想象一下纽约市的乔治·华盛顿大桥、旧金山的金门大桥、佛罗里达的七英里大桥和路易斯安那州庞查特雷恩湖的高速公路。这些桥上一定有数万磅重,但都没有倒塌。

Why It Matters
::为何重要

Bridges are designed to be safe and stable, even in the worst conditions. One of the biggest concerns engineers have are not storms, weather, or natural disasters, but the weight of the vehicles on the bridge. To make sure that weight from vehicles on bridges in the U.S. is distributed evenly, the federal government has rules for the construction of bridges and also for the vehicles traveling on bridges.
::即便在条件最差的情况下,桥梁也是安全稳定的。工程师们最关心的不是风暴、天气或自然灾害,而是桥上车辆的重量。为了确保美国桥上的车辆重量分布均衡,联邦政府对桥梁的建造和桥梁上的车辆都有规则。

There is actually a federal law that states that vehicles with two or more axles cannot exceed the weight computed by the Bridge Formula , designed to make sure the weight within the vehicle is distributed across the axles in such a way that it does not put too much stress on a bridge. This is why some semi-trucks have two trailers—to appropriately distribute the weight across multiple axles. The maximum weight any truck in the U.S. can carry is determined using a rational expression that takes into account the distance between the axles (L) and the number of axles (N) of the truck:
::实际上,有联邦法律规定,有两个或两个以上轴的车辆不能超过“桥梁公式”计算出的重量,其目的是确保车辆的重量分布在轴上,不会给桥梁造成太大的压力。这就是为什么一些半卡车有两辆拖车——在多个轴上适当分配重量的原因。在美国,任何卡车可以携带的最大重量都是用考虑到车轴(L)与车轴(N)之间的距离的合理表达方式确定的。

$\begin{aligned} W = 500 [\frac{L N}{N - 1} + 12 N + 36] . \end{aligned}$

::W=500[LNN-1+12N+36]。

by Barb explains the top 13 facts related to the Federal Bridge Gross Weight Formula.
::Barb解释与联邦桥毛重公式有关的前13个事实。

Summary
::摘要

To add or subtract rational expressions with like denominators, add or subtract the numerators and combine over a common denominator. Simplify the result if possible.
::添加或减去带有类似分母的理性表达式,添加或减去点数并结合到一个共同分母上方。如果可能,简化结果。

Review
::回顾

Add or subtract the following rational expressions:
::添加或减去以下合理表达式:

1. $\begin{aligned} \frac{2}{x} + \frac{5}{x} \end{aligned}$
::1. 2x+5x

2. $\begin{aligned} \frac{5}{2 x} + \frac{7}{2 x} \end{aligned}$
::2. 52x+72x

3. $\begin{aligned} \frac{6}{5 x} + \frac{3 - 2 x}{5 x} \end{aligned}$
::3. 65x+3-2x5x

4. $\begin{aligned} \frac{3}{x} + \frac{x + 1}{x} \end{aligned}$
::4. 3x+x+1x

5. $\begin{aligned} \frac{5}{x + 1} + \frac{x - 4}{x + 1} \end{aligned}$
::5. 5x+1+1+x-4x+1

6. $\begin{aligned} \frac{x + 15}{x - 2} - \frac{10}{x - 2} \end{aligned}$
::6. x+15x-2-10x-2

7. $\begin{aligned} \frac{4 x - 3}{x + 3} + \frac{15}{x + 3} \end{aligned}$
::7. 4x-3x+3+15x+3

8. $\begin{aligned} \frac{3 x + 8}{x^{2} - 4 x - 5} + \frac{2 x + 3}{x^{2} - 4 x - 5} \end{aligned}$
::8. 3x+8x2-4x-5+2x+3x2-4x-5

9. $\begin{aligned} \frac{5 x + 3}{x^{2} - 4} - \frac{2 x + 9}{x^{2} - 4} \end{aligned}$
::9. 5x+3x2 - 4 - 2x+9x2 - 4

10. $\begin{aligned} \frac{3 x^{2} + x}{x^{3} - 8} + \frac{4}{x^{3} - 8} - \frac{2 x^{2} - x}{x^{3} - 8} \end{aligned}$
::10. 3x2+x23-8+4x3-8-8-2x2-x23-8

11. $\begin{aligned} \frac{4 x + 3}{x^{2} + 1} - \frac{x + 2}{x^{2} + 1} + \frac{1 - x}{x^{2} + 1} \end{aligned}$
::11. 4x+3x2+1 - x2x2+1 -x2+1+1 -x2+1

12. $\begin{aligned} \frac{18 x^{2} - 7 x + 2}{8 x^{3} + 4 x^{2} - 18 x - 9} - \frac{3 x^{2} + 13 x - 4}{8 x^{3} + 4 x^{2} - 18 x - 9} + \frac{5 x^{2} - 13}{8 x^{3} + 4 x^{2} - 18 x - 9} \end{aligned}$
::12. 18x2-7x+28x3+4x3+4x2-18x9-3x2+13x-48x3+4x2-18x9+5x2-138x3+4x2-18x9

13. $\begin{aligned} \frac{2 x^{2} + 3 x}{x^{3} + 2 x^{2} - 16 x - 32} + \frac{5 x^{2} - 13}{x^{3} + 2 x^{2} - 16 x - 32} - \frac{4 x^{2} + 9 x + 11}{x^{3} + 2 x^{2} - 16 x - 32} \end{aligned}$
::13. 2x2+3x3+3x3+2x2-16x-32+5x2-13x3+5x2-13x3+2x2-16x-32-44x2+9x+11x3+2x2-16x-3

Explore More
::探索更多

1. Explain why $\begin{aligned} \frac{2}{3} + \frac{4}{5} \neq \frac{3}{4} \end{aligned}$ .
::1. 解释为什么23+45+34。

2. Error Analysis: Identify the error and correctly answer the question.
$\begin{aligned} \frac{12}{x + 4} - \frac{3 - x}{x + 4} = \frac{12 - 3 - x}{x + 4} = \frac{9 - x}{x + 4} \end{aligned}$

::2. 错误分析:查明错误并正确回答问题。 12x+4-3-3-xx+4=12-3-xx+4=12-3-xx+4=9-xx+4=9-xx+4

3. Error Analysis: Find the error and fix it.
::3. 错误分析:找出错误并纠正错误。

$\begin{aligned} \frac{2 x^{2}}{x + 4} + \frac{8 x}{x + 4} = \frac{10 x^{3}}{x + 4} = \frac{10 x^{2}}{4} = \frac{5 x^{2}}{2} \end{aligned}$

::2x2x+4+4+8xx+4=10x3x+4=10x24=5x22

4. The height in feet of a planted tree can be modeled by $\begin{aligned} y = \frac{3 x}{6} + \frac{x^{2}}{6} \end{aligned}$ , where x is the number of years since it was planted. How tall is the tree after 6 years?
::4. 植树面积的高度可由Y=3x6+x26做成模型,X是植树之后的年数,6年后的树有多高?

5. One number is 8 times more than another. Their difference is $\begin{aligned} \frac{21}{20} \end{aligned}$ . Find the two numbers.
::5. 一个数字是另一个数字的8倍,两者之差是2120,找出这两个数字。

6. In triangle ABC, side AB is $\begin{aligned} \frac{x^{2} + 5}{x^{2} + 3 x + 2} \end{aligned}$ units long. Side AC is $\begin{aligned} \frac{3 x^{2} - 3 x}{x^{2} + 3 x + 2} \end{aligned}$ units long. How much longer is side AC than side AB?
::6. 在三角ABC, 侧AB是 x2+5x2+3x+2 单位长。侧AC是 3x2-3x2+3x2 单位长。侧AC比侧AB多长?

Answers to Review and Explore More Problems
::对审查和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces the concepts explored in this section:
::尝试这一互动,强化本节所探讨的概念:

Section outline

General

添加和减减有类似引号的理性表达式

Adding or Subtracting Rational Expressions With Like Denominators ::以类似分号添加或减缩有理表达式

Adding or Subtracting Rational Expressions With Like Denominators ::以类似分号添加或减缩有理表达式

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

WARNING ::警告

Feature: Spread It Out! ::传播它!

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers to Review and Explore More Problems ::对审查和探讨更多问题的答复

PLIX ::PLIX

Adding or Subtracting Rational Expressions With Like Denominators
::以类似分号添加或减缩有理表达式

Adding or Subtracting Rational Expressions With Like Denominators
::以类似分号添加或减缩有理表达式

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

WARNING
::警告

Feature: Spread It Out!
::传播它!

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers to Review and Explore More Problems
::对审查和探讨更多问题的答复

PLIX
::PLIX