节: 与不同排引符添加和减减逻辑表达式 | 9.5 与不同排减器添加和减减逻辑表达式

章节大纲

A ferry travels 30 miles from the mainland to an island in the direction of the wind. I n its return trip, the ferry travels 5 miles per hour slower against the direction of the wind. What is the total time for a round trip? Since the rates of speed are different, this average will involve rational expressions with two different denominators. We discuss how to combine them in this section.
::渡轮从大陆30英里向风方向行驶到一个岛屿。轮渡回程中,每小时绕风行驶慢5英里。往返总时间是多少?由于速度不同,这一平均数将包含两个不同分母的合理表达方式。我们在本节中讨论如何将它们结合起来。

Finding the Least Common Denominator
::寻找最不常见的识别器

The least common denominator (LCD) is the least common multiple of the denominators.
::最小公分母(LCD)是最小公分母的多个。

How to Find the Least Common Denominator (LCD)
::如何找到最不常见识别器(LCD)

1. Factor each expression .
::1. 每种表达方式的因数。

2. The LCD is the product of all the factors, where each is raised to the highest power that appears in any of the factorizations of the expressions.
::2. 液晶显示器是所有因素的产物,每个因素都升至表达式因数中出现的最高权力。

Example 1
::例1

Find the LCD if the denominators are $\begin{aligned} x^{2} - 3 x \end{aligned}$ and $\begin{aligned} x^{2} - 9 \end{aligned}$ .
::如果分母是 x2 - 3x 和 x2 - 9, 请查找 LCD 。

Solution: First, we factor.
::解决方案:第一,我们考虑因素。

$\begin{aligned} x^{2} - 3 x & = x (x - 3) \\ x^{2} - 9 & = (x + 3) (x - 3) \end{aligned}$

::x2-3x=x(x-3-3)x2-9=(x+3)(x-3)

The LCD is $\begin{aligned} x (x + 3) (x - 3), \end{aligned}$ since all the factors appear only once in the expressions.
::LCD为x(x+3)(x-3),因为所有因素只在表达式中出现过一次。

Notice that the LCD is a multiple of each expression.
::注意液晶显示器是每个表达式的倍数。

$\begin{aligned} x^{2} - 3 x & = x (x - 3) & LCD = x (x + 3) (x - 3) \\ x^{2} - 9 & = (x + 3) (x - 3) & LCD = x (x + 3) (x - 3) \end{aligned}$

::x2-3x=x(x-3)LCD=x(x+3)(x-3)(x-3)x2-9=(x+3)(x-3)(x-3)(x-3)LCD=x(x+3)(x-3)

Example 2
::例2

Find the LCD if the denominators are $\begin{aligned} x^{3} - 4 x^{2} + 4 x \end{aligned}$ and $\begin{aligned} x^{5} - x^{4} - 2 x^{3} . \end{aligned}$
::如果分母是 x3 - 4x2+4x和 x5 - x4 - 2x3, 则查找液晶。

Solution: Both of these expressions have a GCF, so we need to factor that out 1st. Then we have quadratics that remain and can be factored.
::解决方案:这两种表达方式都有一个全球合作框架,所以我们需要将这一点从第1个因素中排除出来。然后我们还有可保留和可以考虑的二次方位。

$\begin{aligned} x^{3} - 4 x^{2} + 4 x = x {(x - 2)}^{2} \\ x^{5} - x^{4} - 2 x^{3} = x^{3} (x - 2) (x + 1) \end{aligned}$

::x3-4x2+4x=x(x-2)2x5-x4-2x3=x3(x-2)(x+1)

The LCD is $\begin{aligned} x^{3} (x - 2)^{2} (x + 1) \end{aligned}$ . The factor $\begin{aligned} x \end{aligned}$ appears in both expressions, but it has a higher power in the 2nd one. Likewise, the factor $\begin{aligned} x - 2 \end{aligned}$ appears in both expressions, but it has a higher power in the 1st one.
::LCD 是 x3(x-2)2(x+1) 。系数 x 出现在两个表达式中, 但在第二个表达式中具有更高的功率。同样, 系数 x-2 出现在两个表达式中, 但在第一个表达式中具有更高的功率。

Notice again that the LCD is a multiple of each of these expressions.
::再次提醒注意,液晶显示器是每个表达式的倍数。

$\begin{aligned} x^{3} - 4 x^{2} + 4 x & = x (x - 2)^{2} & LCD = x \cdot x^{2} (x - 2)^{2} (x + 1) \\ x^{5} - x^{4} - 2 x^{3} & = x^{3} (x - 2) (x + 1) & LCD = x^{3} (x - 2) (x - 2) (x + 1) \end{aligned}$

::x3 - 4x2+4x=xx(x- 2)2LCD=xxxx2x2x2x2x2x2(x+1x5-x4-x4--2x3x3x3x3(x-2)(x+1)LCD=x3x3x(x-2)(x-2)(x-2)(x-2)(x+1)

This video by CK-12 demonstrates how to find the Least Common Multiple (LCM) of two .
::这段影片由CK-12拍摄,

Addition and Subtraction of Rational Expressions With Unlike Denominators
::配加和减减异异异控号的理性表达式

Let's review how to add fractions when the denominators are not the same, using an example: $\begin{aligned} \frac{4}{15} + \frac{5}{18} \end{aligned}$ .
::以415+518为例,让我们审查当分母不同时如何添加分数。

First, we find the LCD . $\begin{aligned} 15 = 3 \cdot 5 \end{aligned}$ and $\begin{aligned} 18 = 3 \cdot 6 \end{aligned}$ . The factors are 3, 5, and 6. Since each factor has a power of 1, the LCD is $\begin{aligned} 3 \cdot 5 \cdot 6 = 90 \end{aligned}$ .
::首先,我们发现LCD. 15=3×5和18=3×6。系数是3、5和6。由于每个系数的功率为1,因此LCD是3×5×6=90。

Next, we create equivalent fractions by multiplying the numerator and denominator by any factors that are contained in the LCD, but not in the denominator.
::其次,我们通过将分子和分母乘以液晶体中所含、但不在分母中所含的任何因素来创造等效分数。

$\begin{aligned} \frac{4}{15} + \frac{5}{18} & = \frac{4}{3 \cdot 5} + \frac{5}{3 \cdot 6} \\ \frac{4}{3 \cdot 5} \cdot \frac{6}{6} & = \frac{24}{90} \\ \frac{5}{3 \cdot 6} \cdot \frac{5}{5} & = \frac{25}{90} \end{aligned}$

Then, we replace the original fractions with their equivalent fractions and add .
::然后,我们用它们的等量分数来取代原来的分数,然后添加。

$\begin{aligned} \frac{4}{15} + \frac{5}{18} & = \frac{24}{90} + \frac{25}{90} \\ = \frac{49}{90} \end{aligned}$

We multiplied the first fraction by $\begin{aligned} \frac{6}{6} \end{aligned}$ to obtain 90 in the denominator. Recall that a number divided by itself is 1. Therefore , we have not changed the value of the fraction.
::我们将第一个分数乘以66, 以获得分母中的 90。回顾一个数字本身除以是 1 。因此, 我们没有改变该分数的值。

We will now apply the exact same approach to rational expressions.
::我们现在将对理性表达采用同样的做法。

How to Add and Subtract Rational Expressions With Unlike Denominators
::如何添加和减减与异异控号的理性表达式

1. Find the LCD.
::1. 找到液晶显示器。

2. Create equivalent fractions.
::2. 产生相等的分数。

3. Replace the original fraction with the equivalent fraction and combine the numerators.
::3. 用相等的分数取代原分数,并将分子数合并。

4. Factor the numerator to check if it is possible to simplify the fraction.
::4. 计算点数以检查能否简化分数。

Example 3
::例3

Subtract $\begin{aligned} \frac{7 x + 2}{2 x^{2} + 18 x + 40} - \frac{6}{x + 5} \end{aligned}$ .
::减号 7x+22x2+18x+40-6x+5。

Solution: Factoring the first denominator, we have $\begin{aligned} 2 x^{2} + 18 x + 40 = 2 (x^{2} + 9 x + 20) = 2 (x + 4) (x + 5) \end{aligned}$ . The 2nd denominator is prime. The LCD is $\begin{aligned} 2 (x + 4) (x + 5), \end{aligned}$ or the 1st denominator. We only need to create an equivalent fraction for the 2nd one, since the 1st already has the LCD as its denominator.
::解析度: 乘以第一个分母, 我们有 2x2+18x+40=2( x2+9x+20)=2( x+4)(x+5) =2( x+4)(x+5) 。第二分母是正数。 LCD 是 2x+4 (x+5) 或第一分母。我们只需要为第二个分母创建相等的分母, 因为第一分母已经将 LCD 作为其分母。

$\begin{aligned} \frac{7 x + 2}{2 x^{2} + 18 x + 40} - \frac{6 - x}{x + 5} \\ = \frac{7 x + 2}{2 (x + 5) (x + 4)} - \frac{6 - x}{x + 5} \cdot \frac{2 (x + 4)}{2 (x + 4)} & equivalent fraction \\ = \frac{7 x + 2}{2 (x + 5) (x + 4)} - \frac{2 (6 - x) (x + 4)}{2 (x + 5) (x + 4)} \\ = \frac{7 x + 2}{2 (x + 5) (x + 4)} - \frac{48 + 4 x - 2 x^{2}}{2 (x + 5) (x + 4)} & replaced in problem \\ = \frac{7 x + 2 - (48 + 4 x - 2 x^{2})}{2 (x + 5) (x + 4)} \\ = \frac{7 x + 2 - 48 - 4 x + 2 x^{2}}{2 (x + 5) (x + 4)} \\ = \frac{2 x^{2} + 3 x - 46}{2 (x + 5) (x + 4)} & numerator is prime \end{aligned}$

::7x+22x2+18x+40-6-xx+5=7x(x+5)(x+4)-6-xx+5(x+4)-2(x+4)2(x+4)等当量分数=7x22(x+5)(x+4)-2(x+4)(x+4)(x+4)(x+4)=7x+2(x+5)(x4)48+4+4+4x-2(x+5)(x+5)(x+4)取代问题=7x+2(48+4x-2x-2)(x+5)(x+4)=7x+2-4x+4(x+5)(x+5)(x+4)=2x3x-462(x+5)(x+5)(x+4)=2xx+3x+62(x+5)(x+4)

Example 4
::例4

A ferry travels 30 miles from the mainland to an island in the direction of the wind. I n its return trip, the ferry travels 5 miles per hour slower against the direction of the wind. What is the total time for a round trip?
::渡轮从大陆30英里向风方向行驶到一个岛屿,回程中,渡轮每小时行驶5英里,绕风向行驶较慢。往返旅行的总时间是多少?

Solution: Recall that $\begin{aligned} distance = rate \cdot time \end{aligned}$ . The time is the distance divided by the speed (rate). Since the speed (rate) in the direction of the wind is not specified, we can assign it a variable , $\begin{aligned} r \end{aligned}$ . The slower part of the trip, the part against the wind, is $\begin{aligned} r - 5 \end{aligned}$ miles per hour. To find the total time, we can add up the time it takes to complete each part of the trip.
::解答: 记住距离=时间。时间是速度( 速度) 的距离。由于风向的速度( 速度) 没有指定, 我们可以给它指定一个变量, r。行程中较慢的部分, 与风的相对部分, 每小时为 r-5 英里。要找到总时间, 我们可以将完成每次行程所需的时间加起来。

$\begin{aligned} time from the mainland to the island & = \frac{30}{r} \\ time from the island to the mainland & = \frac{30}{r - 5} \\ total time & = \frac{30}{r} + \frac{30}{r - 5} \\ = \frac{30}{r} \cdot \frac{r - 5}{r - 5} + \frac{30}{r - 5} \cdot \frac{r}{r} \\ = \frac{30 r - 150}{r (r - 5)} + \frac{30 r}{r (r - 5)} \\ = \frac{60 r - 150}{r^{2} - 5 r} \end{aligned}$

::从大陆到该岛的时间=30小时从该岛到大陆的总时间=30-5小时 =30r+30r-5=30r-5r-5r-5r-5_30r-5r-5_5r=30r-150r(r-5)+30r-150r(r-5)+30r-(r-5)=60r-150r-2-5r

For some context, say the ferry travels at 15 miles per hour. Then the time of the round trip would be
::在某种背景下,如果说渡轮每小时15英里的行驶时速,那么往返旅行的时间将是:

$\begin{aligned} \frac{60 r - 150}{r^{2} - 5 r} = \frac{60 (15) - 150}{15^{2} - 5 (15)} = \frac{750}{150} = 5. \end{aligned}$
It would take 5 hours to complete a round trip.
::60-150r2-5r=60(15)-150152-5(15)=750150=5.完成往返旅行需要5小时。

by CK-12 demonstrates an application of adding and subtracting rational expressions.
::CK-12 显示加减合理表达式的应用。

Example 5
::例5

Add $\begin{aligned} \frac{x + 5}{x^{2} - 3 x} + \frac{3}{x^{2} + 2 x} \end{aligned}$ .
::添加 x+5x2-3x+3x2+2x。

Solution: First factor each denominator to find the LCD. The 1st denominator, factored, is $\begin{aligned} x^{2} - 3 x = x (x - 3) \end{aligned}$ . The 2nd denominator is $\begin{aligned} x^{2} + 2 x = x (x + 2) \end{aligned}$ . Both denominators have an $\begin{aligned} x \end{aligned}$ , so we need to list it only once. The LCD is $\begin{aligned} x (x - 3) (x + 2) \end{aligned}$ .
::解析度: 找到液晶体体的每个分母的第一个系数。系数为 x2-3x=x( x-3) 。系数为 1 分母为 x2 - 3x=x( x-3) 。第二分母为 x2+2x=x( x+2 ) 。两个分母都有 x, 所以我们只需要列出一次。 LCD为 x( x-3)( x+2 ) 。

$\begin{aligned} \frac{x + 5}{x^{2} - 3 x} + \frac{3}{x^{2} + 2 x} = \frac{x + 5}{x (x - 3)} + \frac{3}{x (x + 2)} \end{aligned}$

::x+5x2 - 3x+3x2+2x=x+5x(x-3)+3x(x+2)

T he 1st fraction needs to be multiplied by $\begin{aligned} \frac{x + 2}{x + 2} \end{aligned}$ and the 2nd fraction needs to be multiplied by $\begin{aligned} \frac{x + 3}{x + 3} \end{aligned}$ to create equivalent fractions with the LCD as the denominator.
::第一分数需要乘以 x+2x+2, 第二分数需要乘以 x+3x+3, 以液晶分母创建等效分数。

$\begin{aligned} = \frac{x + 2}{x + 2} \cdot \frac{x + 5}{x (x - 3)} + \frac{3}{x (x + 2)} \cdot \frac{x - 3}{x - 3} \\ = \frac{(x + 2) (x + 5)}{x (x + 2) (x - 3)} + \frac{3 (x - 3)}{x (x + 2) (x - 3)} \end{aligned}$

::=x+2x+2x+2xx+5x(x-3)+3x(x+2xx-3x)+3x(x+2x-3x-3)=(x+2)(x+5)(x+5)x(x+2)(x-3)+3(x-3)x(x+2)(x-3)x(x+2)(x-3)

W e need to FOIL the numerator for the 1st expression and distribute the 3 in the numerator of the 2nd. Lastly, we combine the fractions over a common denominator and combine like terms .
::我们需要将第一个表达式的分子FIL作为第一个表达式的分子, 并在第二个表达式的分子中分配 3。最后, 我们将分数组合到一个共同的分母上, 并合并类似术语。

$\begin{aligned} = \frac{x^{2} + 7 x + 10}{x (x + 2) (x - 3)} + \frac{3 x - 9}{x (x + 2) (x - 3)} \\ = \frac{x^{2} + 10 x + 1}{x (x + 2) (x - 3)} \end{aligned}$

::=x2+7x+10x(x+2)(x-3)+3x-9x(x+2)(x-3)=x2+10x+1x(x+2)(x-3)

The quadratic in the numerator is prime , so we are done.
::分子中的二次曲线是质的,所以我们完成了。

Example 6
::例6

Subtract $\begin{aligned} \frac{x - 1}{x^{2} + 5 x + 4} - \frac{x + 2}{2 x^{2} + 13 x + 20} \end{aligned}$ .
::减号 x-1x2+5x+4-x+22x2+13x+20。

Solution: To find the LCD, we need to factor the denominators.
::解答:要找到液晶,我们需要考虑分母。

$\begin{aligned} x^{2} + 5 x + 4 & = (x + 1) (x + 4) \\ 2 x^{2} + 13 x + 20 & = (2 x + 5) (x + 4) \\ LCD & = (x + 1) (2 x + 5) (x + 4) \end{aligned}$

::x2+5x+4=(x+1)(x+4)(x+4)2x2+13x+20=(2x+5)(x+4)LCD=(x+1)(2x+5)(x+4)

Next, we create equivalent fractions and combine the terms in the numerator.
::接下来,我们创造等效的分数, 并在分子中合并术语。

$\begin{aligned} \frac{x - 1}{x^{2} + 5 x + 4} - \frac{x + 2}{2 x^{2} + 13 x + 20} \\ = \frac{x - 1}{(x + 1) (x + 4)} - \frac{x + 2}{(2 x + 5) (x + 4)} \\ = \frac{2 x + 5}{2 x + 5} \cdot \frac{x - 1}{(x + 1) (x + 4)} - \frac{x + 2}{(2 x + 5) (x + 4)} \cdot \frac{x + 1}{x + 1} \\ = \frac{(2 x + 5) (x - 1) - (x + 2) (x + 1)}{(x + 1) (2 x + 5) (x + 4)} \\ = \frac{2 x^{2} + 3 x - 5 - (x^{2} + 3 x + 2)}{(x + 1) (2 x + 5) (x + 4)} \\ = \frac{2 x^{2} + 3 x - 5 - x^{2} - 3 x - 2}{(x + 1) (2 x + 5) (x + 4)} \\ = \frac{x^{2} - 7}{(x + 1) (2 x + 5) (x + 4)} \end{aligned}$

::x - 1x2+5x+4x+4x+4x+4x+4x+2x2x2x2xx+5(x+4)=2x+52x+5x2(2x+4)=2x+52x+5x1x1(x1+1x+4)x+2+2x5(x+4x+4x+4x4x4x4x+4x4x4x+4x+4x2x+2x2x2+2x2x+5)(x1x+5)(2x+4)=2x+52x+5x+5x-2x-2x-3x-2x-3x2(x2x+5)(x2x+5)(x+2x+2x+5)(x+4)x=x2x7(x+1)(2x+5)(2x+5)(x+4)

by Mathispower4u demonstrates how to add and subtract rational expressions with unlike denominators.
::在 Mathispower4u 中演示如何以不同的分母来增减理性表达式。

WARNING
::警告

Make sure to distribute $\begin{aligned} - 1 \end{aligned}$ to ALL of the terms in the numerator of the 2nd fraction. For example, consider $\begin{aligned} \frac{3 x - 5}{2 x + 8} - \frac{x^{2} - 6}{x + 4} \end{aligned}$ .
::确保将 -1 分布到第二分数的分子数中的所有术语中。例如, 考虑 3x-52x+8- x2-6x+4 。

Factoring the denominator of the 1st fraction, we have $\begin{aligned} 2 (x + 4) \end{aligned}$ . The 2nd fraction needs to be multiplied by $\begin{aligned} \frac{2}{2} \end{aligned}$ in order to make the denominators the same.
::乘以第1分数的分母, 我们有2(x+4), 第二分数需要乘以 22 才能使分数相同。

$\begin{aligned} \frac{3 x - 5}{2 x + 8} - \frac{x^{2} - 6}{x + 4} & = \frac{3 x - 5}{2 (x + 4)} - \frac{x^{2} - 6}{x + 4} \cdot \frac{2}{2} \\ = \frac{3 x - 5}{2 (x + 4)} - \frac{2 x^{2} - 12}{2 (x + 4)} \end{aligned}$

::3x-52x+8-x2-6x+4=3x-52(x+4)-x2-6x+4)-x2-6x+4__22=3x-52(x+4)-2x2-122(x+4)

Now that the denominators are the same, subtract . Notice the mistake!
::现在分母是相同的,扣除。注意错误!

$\begin{aligned} = \frac{3 x - 5 - (2 x^{2} - 12)}{2 (x + 4)} \\ \neq \frac{3 x - 5 - 2 x^{2} - 12}{2 (x + 4)} \\ \neq \frac{- 2 x^{2} + 3 x - 17}{2 (x + 4)} \end{aligned}$

::=3x-5-(2x2-12)、2x+4)、3x-5-2x2-122(x+4) -2x2+3x-172(x+4)

Summary
::摘要

To add or subtract rational expressions with unlike denominators, find the LCD, create equivalent fractions, replace the original fraction with its equivalent form, combine, and simplify.
::要添加或减去与分母不同的理性表达式,请找到液晶,创建等效分数,用等效形式替换原分数,合并和简化。

Review
::回顾

Find the LCD.
::找到液晶显示器

1. $\begin{aligned} x^{2} - 9, x^{2} - x - 6 \end{aligned}$
::1. x2-9, x2-x-6

2. $\begin{aligned} x^{2} - 4, x^{2} + 4 x + 4, x^{2} + 3 x - 10 \end{aligned}$
::2. x2-4, x2+4x+4, x2+3x-10

Perform the indicated operation.
::执行指示的操作。

3. $\begin{aligned} \frac{3}{x} - \frac{5}{4 x} \end{aligned}$
::3. 3x-54x

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

5. $\begin{aligned} \frac{x - 5}{4 x} + \frac{3}{x + 2} \end{aligned}$
::5. x-54x+3x+2

6. $\begin{aligned} \frac{x^{2} + 3 x - 10}{x^{2} - 4} - \frac{x}{x + 2} \end{aligned}$
::6. x2+3x-10x2-4-xx2 -xxxx+2

7. $\begin{aligned} \frac{5 x + 14}{2 x^{2} - 7 x - 15} - \frac{3}{x - 5} \end{aligned}$
::7. 5x+142x2-7x-15-3x-5

8. $\begin{aligned} \frac{x - 3}{3 x^{2} + x - 10} + \frac{3}{x + 2} \end{aligned}$
::8. x-33x2+x-10+3x+2

9. $\begin{aligned} \frac{x + 1}{6 x + 2} + \frac{x^{2} - 7 x}{12 x^{2} - 14 x - 6} \end{aligned}$
::9. x+16x+2+x2+x2-7x12x2-14x-6

10. $\begin{aligned} \frac{2}{x + 2} + \frac{3 x + 16}{x^{2} - x - 6} - \frac{2}{x - 3} \end{aligned}$
::10. 2x+2+3x+16x2-x-6-2x-3

11. $\begin{aligned} \frac{6 x^{2} + 4 x + 8}{x^{3} + 3 x^{2} - x - 3} + \frac{x - 4}{x^{2} - 1} - \frac{3 x}{x^{2} + 2 x - 3} \end{aligned}$
::11. 6x2+4x+8x3+3x2-3x2 -3x+x-4x2 -1-3x2+2x-3

12. $\begin{aligned} \frac{3 x}{x^{2} - 3 x - 10} + \frac{x + 1}{x^{2} - 2 x - 15} - \frac{2}{x^{2} + 5 x + 6} \end{aligned}$
::12. 3xx2-3x-3x-10+x1x2-2-2x-2-2x-2x-15-2x2+5x+6

13. $\begin{aligned} \frac{7 + x}{x^{2} - 2 x} - \frac{x - 6}{3 x^{2} + 5 x} - \frac{x + 4}{3 x^{2} - x - 10} \end{aligned}$
::13. 7+xx2-2x-x-x-x-63x2+5x-x-x+43x2-x-10

14. $\begin{aligned} \frac{3 x + 2}{x^{2} - 1} - \frac{10 x - 7}{5 x^{2} + 5 x} + \frac{3}{x - 1} \end{aligned}$
::14. 3x+2x2-1-10x-75x2+5x+3x-1

Explore More
::探索更多

1. Discuss the steps in the process you go through when adding two rational expressions with different denominators. Discuss how you find the least common denominator (LCD) when adding rational expressions, and how you use this LCD to find equivalent rational expressions that you can add.
::1. 讨论在添加两个带有不同分母的理性表达式时所经历的过程的步骤。讨论在添加理性表达式时如何找到最小的公分母(LCD),以及如何使用这个 LCD来找到可以添加的等效的理性表达式。

2. One part of a line segment measures $\begin{aligned} \frac{3}{x - 2} \end{aligned}$ . The other part of the segment measures $\begin{aligned} \frac{2}{x + 1} \end{aligned}$ . What is the total length of the line segment?
::2. 某一行段的某一部分为3x-2;该部分的另一部分为2x+1。该行段的总长度是多少?

3. The length of a garden plot is $\begin{aligned} \frac{6 x^{2} - 5}{2 x^{2} + 4 x - 6} \end{aligned}$ . The width of the plot is $\begin{aligned} \frac{2 x - 7}{x + 3} \end{aligned}$ . How much longer is the garden plot than it is wide?
::3. 园艺的长度为6x2-52x2+4x-6。园艺的宽度为2x-7x+3。园艺的宽度比宽多久?

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

章节大纲

Finding the Least Common Denominator ::寻找最不常见的识别器

How to Find the Least Common Denominator (LCD) ::如何找到最不常见识别器(LCD)

Example 1 ::例1

Example 2 ::例2

Addition and Subtraction of Rational Expressions With Unlike Denominators ::配加和减减异异异控号的理性表达式

How to Add and Subtract Rational Expressions With Unlike Denominators ::如何添加和减减与异异控号的理性表达式

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

WARNING ::警告

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

Finding the Least Common Denominator
::寻找最不常见的识别器

How to Find the Least Common Denominator (LCD)
::如何找到最不常见识别器(LCD)

Example 1
::例1

Example 2
::例2

Addition and Subtraction of Rational Expressions With Unlike Denominators
::配加和减减异异异控号的理性表达式

How to Add and Subtract Rational Expressions With Unlike Denominators
::如何添加和减减与异异控号的理性表达式

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

WARNING
::警告

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复