Course: 9.11 添加和减减液晶显示器为一个分计器的逻辑表达式

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添加和减减逻辑表达式,其中LCD为“一个分计器”
The length of a garden plot is $\begin{align*}\frac{6x^2-5}{2x^2 + 4x - 6}\end{align*}$ . The width of the plot is $\begin{align*}\frac{2x-7}{x+3}\end{align*}$ . How much longer is the garden plot than it is wide?
::园艺的长度是 6x2 - 52x2+4x-6。园艺的宽度是 2x-7x+3。园艺的宽度比宽长多长?

Adding and Subtracting Rational Expressions
::添加和减减理性表达式

Recall when two fractions do not have the same denominator. You have to multiply one or both fractions by a number to create equivalent fractions in order to combine them.
::当两个分数不具有相同的分母时提醒。您需要将一个或两个分数乘以一个或两个分数, 才能创建等效分数, 才能合并。

$\begin{align*}\frac{1}{2} + \frac{3}{4}\end{align*}$

Here, 2 goes into 4 twice. So, we will multiply the first fraction by $\begin{align*}{\color{red}\frac{2}{2}}\end{align*}$ to get a denominator of 4. Then, the two fractions can be added.
::这里, 2 进到 4 中两次。因此, 我们将第一个分数乘以 22 以获得 4 分母。然后, 两个分数可以添加。

$\begin{align*}{\color{red}\frac{2}{2} \cdot} \frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4}\end{align*}$

Once the denominators are the same, the fractions can be combined. We will apply this idea to rational expressions in order to add or subtract ones without like denominators.
::一旦分母相同, 分数可以合并。我们将把这个想法应用到理性表达中, 以便添加或减减那些没有相似分母的分母。

Let's add or subtract the following rational expressions.
::我们加或减以下合理表达式。

Subtract $\begin{align*}\frac{3x-5}{2x+8} - \frac{x^2-6}{x+4}\end{align*}$ .
::减号 3x-52x+8-x2-6x+4。

Factoring the denominator of the first fraction, we have $\begin{align*}2(x+4)\end{align*}$ . The second fraction needs to be multiplied by $\begin{align*}{\color{red}\frac{2}{2}}\end{align*}$ in order to make the denominators the same.
::在计算第一个分数的分母时,我们有2(x+4),第二个分数需要乘以22才能使分数相同。

$\begin{aligned} \frac{3 x - 5}{2 x + 8} - \frac{x^{2} - 6}{x + 4} & = \frac{3 x - 5}{2 (x + 4)} - \frac{x^{2} - 6}{x + 4} \cdot \frac{2}{2} \\ = \frac{3 x - 5}{2 (x + 4)} - \frac{2 x^{2} - 12}{2 (x + 4)} \end{aligned}$
$\begin{align*}\frac{3x-5}{2x+8} - \frac{x^2-6}{x+4} &= \frac{3x-5}{2(x+4)} - \frac{x^2-6}{x+4} \cdot {\color{red}\frac{2}{2}} \\ &= \frac{3x-5}{2(x+4)} - \frac{2x^2-12}{2(x+4)}\end{align*}$
::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 the second rational expression just like you've done before.
::现在分母是相同的, 减去第二个理性表达式, 就像你以前所做的那样。

$\begin{aligned} = \frac{3 x - 5 - (2 x^{2} - 12)}{2 (x + 4)} \\ = \frac{3 x - 5 - 2 x^{2} + 12}{2 (x + 4)} \\ = \frac{- 2 x^{2} + 3 x + 7}{2 (x + 4)} \end{aligned}$
$\begin{align*}&= \frac{3x-5-(2x^2-12)}{2(x+4)} \\ &= \frac{3x-5-2x^2+12}{2(x+4)} \\ &= \frac{-2x^2+3x+7}{2(x+4)}\end{align*}$
::=3x-5-(2x2-12)(x+4)=3x-5-2x2+122(x+4)2x2+3x+72(x+4)

The numerator is not factorable, so we are done.
::分子是不可计数的,所以我们完成了。

Add $\begin{align*}\frac{2x-3}{x+5} + \frac{x^2+1}{x^2-2x-35}\end{align*}$ .
::添加 2x-3x+5+x2+1x2-2x-2x-35。

Factoring the second denominator, we have $\begin{align*}x^2-2x-35=(x+5)(x-7)\end{align*}$ . So, we need to multiply the first fraction by $\begin{align*}{\color{red}\frac{x-7}{x-7}}\end{align*}$ .
::在计算第二个分母时,我们有 x2-2x-35=(x+5)(x-7) 所以,我们需要将第一个分数乘以 x-7x-7。

$\begin{aligned} \overset{F O I L}{\overset{⏞}{\frac{(x - 7)}{(x - 7)} \cdot \frac{(2 x - 3)}{(x + 5)}}} + \frac{x^{2} + 1}{(x - 7) (x + 5)} & = \frac{2 x^{2} - 17 x + 21}{(x - 7) (x + 5)} + \frac{x^{2} + 1}{(x - 7) (x + 5)} \\ = \frac{3 x^{2} - 17 x + 22}{(x - 7) (x + 5)} \end{aligned}$
$\begin{align*}\overbrace{{\color{red}\frac{(x-7)}{(x-7)}} \cdot \frac{(2x-3)}{(x+5)}}^{FOIL} + \frac{x^2+1}{(x-7)(x+5)} &= \frac{2x^2-17x+21}{(x-7)(x+5)} + \frac{x^2+1}{(x-7)(x+5)} \\ &= \frac{3x^2-17x+22}{(x-7)(x+5)}\end{align*}$
:x-7(x-7)(x-7)(x-7)(x-3)(x+5) =2x2-7(x+5)+21(x-7(x+5)+x2+1(x-7)(x-7)(x+5)+x2+1(x-7)(x+5)=3x2-17x+22(x-7)(x+5))

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

Factoring the first denominator, we have $\begin{align*}2x^2+18x+40=2(x^2+9x+20)=2(x+4)(x+5)\end{align*}$ . This is the Lowest Common Denominator, or LCD. The second fraction needs the 2 and the $\begin{align*}(x+4)\end{align*}$ .
::在计算第一个分母时,我们有2x2+18x+40=2(2x2+9x+20)=2(2x+4)(x+5),这是最低的常见分母,或LCD。第二个分母需要2和(x+4)。

$\begin{aligned} \frac{7 x + 2}{2 x^{2} + 18 x + 40} - \frac{6 - x}{x + 5} & = \frac{7 x + 2}{2 (x + 5) (x + 4)} - \frac{6 - x}{x + 5} \cdot \frac{2 (x + 4)}{2 (x + 4)} \\ = \frac{7 x + 2}{2 (x + 5) (x + 4)} - \frac{2 (6 - x) (x + 4)}{2 (x + 5) (x + 4)} \\ = \frac{7 x + 2}{2 (x + 5) (x + 4)} - \frac{48 + 4 x - 2 x^{2}}{2 (x + 5) (x + 4)} \\ = \frac{7 x + 2 - (48 + 4 x - 2 x^{2})}{2 (x + 5) (x + 4)} \\ = \frac{7 x + 2 - 48 - 4 x + 2 x^{2}}{2 (x + 5) (x + 4)} \\ = \frac{2 x^{2} + 3 x - 46}{2 (x + 5) (x + 4)} \end{aligned}$
$\begin{align*}\frac{7x+2}{2x^2+18x+40} - \frac{6-x}{x+5} &= \frac{7x+2}{2(x+5)(x+4)} - \frac{6-x}{x+5}{\color{red}\cdot \frac{2(x+4)}{2(x+4)}} \\ &= \frac{7x+2}{2(x+5)(x+4)} - \frac{2(6-x)(x+4)}{2(x+5)(x+4)} \\ &= \frac{7x+2}{2(x+5)(x+4)} - \frac{48+4x-2x^2}{2(x+5)(x+4)} \\ &= \frac{7x+2-(48+4x-2x^2)}{2(x+5)(x+4)} \\ &= \frac{7x+2-48-4x+2x^2}{2(x+5)(x+4)} \\ &= \frac{2x^2+3x-46}{2(x+5)(x+4)}\end{align*}$
::7x+22x2+18x+4x+4x+4x-6x6-x6+5=7x2x(x+5)(x+4)-6xxx+5(x+4)(2x+4)(x+4)=7x22(x+5)(x+4)(x+4)(x+4)(x+4)(x+4)(x+4)(7x4)(x+4)(x+5(x+5)(x+4)(x+4)(48+4x-2x2(x2)(x+5)(x+4)=7x-2-48-4x2x(x+5)(x+4)(x+4)(2x+5)(x+4)(x+4)(2xx2x2x2x2+3x-462(x+5)(x+4)

Examples
::实例

Example 1
::例1

Earlier, you were asked to find how much longer the garden plot is than it is wide.
::更早之前,有人要求你找到花园的地皮比宽的多长。

We need to subtract the width from the length.
::我们需要从长度中减去宽度。

$\begin{align*}\frac{6x^2-5}{2x^2 + 4x - 6} - \frac{2x-7}{x+3}\end{align*}$
::6x2-52x2+4x-6-2x-7x+3

Factoring the first denominator, we have $\begin{align*}2x^2+4x-6=(2x-2)(x+3)\end{align*}$ . So, we need to multiply the second fraction by $\begin{align*}{\color{red}\frac{2x-2}{2x-2}}\end{align*}$ .
::乘以第一个分母, 我们有 2x2+4x-6=( 2x-2)( x+3) 所以我们需要将第二个分数乘以 2x- 22x-2。

$\begin{aligned} \frac{6 x^{2} - 5}{2 x^{2} + 4 x - 6} - \overset{F O I L}{\overset{⏞}{\frac{(2 x - 2)}{(2 x - 2)} \cdot \frac{(2 x - 7)}{(x + 3)}}} & = \frac{4 x^{2} - 18 x + 14}{2 x^{2} + 4 x - 6} \\ \frac{6 x^{2} - 5}{2 x^{2} + 4 x - 6} - \frac{4 x^{2} - 18 x + 14}{2 x^{2} + 4 x - 6} \\ \frac{6 x^{2} - 5 - (4 x^{2} - 18 x + 14)}{2 x^{2} + 4 x - 6} \\ \frac{6 x^{2} - 5 - 4 x^{2} + 18 x - 14}{2 x^{2} + 4 x - 6} \\ \frac{2 x^{2} + 18 x - 19}{2 x^{2} + 4 x - 6} \end{aligned}$
$\begin{align*}\frac{6x^2-5}{2x^2 + 4x - 6} -\overbrace{{\color{red}\frac{(2x-2)}{(2x-2)}} \cdot \frac{(2x-7)}{(x+3)}}^{FOIL} &= \frac{4x^2-18x+14}{2x^2+4x-6} \\ \frac{6x^2-5}{2x^2 + 4x - 6} - \frac{4x^2-18x+14}{2x^2+4x-6}\\ \frac{6x^2-5-(4x^2-18x+14)}{2x^2+4x-6}\\ \frac{6x^2-5-4x^2+18x-14}{2x^2+4x-6}\\ \frac{2x^2+18x-19}{2x^2+4x-6}\end{align*}$
::6x2 - 522x2+4x4x-2-6-(2x-2-2)(2x-2-2)-2)-(2x-7(x+3)-(2x-7)(x+3)-(2x-7(7)-7(x+3)-(2x)-2x-18x)-(FoIL=4x2-18x2+142x2+4x4x-66x2-52x2+4x4x-4x6-4x2+4x-4x4x-4x2-(4x2-2-18x-66x2-4x)-(4x2-2-18x2-4x-2-18x-192x2+4x-6)

Therefore , the garden plot is $\begin{align*}\frac{2x^2+18x-19}{2x^2+4x-6}\end{align*}$ longer than it is wide
::因此,园艺地的面积比宽2x2+18x-192x2+4x-6长2x2+18x-192x2+4x-6

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

Example 2
::例2

$\begin{align*}\frac{2}{x+1} - \frac{x}{3x+3}\end{align*}$
::2x+1-x3x+3

The LCD is $\begin{align*}3x+3\end{align*}$ or $\begin{align*}3(x+1)\end{align*}$ . Multiply the first fraction by $\begin{align*}\frac{3}{3}\end{align*}$ .
::液晶显示器为3x+3或3(x+1),乘以第一组33。

$\begin{aligned} \frac{2}{x + 1} - \frac{x}{3 x + 3} & = \frac{3}{3} \cdot \frac{2}{x + 1} - \frac{x}{3 (x + 1)} \\ = \frac{6}{3 (x + 1)} - \frac{x}{3 (x + 1)} \\ = \frac{6 - x}{3 (x + 1)} \end{aligned}$
$\begin{align*}\frac{2}{x+1} - \frac{x}{3x+3} &= \frac{3}{3} \cdot \frac{2}{x+1} - \frac{x}{3(x+1)} \\ &= \frac{6}{3(x+1)} - \frac{x}{3(x+1)} \\ &= \frac{6-x}{3(x+1)}\end{align*}$
::2x+1-x3x+3=332x+1-x3(x+1)=63(x+1)-x3(x+1)=6-x3(x+1)

Example 3
::例3

$\begin{align*}\frac{x-10}{x^2+4x-24} + \frac{x+3}{x+6}\end{align*}$
::x- 10x2+4x- 24+x+3x+6

Here, the LCD $\begin{align*}x^2+4x-24\end{align*}$ or $\begin{align*}(x+6)(x-4)\end{align*}$ . Multiply the second fraction by $\begin{align*}\frac{x-4}{x-4}\end{align*}$ .
::在此, LCD x2+4x-24 或 (x+6)(x-4) 乘以 X-4x-4。

$\begin{aligned} \frac{x - 10}{x^{2} + 4 x - 24} + \frac{x + 3}{x + 6} & = \frac{x - 10}{(x + 6) (x - 4)} + \frac{x + 3}{x + 6} \cdot \frac{x - 4}{x - 4} \\ = \frac{x - 10}{(x + 6) (x - 4)} + \frac{x^{2} - x - 12}{(x + 6) (x - 4)} \\ = \frac{x - 10 + x^{2} - x - 12}{(x + 6) (x - 4)} \\ = \frac{x^{2} - 22}{(x + 6) (x - 4)} \end{aligned}$
$\begin{align*}\frac{x-10}{x^2+4x-24} + \frac{x+3}{x+6} &= \frac{x-10}{(x+6)(x-4)} + \frac{x+3}{x+6} \cdot \frac{x-4}{x-4} \\ &= \frac{x-10}{(x+6)(x-4)} + \frac{x^2-x-12}{(x+6)(x-4)} \\ &= \frac{x-10+x^2-x-12}{(x+6)(x-4)} \\ &= \frac{x^2-22}{(x+6)(x-4)}\end{align*}$
::x- 10x2+4x- 4x- 4x- 4x- 4x+3x- 10x(x+6(x-4)+3x+6x+6-x- 4x-4x- 4x- x- 10(x+6(x- 6)(x- 4)+x-2- x- 12(x+6)(x- 4)=x2- 22(x+6)(x-4)

Example 4
::例4

$\begin{align*}\frac{3x^2-5}{3x^2-12} + \frac{x+8}{3x+6}\end{align*}$
::3x2 - 53x2 - 12+x+83x+6

The LCD is $\begin{align*}3x^2-12=3(x-2)(x+2)\end{align*}$ . The second fraction’s denominator factors to be $\begin{align*}3x+6=3(x+2)\end{align*}$ , so it needs to be multiplied by $\begin{align*}\frac{x-2}{x-2}\end{align*}$ .
::LCD为 3x2 - 12=3(x-2)(x+2)x+2),第二个分数分母因数为 3x+6=3(x+2),因此需要乘以 x-2x-2。

$\begin{aligned} \frac{3 x^{2} - 5}{3 x^{2} - 12} + \frac{x + 8}{3 x + 6} & = \frac{3 x^{2} - 5}{3 (x - 2) (x + 2)} + \frac{x + 8}{3 (x + 2)} \cdot \frac{x - 2}{x - 2} \\ = \frac{3 x^{2} - 5}{3 (x - 2) (x + 2)} + \frac{x^{2} + 6 x - 16}{3 (x - 2) (x + 2)} \\ = \frac{3 x^{2} - 5 + x^{2} + 6 x - 16}{3 (x - 2) (x + 2)} \\ = \frac{4 x^{2} + 6 x - 21}{3 (x - 2) (x + 2)} \end{aligned}$
$\begin{align*}\frac{3x^2-5}{3x^2-12} + \frac{x+8}{3x+6} &= \frac{3x^2-5}{3(x-2)(x+2)} + \frac{x+8}{3(x+2)} \cdot \frac{x-2}{x-2} \\ &= \frac{3x^2-5}{3(x-2)(x+2)} + \frac{x^2+6x-16}{3(x-2)(x+2)} \\ &= \frac{3x^2-5+x^2+6x-16}{3(x-2)(x+2)} \\ &= \frac{4x^2+6x-21}{3(x-2)(x+2)}\end{align*}$
::3x2 - 532 - 532 - 532 - 532 - 12+12+12+x+83x+6=3x2 - 532 (x-2(x+2)(x+2)+x2 - 2x-2x-2=3x2 - 532(x-2(x+2)(x+2)+x2+6x-1663(x-2)(x+2)=3x2 - 5+x2+6x-163(x-2)(x+2)=4x2+6x-213(x-2)(x+2)+2)

Review
::回顾

Find the LCD.
::找到液晶显示器

$\begin{align*}x, \ 6x\end{align*}$
::x, 6xxx, 6x

$\begin{align*}x, \ x+1\end{align*}$
::x, x+1 x, x+1

$\begin{align*}x+2, \ x-4\end{align*}$
::x+2, x- 4 x+2, x- 4

$\begin{align*}x, \ x-1, \ x^2 - 1\end{align*}$
::x, x - 1, x2 - 1

Perform the indicated operations.
::执行指定的操作。

$\begin{align*}\frac{3}{x} - \frac{5}{4x}\end{align*}$
::3x-54x

$\begin{align*}\frac{x+2}{x+3} + \frac{x-1}{x^2+3x}\end{align*}$
::x+2x+3+x-1x2+3x

$\begin{align*}\frac{x}{x-7} - \frac{2x+7}{3x-21}\end{align*}$
::xx-7-2x+73x-21

$\begin{align*}\frac{x^2+3x-10}{x^2-4} - \frac{x}{x+2}\end{align*}$
::x2+3x - 10x2 - 4 - xxx+2

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

$\begin{align*}\frac{x-3}{3x^2+x-10} + \frac{3}{x+2}\end{align*}$
::x-3x2+x-10+3x+2

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

$\begin{align*}\frac{-3x^2-10x+15}{10x^2-x-3} + \frac{x+4}{2x+1}\end{align*}$
::-3x2-10x+1510x2-x-3+x+42x+1

$\begin{align*}\frac{8}{2x-5} - \frac{x+5}{2x^2+x-15}\end{align*}$
::82x-5-x+52x2+x-15

$\begin{align*}\frac{2}{x+2} + \frac{3x+16}{x^2-x-6} - \frac{2}{x-3}\end{align*}$
::2x+2+3x+16x2-x-6-2x-3

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

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

添加和减减逻辑表达式,其中LCD为“一个分计器”

Adding and Subtracting Rational Expressions ::添加和减减理性表达式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Adding and Subtracting Rational Expressions
::添加和减减理性表达式

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)