2018年9月4日火曜日

MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical &

MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical &

Presentation on theme: "MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical &"— Presentation transcript:

1 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §6.6 Rational Equations 
2 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §6.4 → Complex Rational Expressions  Any QUESTIONS About HomeWork §6.4 → HW-21 6.4 MTH 55 
3 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 3 Bruce Mayer, PE Chabot College Mathematics Solving Rational Equations  In previous Lectures, we learned how to simplify expressions. We now learn to solve a new type of equation. A rational equation is an equation that contains one or more rational expressions. Some examples:  We want determine the value(s) for x that make these Equations TRUE 
4 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 4 Bruce Mayer, PE Chabot College Mathematics To Solve a Rational Equation 1.List any restrictions that exist. Numbers that make a denominator equal 0 canNOT possibly be solutions. 2.CLEAR the equation of FRACTIONS by multiplying both sides by the LCM of ALL the denominators present 3.Solve the resulting equation using the addition principle, the multiplication principle, and the Principle of Zero Products, as needed. 4.Check the possible solution(s) in the original equation. 
5 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION - Because no variable appears in the denominator, no restrictions exist. The LCM of 5, 2, and 4 is 20, so we multiply both sides by 20 Using the multiplication principle to multiply both sides by the LCM. Parentheses are important! Using the distributive law. Be sure to multiply EACH term by the LCM Simplifying and solving for x. If fractions remain, we have either made a mistake or have not used the LCM of ALL the denominators. 
6 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 6 Bruce Mayer, PE Chabot College Mathematics Checking Answers  Since a variable expression could represent 0, multiplying both sides of an equation by a variable expression does NOT always produce an Equivalent Equation COULD be Multiplying by Zero and Not Know it  Thus checking each solution in the original equation is essential. 
7 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION - Note that x canNOT equal 0. The Denominator LCM is 15x. 
8 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Solve  CHECK tentative Solution, x = 5  The Solution x = 5 CHECKS  
9 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION - Note that x canNOT equal 0. The Denom LCM is x  Thus by Zero Products: x = 3 or x = 4 
10 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Solve  CHK: For x = 3 For x = 4  Both of these check, so there are two solutions; 3 and 4 
11 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION  Note that y canNOT equal 3 or −3. We multiply both sides of the equation by the Denom LCM. 
12 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION - Note that x canNOT equal 1 or −1. Multiply both sides of the eqn by the LCM Because of the restriction above, 1 must be rejected as a solution. This equation has NO solution. 
13 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION: Because the left side of this equation is undefined when x is 0, we state at the outset that x  0.  Next, we multiply both sides of the equation by the LCD, 4x: Multiplying by the LCD to clear fractions 
14 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLN cont. Using the distributive law Locating factors equal to 1 Removing factors equal to 1 Using the distributive law 
15 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 15 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLN cont. This should check since x  0.  CHECK 8 
16 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 16 Bruce Mayer, PE Chabot College Mathematics Rational Eqn CAUTION  When solving rational equations, be sure to list any Division-by-Zero restrictions as part of the first step.  Refer to the restriction(s) as you proceed 
17 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION: To find all restrictions and to assist in finding the LCD, we factor:  Note that to prevent division by zero x  3 and x  −3.  Next multiply by the LCD, (x + 3)(x – 3), and then use the distributive law 
18 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLUTION: By LCD Multiplication  Remove factors Equal to One and solve the resulting Eqn Keep in Mind any restrictions 
19 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Solve  SOLN cont.: Multiply and Collect Similar terms  A check will confirm that 22 is the solution 
20 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Eqn with NO Soln To avoid division by zero, exclude from the expression domain 1 and –1, since these values make one or more of the denominators in the equation equal 0. Distributive property Solve. 3 x – 1 = 2 x + 1 – 6 x 2 – 1 = 3 x – 1 2 x + 1 – 6 x 2 – 1 ( x – 1)( x + 1) = 3 x – 1 2 x + 1 – 6 x 2 – 1 ( x – 1)( x + 1) = – 63( x + 1)2( x – 1) = – 63 x + 32 x + 2 = 6 x + 5 = 1 x Multiply each side by the LCD, ( x –1)( x + 1). Multiply. Distributive property Combine terms. Subtract 5. 
21 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Eqn with NO Soln Solve. 3 x – 1 = 2 x + 1 – 6 x 2 – 1 Since 1 is not in the domain, it cannot be a solution of the equation. Substituting 1 in the original equation shows why. Check: = 3 x – 1 2 x + 1 – 6 x 2 – 1 = 3 1 – 1 2 1 + 1 – 6 1 2 – 1 = 3 0 2 2 – 6 0 Since division by 0 is undefined, the given equation has no solution, and the solution set is ∅. 
22 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Fcn to Eqn  Given Function:  Find all values of a for which On Board  SOLUTION  On Board  By Function Notation:  Thus Need to find all values of a for which f(a) = 4 
23 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Fcn to Eqn  Solve for a:  First note that a  0. To solve for a, multiply both sides of the equation by the LCD, a: Multiplying both sides by a. Parentheses are important. Using the distributive law 
24 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Fcn to Eqn  CarryOut Solution  CHECK Simplifying Getting 0 on one side Factoring Using the principle of zero products  STATE: The solutions are 5 and −1. For a = 5 or a = −1, we have f(a) = 4. 
25 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 25 Bruce Mayer, PE Chabot College Mathematics Rational Equations and Graphs  One way to visualize the solution to the last example is to make a graph. This can be done by graphing; e.g., Given  Find x such that f(x) = 4 
26 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 26 Bruce Mayer, PE Chabot College Mathematics Rational Equations and Graphs  Graph the function, and on the same grid graph y = g(x) = 4  We then inspect the graph for any x-values that are paired with 4. It appears from the graph that f(x) = 4 when x = 5 or x = −1. 4 5 
27 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 27 Bruce Mayer, PE Chabot College Mathematics Rational Equations and Graphs  Graphing gives approximate solutions  Although making a graph is not the fastest or most precise method of solving a rational equation, it provides visualization and is useful when problems are too difficult to solve algebraically 
28 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 28 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §6.6 Exercise Set 34, 38, 62  Rational Expressions 
29 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 29 Bruce Mayer, PE Chabot College Mathematics All Done for Today Remember: can NOT Divide by ZERO 
30 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 30 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix – 
31 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 31 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table 
32 BMayer@ChabotCollege.edu MTH55_Lec-34_sec_6-6_Rational_Equations.ppt 32 Bruce Mayer, PE Chabot College Mathematics 

ゼロ除算の発見は日本です：
∞？？？    
∞は定まった数ではない・
人工知能はゼロ除算ができるでしょうか：

とても興味深く読みました：2014年2月2日　4周年を超えました：
ゼロ除算の発見と重要性を指摘した：日本、再生核研究所


ゼロ除算関係論文・本

ダ・ヴィンチの名言 格言｜無こそ最も素晴らしい存在
                     

ゼロ除算の発見はどうでしょうか：
Black holes are where God divided by zero：

再生核研究所声明371（2017.6.27）ゼロ除算の講演―　国際会議　
https://ameblo.jp/syoshinoris/entry-12287338180.html

1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12276045402.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12263708422.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12272721615.html
Division By Zero（ゼロ除算）1/0=0、0/0=0、z/0=0
ゼロ除算（ゼロじょざん、division by zero）1/0=0、0/0=0、z/0=0

ソクラテス・プラトン・アリストテレス　その他
https://ameblo.jp/syoshinoris/entry-12328488611.html

ドキュメンタリー 2017: 神の数式 第２回 宇宙はなぜ生まれたのか
https://www.youtube.com/watch?v=iQld9cnDli4
〔NHKスペシャル〕神の数式 完全版 第3回 宇宙はなぜ始まったのか
https://www.youtube.com/watch?v=DvyAB8yTSjs&t=3318s
〔NHKスペシャル〕神の数式 完全版 第1回 この世は何からできているのか
https://www.youtube.com/watch?v=KjvFdzhn7Dc
NHKスペシャル 神の数式 完全版 第4回 異次元宇宙は存在するか
https://www.youtube.com/watch?v=fWVv9puoTSs

再生核研究所声明 411(2018.02.02): 　ゼロ除算発見4周年を迎えて
https://ameblo.jp/syoshinoris/entry-12348847166.html

再生核研究所声明 416(2018.2.20): 　ゼロ除算をやってどういう意味が有りますか。何か意味が有りますか。何になるのですか　－　回答
再生核研究所声明 417(2018.2.23): 　ゼロ除算って何ですか　－　中学生、高校生向き　回答
再生核研究所声明 418(2018.2.24): 　割り算とは何ですか？　ゼロ除算って何ですか　－　小学生、中学生向き　回答
再生核研究所声明 420(2018.3.2): ゼロ除算は正しいですか，合っていますか、信用できますか　－　回答

2018.3.18．午前中　最後の講演：　日本数学会　東大駒場、函数方程式論分科会　講演書画カメラ用　原稿
The Japanese Mathematical Society, Annual Meeting at the University of Tokyo. 2018.3.18.
https://ameblo.jp/syoshinoris/entry-12361744016.html より
再生核研究所声明 424(2018.3.29): 　レオナルド・ダ・ヴィンチとゼロ除算
再生核研究所声明 427(2018.5.8): 神の数式、神の意志　そしてゼロ除算

Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
私は数学を信じない。　アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
1423793753.460.341866474681。

Einstein's Only Mistake: Division by Zero

 

God’s most important commandment

never-divide-by-zero-meme-66

Even more important than “thou shalt not eat seafood”
Published by admin, on October 18th, 2011 at 3:47 pm. Filled under: Never Divide By Zero Tags: commandment, Funny, god, zero • Comments Off on God’s most important commandment
http://thedistractionnetwork.com/.../never-divide.../page/4/

1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12276045402.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12263708422.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12272721615.html
Division By Zero（ゼロ除算）1/0=0、0/0=0、z/0=0
ゼロ除算（ゼロじょざん、division by zero）1/0=0、0/0=0、z/0=0

再生核研究所声明371（2017.6.27）ゼロ除算の講演―　国際会議　https://sites.google.com/site/sandrapinelas/icddea-2017　報告

ソクラテス・プラトン・アリストテレス　その他
https://ameblo.jp/syoshinoris/entry-12328488611.html

Ten billion years ago　DIVISION By ZERO：
https://www.facebook.com/notes/yoshinori-saito/ten-billion-years-ago-division-by-zero/1930645683923690/

One hundred million years ago　DIVISION By ZERO
https://www.facebook.com/.../one-hundred-million-years-ago


ゼロ除算は定義が問題です：

再生核研究所声明　１４８（2014.2.12）　100/0=0,  0/0=0　－　割り算の考えを自然に拡張すると　―　神の意志 https://blogs.yahoo.co.jp/kbdmm360/69056435.html

再生核研究所声明１７１（2014.7.30）掛け算の意味と割り算の意味　―　ゼロ除算100/0=0は自明である？http://reproducingkernel.blogspot.jp/2014/07/201473010000.html


Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.

私は数学を信じない。　アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。1423793753.460.341866474681。

Einstein's Only Mistake: Division by Zero

#divide by zero

TOP DEFINITION
  
A super-smart math teacher that teaches at HTHS and can divide by zero.
Hey look, that genius’s IQ is over 9000!
by Lawlbags! October 21, 2009


Dividing by zero is the biggest epic fail known to mankind. It is a proven fact that a succesful division by zero will constitute in the implosion of the universe.
You are dividing by zero there, Johnny. Captain Kirk is not impressed.

Divide by zero?!?!! OMG!!! Epic failzorz

3
  
Divide by zero is undefined.
Divide by zero is undefined.
by JaWo October 28, 2006

1) The number one ingredient for a catastrophic event in which the universe enfolds and collapses on itself and life as we know it ceases to exist.

2) A mathematical equation such as a/0 whereas a is some number and 0 is the divisor. Look it up on Wikipedia or something. Pretty confusing shit.

3) A reason for an error in programming
Hey, I divided by zero! ...Oh shi-

a/0

Run-time error: '11': Division by zero
by DefectiveProduct September 08, 2006

When even math shows you that not everything can be figured out with math. When you divide by zero, math kicks you in the shins and says "yeah, there's kind of an answer, but it ain't just some number."

It's when mathematicians become philosophers.
Math:
Let's say you have ZERO apples, and THREE people. How many apples does each person get? ZERO, cause there were no apples to begin with

Not-math because of dividing by zero:
Let's say there are THREE apples, and ZERO people. How many apples does each person get? Friggin... How the Fruitcock should I know! How can you figure out how many apples each person gets if there's no people to get them?!? You'd think it'd be infinity, but not really. It could almost be any number, cause you could be like "each person gets 400 apples" which would be true, because all the people did get 400 apples, because there were no people. So all the people also got 42 apples, and a million and 7 apples. But it's still wrong.
by Zacharrie February 15, 2010


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