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Addition and Subtraction II
------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0137.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Addition and Subtraction
Addition and Subtraction II
Often we multiply the numerator and the denominator by –1 = to change= the denominator to it’s opposite,
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a
b
= à <= /div>
Addition and Subtraction II
Often we multiply the numerator and the denominator by –1 = to change= the denominator to it’s opposite, i.e.
–a 
–b 
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Example A. Switch the following fractions to opposite
denominator. 
a.
2
3
=3D
a
b
= à <= /div>
b.
a – b 
=3D
d.
x + y  
2x – y 
=3D
Addition and Subtraction II
Often we multiply the numerator and the denominator by –1 = to change= the denominator to it’s opposite, i.e.
–3 
–a 
–b 
------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0140_image002.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhGQACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAX AAEAgAAAAAAAAAIEjI+pVgA7 ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0140_image003.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAB AAEAgAAAAAECAwICRAEAOw== ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0140_image004.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMwACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAx AAEAgAAAAAAAAAIGjI+py40FADs= ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0140_image005.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA/ AAIAgAAAAAAAAAIOhI+py40Bo5y02outQwUAOw== ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0141.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Addition and Subtraction
Example A. Switch the following fractions to opposite
denominator. 
a.
2
3
=3D
a
b
= à <= /div>
–2 
b.
a – b 
=3D
d.
x + y  
2x – y 
=3D
Addition and Subtraction II
Often we multiply the numerator and the denominator by –1 = to change= the denominator to it’s opposite, i.e.
–3 
–3 
–a 
–b 
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Example A. Switch the following fractions to opposite
denominator. 
a.
2
3
=3D
a
b
= à <= /div>
–2 
b.
a – b 
=3D
–(b  – a) 
 3
d.
x + y  
2x – y 
=3D
Addition and Subtraction II
Often we multiply the numerator and the denominator by –1 = to change= the denominator to it’s opposite, i.e.
–3 
–3 
–a 
–b 
=3D
b – a 
 3
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Example A. Switch the following fractions to opposite
denominator. 
a.
2
3
=3D
a
b
= à <= /div>
–2 
b.
a – b 
=3D
–(b  – a) 
 3
d.
x + y  
2x – y 
=3D
–x – y
y – 2x
Addition and Subtraction II
Often we multiply the numerator and the denominator by –1 = to change= the denominator to it’s opposite, i.e.
–3 
–3 
–a 
–b 
=3D
b – a 
 3
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Example A. Switch the following fractions to opposite
denominator. 
a.
2
3
=3D
a
b
= à <= /div>
–2 
b.
a – b 
=3D
–(b  – a) 
 3
d.
x + y  
2x – y 
=3D
–x – y
y – 2x
Addition and Subtraction II
Often we multiply the numerator and the denominator by –1 = to change= the denominator to it’s opposite, i.e.
–3 
–3 
–a 
–b 
=3D
b – a 
 3
When combining two fra= ctions with opposite denominators, we may switch one of t= hem to make their denominators the same.
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Example B. Switch one of the denominators then combine.
x + y  
2x – y 
+
x – 3y
y – 2x
Addition and Subtraction II
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Example B. Switch one of the denominators then combine.
x + y  
2x – y 
+
x – 3y
y – 2x
Opposites=
denominat= ors
Addition and Subtraction II
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Example B. Switch one of the denominators then combine.
x + y  
2x – y 
+
x – 3y
y – 2x
Opposites=
denominat= ors
=3D
x + y  
2x – y 
+
3y – x
2x – y 
switched
Addition and Subtraction II
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Example B. Switch one of the denominators then combine.
x + y  
2x – y 
+
x – 3y
y – 2x
Opposites=
denominat= ors
=3D
x + y  
2x – y 
+
3y – x
2x – y 
switched
x + y + 3y – x   
2x – y 
=3D
4y
2x – y 
=3D
Addition and Subtraction II
Example C. combine
 x + 3  
(4 – x2)
+
  2x – 1
(x2 – 3x + 2)
Factor each denominator.<= /div>
(4 – x2) =3D (2 – x )(2 + x)=  
(x2 – 3x + 2) =3D (x – 2) = (x – 1)
opposites
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Example B. Switch one of the denominators then combine.
x + y  
2x – y 
+
x – 3y
y – 2x
Opposites=
denominat= ors
=3D
x + y  
2x – y 
+
3y – x
2x – y 
switched
x + y + 3y – x   
2x – y 
=3D
Addition and Subtraction II
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Example B. Switch one of the denominators then combine.
x + y  
2x – y 
+
x – 3y
y – 2x
Opposites=
denominat= ors
=3D
x + y  
2x – y 
+
3y – x
2x – y 
switched
x + y + 3y – x   
2x – y 
=3D
4y
2x – y 
=3D
Addition and Subtraction II
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Example B. Switch one of the denominators then combine.
x + y  
2x – y 
+
x – 3y
y – 2x
Opposites=
denominat= ors
=3D
x + y  
2x – y 
+
3y – x
2x – y 
switched
x + y + 3y – x   
2x – y 
=3D
4y
2x – y 
=3D
Addition and Subtraction II
Example C. combine
 x + 3  
(4 – x2)
+
  2x – 1
(x2 – 3x + 2)
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Example B. Switch one of the denominators then combine.
x + y  
2x – y 
+
x – 3y
y – 2x
Opposites=
denominat= ors
=3D
x + y  
2x – y 
+
3y – x
2x – y 
switched
x + y + 3y – x   
2x – y 
=3D
4y
2x – y 
=3D
Addition and Subtraction II
Example C. combine
 x + 3  
(4 – x2)
+
  2x – 1
(x2 – 3x + 2)
Factor each denominator.<= /div>
(4 – x2) 
(x2 – 3x + 2)
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Example B. Switch one of the denominators then combine.
x + y  
2x – y 
+
x – 3y
y – 2x
Opposites=
denominat= ors
=3D
x + y  
2x – y 
+
3y – x
2x – y 
switched
x + y + 3y – x   
2x – y 
=3D
4y
2x – y 
=3D
Addition and Subtraction II
Example C. combine
 x + 3  
(4 – x2)
+
  2x – 1
(x2 – 3x + 2)
Factor each denominator.<= /div>
(4 – x2) =3D (2 – x )(2 + x)=  
(x2 – 3x + 2) =3D
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Example B. Switch one of the denominators then combine.
x + y  
2x – y 
+
x – 3y
y – 2x
Opposites=
denominat= ors
=3D
x + y  
2x – y 
+
3y – x
2x – y 
switched
x + y + 3y – x   
2x – y 
=3D
4y
2x – y 
=3D
Addition and Subtraction II
Example C. combine
 x + 3  
(4 – x2)
+
  2x – 1
(x2 – 3x + 2)
Factor each denominator.<= /div>
(4 – x2) =3D (2 – x )(2 + x)=  
(x2 – 3x + 2) =3D (x – 2) = (x – 1)
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Example B. Switch one of the denominators then combine.
x + y  
2x – y 
+
x – 3y
y – 2x
Opposites=
denominat= ors
=3D
x + y  
2x – y 
+
3y – x
2x – y 
switched
x + y + 3y – x   
2x – y 
=3D
4y
2x – y 
=3D
Addition and Subtraction II
Example C. combine
 x + 3  
(4 – x2)
+
  2x – 1
(x2 – 3x + 2)
Factor each denominator.<= /div>
(4 – x2) =3D (2 – x= )(2 + x) 
(x2 – 3x + 2) =3D (x – 2) (x – 1)
opposites
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 x + 3  
(4 – x2)
+
  2x – 1
(x2 – 3x + 2)
Switch to opposite denominator
Addition and Subtraction II
------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0155.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Addition and Subtraction
 x + 3  
(4 – x2)
+
  2x – 1
(x2 – 3x + 2)
Switch to opposite denominator
=3D
–x – 3
(x2 – 4)
+
  2x – 1
(x2 – 3x + 2)
Addition and Subtraction II
------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0155_image107.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwAH ABIAgAAAAAAAAAIYhGMJqOH9jovz1SnVYil2nkGaF34juCUFADs= ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0155_image108.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA/ AAIAgAAAAAAAAAIOhI8Yy+0Po5wx2YuzRgUAOw== ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0155_image109.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdAAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBy AAIAgAAAAAAAAAIUhI+py+1vgpy02ouz3rz7DIXiUgAAOw== ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0155_image110.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhsgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACw AAEAgAAAAAAAAAIPjI+py+0Hopy0yoez3jwXADs= ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0156.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Addition and Subtraction
 x + 3  
(4 – x2)
+
  2x – 1
(x2 – 3x + 2)
Switch to opposite denominator
=3D
–x – 3
(x2 – 4)
+
  2x – 1
(x2 – 3x + 2)
=3D
Addition and Subtraction II
(x – 2)(x + 2)
+
  2x – 1
(x – 2)(x – 1)
–x – 3
------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0156_image111.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwAH ABIAgAAAAAAAAAIYhGMJqOH9jovz1SnVYil2nkGaF34juCUFADs= ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0156_image112.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA/ AAIAgAAAAAAAAAIOhI8Yy+0Po5wx2YuzRgUAOw== ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0156_image113.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdAAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBy AAIAgAAAAAAAAAIUhI+py+1vgpy02ouz3rz7DIXiUgAAOw== ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0156_image114.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhsgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACw AAEAgAAAAAAAAAIPjI+py+0Hopy0yoez3jwXADs= ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0156_image115.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhygACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAADJ AAEAgAAAAAAAAAIPjI+py+0PIZi02huz3lwXADs= ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0157.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Addition and Subtraction
 x + 3  
(4 – x2)
+
  2x – 1
(x2 – 3x + 2)
Switch to opposite denominator
=3D
–x – 3
(x2 – 4)
+
  2x – 1
(x2 – 3x + 2)
=3D
The LCD is (x – 2)(x + 2)(x – 1),
Addition and Subtraction II
(x – 2)(x + 2)
+
  2x – 1
(x – 2)(x – 1)
–x – 3
------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0157_image116.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDAAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwAH ABIAgAAAAAAAAAIYhGMJqOH9jovz1SnVYil2nkGaF34juCUFADs= ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0157_image117.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhQAADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQA/ AAIAgAAAAAAAAAIOhI8Yy+0Po5wx2YuzRgUAOw== ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0157_image118.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdAAEAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQBy AAIAgAAAAAAAAAIUhI+py+1vgpy02ouz3rz7DIXiUgAAOw== ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0157_image119.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhsgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACw AAEAgAAAAAAAAAIPjI+py+0Hopy0yoez3jwXADs= ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0157_image120.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhygACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAADJ AAEAgAAAAAAAAAIPjI+py+0PIZi02huz3lwXADs= ------=_NextPart_01C95D53.9BA6ABF0 Content-Location: file:///C:/D9279D69/7AdditionandSubtractionII_files/slide0158.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Addition and Subtraction
 x + 3  
(4 – x2)
+
  2x – 1
(x2 – 3x + 2)
Switch to opposite denominator
=3D
–x – 3
(x2 – 4)
+
  2x – 1
(x2 – 3x + 2)
=3D
(x – 2)(x + 2)
+
  2x – 1
(x – 2)(x – 1)
The LCD is (x – 2)(x + 2)(x – 1), hence
Addition and Subtraction II
–x – 3
(x – 2)(x + 2)
+
  2x – 1
(x – 2)(x – 1)
–x – 3
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