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Solving Equation= s
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Two expressions set eq= ual to each other is called
an equati= on. <= /div>
Solving Equation= s
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Two expressions set eq= ual to each other is called
an equati= on. To solve= an equation means to find vealue(s) for the vari= ables that makes the equation true. 
Solving Equation= s
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Two expressions set eq= ual to each other is called
an equati= on. To solve= an equation means to find vealue(s) for the vari= ables that makes the equation true.  Equations made from polynomial expressions, rational expressions, or algebraic expressions= are called polynomial equations, rational equations, or = algebraic equations respectively.
Solving Equation= s
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Two expressions set eq= ual to each other is called
an equati= on. To solve= an equation means to find vealue(s) for the vari= ables that makes the equation true.  Equations made from polynomial expressions, rational expressions, or algebraic expressions= are called polynomial equations, rational equations, or = algebraic equations respectively.
Solving Equation= s
We solve polynomial equations by factoring.
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Two expressions set eq= ual to each other is called
an equati= on. To solve= an equation means to find vealue(s) for the vari= ables that makes the equation true.  Equations made from polynomial expressions, rational expressions, or algebraic expressions= are called polynomial equations, rational equations, or = algebraic equations respectively.
Solving Equation= s
We solve polynomial equations by factoring.  = ;
To solve other algebra= ic equations, we have to transform them into polynomial equations.
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Two expressions set eq= ual to each other is called
an equati= on. To solve= an equation means to find vealue(s) for the vari= ables that makes the equation true.  Equations made from polynomial expressions, rational expressions, or algebraic expressions= are called polynomial equations, rational equations, or = algebraic equations respectively.
Solving Equation= s
Example A: 3x2 – 2x =3D 8
We solve polynomial equations by factoring.  = ;
To solve other algebra= ic equations, we have to transform them into polynomial equations.
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Two expressions set eq= ual to each other is called
an equati= on. To solve= an equation means to find vealue(s) for the vari= ables that makes the equation true.  Equations made from polynomial expressions, rational expressions, or algebraic expressions= are called polynomial equations, rational equations, or = algebraic equations respectively.
Solving Equation= s
Example A: 3x2 – 2x =3D 8
We solve polynomial equations by factoring.  = ;
To solve other algebra= ic equations, we have to transform them into polynomial equations.
Set one side to 0,  3x2 – 2x – 8 =3D 0  
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Two expressions set eq= ual to each other is called
an equati= on. To solve= an equation means to find vealue(s) for the vari= ables that makes the equation true.  Equations made from polynomial expressions, rational expressions, or algebraic expressions= are called polynomial equations, rational equations, or = algebraic equations respectively.
Solving Equation= s
Example A: 3x2 – 2x =3D 8
We solve polynomial equations by factoring.  = ;
To solve other algebra= ic equations, we have to transform them into polynomial equations.
Set one side to 0,  3x2 – 2x – 8 =3D 0    factor this;
(3x + 4)(x – 2) =3D 0 
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Two expressions set eq= ual to each other is called
an equati= on. To solve= an equation means to find vealue(s) for the vari= ables that makes the equation true.  Equations made from polynomial expressions, rational expressions, or algebraic expressions= are called polynomial equations, rational equations, or = algebraic equations respectively.
Solving Equation= s
Example A: 3x2 – 2x =3D 8
We solve polynomial equations by factoring.  = ;
To solve other algebra= ic equations, we have to transform them into polynomial equations.
Set one side to 0,  3x2 – 2x – 8 =3D 0    factor this;
(3x + 4)(x – 2) =3D 0  extract answers;
x =3D -4/3, 2
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Equations= of the Form xp/q =3D c
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Equations= of the Form xp/q =3D c
Solve xp/q =3D c by raising both sides to the reciprocal exponent q/p and check your answers afterwards.
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Equations= of the Form xp/q =3D c
Solve xp/q =3D c by raising both sides to the reciprocal exponent q/p and check your answers afterwards.
Example B. Solve x=3D 16
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Equations= of the Form xp/q =3D c
Solve xp/q =3D c by raising both sides to the reciprocal exponent q/p and check your answers afterwards.
Example B. Solve x=3D 16
Raise both sides to -3/2 power.
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Equations= of the Form xp/q =3D c
Solve xp/q =3D c by raising both sides to the reciprocal exponent q/p and check your answers afterwards.
Example B. Solve x=3D 16
x-2/3 =3D 16 <= /span>à
(x-2/3)= -3/2 =3D (16)= -3/2
Raise both sides to -3/2 power.
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Equations= of the Form xp/q =3D c
Solve xp/q =3D c by raising both sides to the reciprocal exponent q/p and check your answers afterwards.
Example B. Solve x=3D 16
x-2/3 =3D 16 <= /span>à
(x-2/3)= -3/2 =3D (16)= -3/2
x=3D 1/64  and it's a solution.
Raise both sides to -3/2 power.
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Equations= of the Form xp/q =3D c
Solve xp/q =3D c by raising both sides to the reciprocal exponent q/p and check your answers afterwards.
Example B. Solve x=3D 16
x-2/3 =3D 16 <= /span>à
(x-2/3)= -3/2 =3D (16)= -3/2
x=3D 1/64  and it's a solution.
Example C. Solve (2x – 3)3/2 =3D -8
Raise both sides to -3/2 power.
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Equations= of the Form xp/q =3D c
Solve xp/q =3D c by raising both sides to the reciprocal exponent q/p and check your answers afterwards.
Example B. Solve x=3D 16
x-2/3 =3D 16 <= /span>à
(x-2/3)= -3/2 =3D (16)= -3/2
x=3D 1/64  and it's a solution.
Example C. Solve (2x – 3)3/2 =3D -8
Raise both sides to -3/2 power.
Raise both sides to 2/3 power.
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Equations= of the Form xp/q =3D c
Solve xp/q =3D c by raising both sides to the reciprocal exponent q/p and check your answers afterwards.
Example B. Solve x=3D 16
x-2/3 =3D 16 <= /span>à
(x-2/3)= -3/2 =3D (16)= -3/2
x=3D 1/64  and it's a solution.
Example C. Solve (2x – 3)3/2 =3D -8
Raise both sides to -3/2 power.
Raise both sides to 2/3 power.
(2x – 3)=3D -8 &agrav= e;
[(2x – 3)]2/3 =3D (-8)2/3 
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Equations= of the Form xp/q =3D c
Solve xp/q =3D c by raising both sides to the reciprocal exponent q/p and check your answers afterwards.
Example B. Solve x=3D 16
x-2/3 =3D 16 <= /span>à
(x-2/3)= -3/2 =3D (16)= -3/2
x=3D 1/64  and it's a solution.
Example C. Solve (2x – 3)3/2 =3D -8
Raise both sides to -3/2 power.
Raise both sides to 2/3 power.
(2x – 3)=3D -8 &agrav= e;
[(2x – 3)]2/3 =3D (-8)2/3 
(2x – 3) =3D 4
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Equations= of the Form xp/q =3D c
Solve xp/q =3D c by raising both sides to the reciprocal exponent q/p and check your answers afterwards.
Example B. Solve x=3D 16
x-2/3 =3D 16 <= /span>à
(x-2/3)= -3/2 =3D (16)= -3/2
x=3D 1/64  and it's a solution.
Example C. Solve (2x – 3)3/2 =3D -8
Raise both sides to -3/2 power.
Raise both sides to 2/3 power.
(2x – 3)=3D -8 &agrav= e;
[(2x – 3)]2/3 =3D (-8)2/3 
(2x – 3) =3D 4
2x =3D 7 à x =3D 7/2
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Equations= of the Form xp/q =3D c
Solve xp/q =3D c by raising both sides to the reciprocal exponent q/p and check your answers afterwards.
Example B. Solve x=3D 16
x-2/3 =3D 16 <= /span>à
(x-2/3)= -3/2 =3D (16)= -3/2
x=3D 1/64  and it's a solution.
Example C. Solve (2x – 3)3/2 =3D -8
Raise both sides to -3/2 power.
Raise both sides to 2/3 power.
(2x – 3)=3D -8 &agrav= e;
[(2x – 3)]2/3 =3D (-8)2/3 
(2x – 3) =3D 4
2x =3D 7 à x =3D 7/2
Since x =3D 7/2 doesn't work because 43/2 = =3D -8,
there is no solution.
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Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD.
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Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
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Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
Example D.  Solve
x – 2
2
=3D
x + 1
4
+ 1
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Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
Example D.  Solve
LCD =3D = (x – 2)(x + 1), multiple the LCD to both sides = of the equation:
x – 2
2
=3D
x + 1
4
+ 1
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Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
Example D.  Solve
LCD =3D<= /span> (x – 2)(x + 1), multiple the LCD to both sides = of the equation:
 
(x – 2= )(x + 1) * [        = ;            &n= bsp;    ]
x – 2
2
=3D
x + 1
4
+ 1
x – 2
2
=3D
x + 1
4
+ 1
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Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
Example D.  Solve
LCD =3D = (x – 2)(x + 1), multiple the LCD to both sides = of the equation:
 
(x ̵= 1; 2)(x + 1) * [        = ;            &n= bsp;    ]
x – 2
2
=3D
x + 1
4
+ 1
(x + 1)
x – 2
2
=3D
x + 1
4
+ 1
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Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
Example D.  Solve
LCD =3D = (x – 2)(x + 1), multiple the LCD to both sides = of the equation:
 
(x ̵= 1; 2)(x + 1) * [        = ;            &n= bsp;    ]
x – 2
2
=3D
x + 1
4
+ 1
(x + 1)
(x – 2)
x – 2
2
=3D
x + 1
4
+ 1
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Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
Example D.  Solve
LCD =3D = (x – 2)(x + 1), multiple the LCD to both sides = of the equation:
 
(x ̵= 1; 2)(x + 1) * [        = ;            &n= bsp;    ]
x – 2
2
=3D
x + 1
4
+ 1
(x + 1)
(x – 2)
(x + 1)(x – 2)
x – 2
2
=3D
x + 1
4
+ 1
------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0169_image013.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdQAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB1 ABMAgAAAAAAAAAJOhI+py+0Po2Shhomz3hzY34XiKH0Viaaoearui7HwTFNyjdd3zrvs1QuuTMKi 6GdMbnbKJgTpjDqY0uoBas16iNostiulgpvfcbVlThcAADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0169_image014.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAB5 AAEAgAAAAAAAAAINjI+py50Ao5zU2YtRAQA7 ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0169_image015.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhhgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACC AA4AgAAAAAAAAAJIhI+py+0Po5wn2Buo3rz7hF3fSJZfaJnqyipo1sZaKL/yLdmsjvcMrwL6hoAX bCUk9pJBjFLJfEqRqKk1Fr1qOdmtt3P8ilkFADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0169_image016.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAB AAEAgAAAAAECAwICRAEAOw== ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0170.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Exponents
Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
Example D.  Solve
LCD =3D = (x – 2)(x + 1), multiple the LCD to both sides = of the equation:
 
(x ̵= 1; 2)(x + 1) * [        = ;            &n= bsp;    ]
2(x + 1) =3D 4(x – 2) + 1*(x + 1)(x &#= 8211; 2)
x – 2
2
=3D
x + 1
4
+ 1
(x + 1)
(x – 2)
(x + 1)(x – 2)
x – 2
2
=3D
x + 1
4
+ 1
------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0170_image017.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdQAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB1 ABMAgAAAAAAAAAJOhI+py+0Po2Shhomz3hzY34XiKH0Viaaoearui7HwTFNyjdd3zrvs1QuuTMKi 6GdMbnbKJgTpjDqY0uoBas16iNostiulgpvfcbVlThcAADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0170_image018.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAB5 AAEAgAAAAAAAAAINjI+py50Ao5zU2YtRAQA7 ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0170_image019.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhhgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACC AA4AgAAAAAAAAAJIhI+py+0Po5wn2Buo3rz7hF3fSJZfaJnqyipo1sZaKL/yLdmsjvcMrwL6hoAX bCUk9pJBjFLJfEqRqKk1Fr1qOdmtt3P8ilkFADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0170_image020.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAB AAEAgAAAAAECAwICRAEAOw== ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0171.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Exponents
Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
Example D.  Solve
LCD =3D = (x – 2)(x + 1), multiple the LCD to both sides = of the equation:
 
(x ̵= 1; 2)(x + 1) * [        = ;            &n= bsp;    ]
2= (x + 1) =3D 4(x – 2) + 1*(x + 1)(x &#= 8211; 2)
2x + 2 = =3D 4x – 8 + x– x – 2
x – 2
2
=3D
x + 1
4
+ 1
(x + 1)
(x – 2)
(x + 1)(x – 2)
x – 2
2
=3D
x + 1
4
+ 1
------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0171_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdQAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB1 ABMAgAAAAAAAAAJOhI+py+0Po2Shhomz3hzY34XiKH0Viaaoearui7HwTFNyjdd3zrvs1QuuTMKi 6GdMbnbKJgTpjDqY0uoBas16iNostiulgpvfcbVlThcAADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0171_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAB5 AAEAgAAAAAAAAAINjI+py50Ao5zU2YtRAQA7 ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0171_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhhgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACC AA4AgAAAAAAAAAJIhI+py+0Po5wn2Buo3rz7hF3fSJZfaJnqyipo1sZaKL/yLdmsjvcMrwL6hoAX bCUk9pJBjFLJfEqRqKk1Fr1qOdmtt3P8ilkFADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0171_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAB AAEAgAAAAAECAwICRAEAOw== ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0172.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Exponents
Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
Example D.  Solve
LCD =3D = (x – 2)(x + 1), multiple the LCD to both sides = of the equation:
 
(x ̵= 1; 2)(x + 1) * [        = ;            &n= bsp;    ]
2= (x + 1) =3D 4(x – 2) + 1*(x + 1)(x &#= 8211; 2)
2x + 2 = =3D 4x – 8 + x– x – 2
2x + 2 = =3D x2 + 3x – 10
x – 2
2
=3D
x + 1
4
+ 1
(x + 1)
(x – 2)
(x + 1)(x – 2)
x – 2
2
=3D
x + 1
4
+ 1
------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0172_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdQAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB1 ABMAgAAAAAAAAAJOhI+py+0Po2Shhomz3hzY34XiKH0Viaaoearui7HwTFNyjdd3zrvs1QuuTMKi 6GdMbnbKJgTpjDqY0uoBas16iNostiulgpvfcbVlThcAADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0172_image026.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAB5 AAEAgAAAAAAAAAINjI+py50Ao5zU2YtRAQA7 ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0172_image027.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhhgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACC AA4AgAAAAAAAAAJIhI+py+0Po5wn2Buo3rz7hF3fSJZfaJnqyipo1sZaKL/yLdmsjvcMrwL6hoAX bCUk9pJBjFLJfEqRqKk1Fr1qOdmtt3P8ilkFADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0172_image028.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAB AAEAgAAAAAECAwICRAEAOw== ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0173.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Exponents
Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
Example D.  Solve
LCD =3D = (x – 2)(x + 1), multiple the LCD to both sides = of the equation:
 
(x ̵= 1; 2)(x + 1) * [        = ;            &n= bsp;    ]
2= (x + 1) =3D 4(x – 2) + 1*(x + 1)(x &#= 8211; 2)
2x + 2 = =3D 4x – 8 + x– x – 2
2x + 2 = =3D x2 + 3x – 10
       0 = =3D x2 + x – 12
x – 2
2
=3D
x + 1
4
+ 1
(x + 1)
(x – 2)
(x + 1)(x – 2)
x – 2
2
=3D
x + 1
4
+ 1
------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0173_image029.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdQAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB1 ABMAgAAAAAAAAAJOhI+py+0Po2Shhomz3hzY34XiKH0Viaaoearui7HwTFNyjdd3zrvs1QuuTMKi 6GdMbnbKJgTpjDqY0uoBas16iNostiulgpvfcbVlThcAADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0173_image030.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAB5 AAEAgAAAAAAAAAINjI+py50Ao5zU2YtRAQA7 ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0173_image031.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhhgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACC AA4AgAAAAAAAAAJIhI+py+0Po5wn2Buo3rz7hF3fSJZfaJnqyipo1sZaKL/yLdmsjvcMrwL6hoAX bCUk9pJBjFLJfEqRqKk1Fr1qOdmtt3P8ilkFADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0173_image032.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAB AAEAgAAAAAECAwICRAEAOw== ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0174.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Exponents
Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
Example D.  Solve
LCD =3D = (x – 2)(x + 1), multiple the LCD to both sides = of the equation:
 
(x ̵= 1; 2)(x + 1) * [        = ;            &n= bsp;    ]
2= (x + 1) =3D 4(x – 2) + 1*(x + 1)(x &#= 8211; 2)
2x + 2 = =3D 4x – 8 + x– x – 2
2x + 2 = =3D x2 + 3x – 10
       0 = =3D x2 + x – 12
       0 = =3D (x + 4)(x – 3) à x =3D -4, 3 
x – 2
2
=3D
x + 1
4
+ 1
(x + 1)
(x – 2)
(x + 1)(x – 2)
x – 2
2
=3D
x + 1
4
+ 1
------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0174_image033.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdQAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB1 ABMAgAAAAAAAAAJOhI+py+0Po2Shhomz3hzY34XiKH0Viaaoearui7HwTFNyjdd3zrvs1QuuTMKi 6GdMbnbKJgTpjDqY0uoBas16iNostiulgpvfcbVlThcAADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0174_image034.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAB5 AAEAgAAAAAAAAAINjI+py50Ao5zU2YtRAQA7 ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0174_image035.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhhgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACC AA4AgAAAAAAAAAJIhI+py+0Po5wn2Buo3rz7hF3fSJZfaJnqyipo1sZaKL/yLdmsjvcMrwL6hoAX bCUk9pJBjFLJfEqRqKk1Fr1qOdmtt3P8ilkFADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0174_image036.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAB AAEAgAAAAAECAwICRAEAOw== ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0162.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Exponents
Rational Equatio= ns
Solve rational equations by clearing all denominators = using the= LCD. Check the answers afterwards.
Example D.  Solve
LCD =3D = (x – 2)(x + 1), multiple the LCD to both sides = of the equation:
 
(x ̵= 1; 2)(x + 1) * [        = ;            &n= bsp;    ]
2= (x + 1) =3D 4(x – 2) + 1*(x + 1)(x &#= 8211; 2)
2x + 2 = =3D 4x – 8 + x– x – 2
2x + 2 = =3D x2 + 3x – 10
       0 = =3D x2 + x – 12
       0 = =3D (x + 4)(x – 3) à x =3D -4, 3 
        = ;            &n= bsp;            Both are good.
x – 2
2
=3D
x + 1
4
+ 1
(x + 1)
(x – 2)
(x + 1)(x – 2)
x – 2
2
=3D
x + 1
4
+ 1
------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0162_image037.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhdQAUAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQB1 ABMAgAAAAAAAAAJOhI+py+0Po2Shhomz3hzY34XiKH0Viaaoearui7HwTFNyjdd3zrvs1QuuTMKi 6GdMbnbKJgTpjDqY0uoBas16iNostiulgpvfcbVlThcAADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0162_image038.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhVwAcAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAABW ABwAgAAAAAAAAAJiRI6py+0PYwO02ouz3rx7fHziSJZbaKbqeqHsC39uTNfVbOcvrvcm7wvKAsLi CGhMWpBKJbNpfEKF0qmvatVhs7Ytl+b9wsJiFrmsOqN/xDW17b7C49o5vWu/gyX8vv8PUQAAOw== ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0162_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAB5 AAEAgAAAAAAAAAINjI+py50Ao5zU2YtRAQA7 ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0162_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhhgAQAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAACC AA4AgAAAAAAAAAJIhI+py+0Po5wn2Buo3rz7hF3fSJZfaJnqyipo1sZaKL/yLdmsjvcMrwL6hoAX bCUk9pJBjFLJfEqRqKk1Fr1qOdmtt3P8ilkFADs= ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0162_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhegACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAB AAEAgAAAAAECAwICRAEAOw== ------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0175.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Exponents
Rational Equatio= ns
Proportio= nal equations are rational equations of the
form        = ;         .
A
B
=3D
C
D
------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0176.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Exponents
Rational Equatio= ns
Proportio= nal equations are rational equations of the
form        = ;         .
A
B
=3D
C
D
To solve proportional equations, we cross-multiply,   
A
B
=3D
C
D
------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0182.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Exponents
Rational Equatio= ns
Proportio= nal equations are rational equations of the
form        = ;         .
A
B
=3D
C
D
To solve proportional equations, we cross-multiply,   
A
B
=3D
C
D
à  AD =3D BC
then solve.
------=_NextPart_01C957B6.A88606B0 Content-Location: file:///C:/09764A33/6SolvingEquations_files/slide0177.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Exponents
Example E. Solve for x.&#= 13;
2
3
(x + 2)
(x – 5)=  
=3D
Rational Equatio= ns
Proportio= nal equations are rational equations of the
form        = ;         .