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$ 4 − x 5x + 3 = 7 5

&

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&

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%

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2 − 2x =

3x − 3 −7 2

2( 2 − 2 x =

3x − 3 − 7) L L L 2

G ; G ) 0 ; 0 ; /G G ; G ) 0 ; /6 G 3 /6 ) 0 3 G ,/ ) 6 ,/< 6 ) 0)

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3600x + 2000 = 20000 3600x = 20000 − 2000 3600x = 18000 x = 18000 / 3600 x=5

=

8 "

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y

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P

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2 x 3 5x 1 − = − ! 3 4 6 2

=

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2 x 3 5x 1 − = − & 3 4 6 2

) /,

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%

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x 2 + 14x + 48 = 0

(x + 8)(x + 6) = 0

x +8 = 0∧x +6 = 0 x = −8 ∧ x = −6

=

/G 3 G@ ) +!

%

) ,G

0

%

'

"=

x=

x=

=

=

8

− b ± b2 − 4ac 2a − 14 ±

(14)2 − 4(1)(48) − 14 ± = 2(1)

a=1 b = 14 c = 48

196 − 192 2

− 14 ± 4 2

− 14 + 2 − 12 = −6 = 2 2 − 14 − 2 − 16 = = −8 2 2

− 14± 2 2

2x + 3 y = 2

(

10 x + 9 y = 8

!

%

& x = 1 / 2; y = 1 / 3 & x = 1; y = 1 & x = 2; y = 3 & x = −1 / 2; y − 1 / 3 (

= =

8 =

!

D =

!

=

(

=

=

x=

E F 3y = 2 − 1

2x + 3y = 2

3y = 1

1 +3= 2 2 1 + 3y = 2 2

'

y=

1 3

= B

E F

2 x + 3y = 2 2 x = 2 − 3y 2 − 3y x= 2 >0&

=

/

Ecuación 3

1 2

=

/

=

0

=

10 − 15 y + 9 y = 8 10 − 6 y = 8 10 − 8 = 6 y

10x + 9 y = 8 2 − 3y + 9y = 8 2 20 − 30 y = + 9y = 8 2

10

2 = 6y 2 =y 6 1 =y 3

D

E F

2 − 3y 2 2 − 3(1 / 3) x= 2 2 −1 1 x= = 2 2

x=

'

= B

E F

2 x + 3y = 2 2 − 3y 2x = 2 0

,

/

10x + 9 y = 8 G

x=

8 − 9y 10

,

=

0

0

G

2 − 3y 8 − 9y − 2 10 E F =

G

10(2 − 3y ) = 2(8 − 9 y )

8 − 9y 10 8 − 9(1 / 3) 8 − 3 5 1 x= = = = 10 10 10 2 1 x= 2

20 − 30y = 16 − 18y

x=

20 − 16 = −18 + 30y 4 = 12y 4 1 = 12 3 1 =y 3

E

=

D F 0+

#

#

!

%

& 0?+ & 0++ & /5+ & /@+ "

/,

12 * 30 = 360

!

%

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" >/< 0&!

%

1 1 f ( ) = 3( ) − 1 = 1 − 1 = 0 3 3

& 0 & / & +

E F & 5 & /< 5 & G & 0

Solución. x 2 = 32 + 4 2 x = 9 + 16 x = 25 x =5

"

%

,0+

!

J(

$ " 8

"

0+OK

y 230cm 230cm(sen 30º ) = y

Sen 30º =

& //5 & ,0+

230cm(1 / 2 ) = y

& /,5

115cm = y

& ,++

7

7

,+

,+ G++

J K

& ,+ & ?+ & G+ & @+

!

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T 7

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(

) , ; ,+

7

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B

#

=

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4

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D

C $ ,< / ) ,! G< , ) ,! @< G ) , R> &) ,

V

4 $

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/ $

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N %

•

M /, 5+

• J

$

8

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= =

$

K

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3 /G 5+

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%

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M ,5+ ++

x2 + x 4 x −1

=

&

03 ,

&

3/<

%

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,

0

%

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3

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3

,

= =

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;

& 1G 3 &

;

; & ; >,

3G

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= ; >0 ;

,

; & ; >,

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3

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3

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30

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30; 3

30

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" %

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=

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& /< , & 5< 6 & 5< 0 & 5< , = ,>0 ; & ; G ) 0 ; ,>/ ; , & ?;, ;G ) 0;,3G ?;? ) /3G ?;/) G 3? 5 ) /+ 5< /+ )

5Y

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=

=

2x − 1 = 1! x −1

%

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: 1,

&

:,

&

A 1,

&

A,

=

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= /35 :@ 36 / ; 6 3 5 : @ 36 ; 6 1? 3 5 : @ 5 ; 5 1? : @ ; 5 1?< 0 : 0 < 0 1, : A 1,

" & *1,! ,. & *1,! ,* & .12 ! 1,*

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& .1 2 ! 1,.

*,! 2 3*

= G;

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+

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) 1,

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=

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4 − x2

%

%

%

B

=

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7

3

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$

%

& 5 & ] 5 & / & ] 0 =

%

(− 2)2 + 1

R >1,& ) )

4 +1

)

5 D

>

% &

3

& ,

35

& 6 & =

% $ =

>

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5&

=

T $ T %

>

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5&

$

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5&!

"

=

f ( x ) = x − 4 , es :

& *1G! G. & .1G! G* & .1∞! G. & *G! ∞3* = x−4≥0 x−4+4≥0+4 x≥4

7 > &)

,3

/!

7 > 3 /&!

& x 2 + 2x + 3

h ( x + 1) = ( x + 1) 2 + 1

& x2 + 3

= ( x 2 + 2x + 1) + 1

& x 2 + 2x + 2

= x 2 + 2x + 1 + 1

& x 2 + 6x + 9

= x 2 + 2x + 2

B & ./! 5* & .1/! 5* & .12 ! /* ^ .5! 2 3* & .12 ! 1/. ^ *5! 2 3*

=

%

,1

? A15

%

=

=

x 2 − 6 x + 5 > −5 + 5

x 2 − 6 x + 5 > 0(+, a )w (x − 5)(x − 1) > 0 x − 5 = 0 ∧ x −1= 0 x = 5 ∧ x =1

'

)

+! 0

x − 5 = 0 − 5 = −5 x − 5 = 3 − 5 = −2

B

=

7

_ !

c.s = ]− ∞,1[ ∪ ]5, ∞ +[ %

#

E` F

D

$ 3 $

$ "

$

E F$

B

A∩ B∩C & */! @. & */! @* & .122 ! /5* & ./! @*

( ) */! @*! - ) .12 ! /,*! %

) *10! /5

%

6

"

7

E F =

8

(

!

7

5!+++ 8 =

8

0 =

"

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=

> & ) > ; 0& >5!+++&! &

"

%

=

%

=

&

=

&

"

=

&

"

=

8

= "

=

$

E F

E F

=

=

3x + 1 8 x − 9 = 2 3

Solucion : &

) ,

&

) 1/,

&

) /I < /0

&

) 0

3x + 1 8 x − 9 = 2 3 3(3x + 1) = 2(8 x − 9 ) 9 x + 3 = 16 x − 18 3 + 18 = 16 x − 9 x 21 = 7 x 21 / 7 = x 3=x

%

"

,++

/+X

B

"

"

$

,+X #

J$ K

& ,+ & ,5 & G+ & G5 = ) ,++ >+ I & ) /@+ ,++ 3

)

+ @ >,++ 3 & ) /@+ ,++ 3

) /@+< + @

,++ 3

) ,,5

) ,,5 ; ,++ ) ,5

!

a

,+

D

B B

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8 "

J

8

%

"

K

& &

P

& & (

P #

8 "

T 8

B =

E F! "

=

(1 / 3x − 7 )(1 / 3x + 7 )

( "

7 & /< 0

,;

& /< I

,1

GI

& GI ; /< 0 & /< I

Solucion.

6 ,

(1 / 3x − 7 )(1 / 3x + 7 ) = (1 / 3 x )2 − (7 )2 = 1 / 9 x 2 − 49

, 3GI

8 "

%

&

) /<

!

&

) /<

!

; *+.

&

) /<

!

; *+.

&

) /<

!

; *+. ; *+.

"

%

5 9

,++

/++!

9 G 9

% Solucion.

& /++ & ,++

x x ;200 = n 5 200(5) = x + x + x + x + x 1000 = 4 x + 100 ×=

& ,,5 & 0++

1000 − 100 = 4 x 900 / 4 = x 225 = x

I ++ G /+

,,5

,+

( "

# %

& >, ; /& >0 3 ,& & >, ; /& >0 ; ,& & >, 3 /& >0 3 ,& & >, 3 /& >0 ; ,& =

6x 2 + x − 2

(

=

=

!

=

8

$ $

!$

T 7

T

7

T 7

T

T

T

T

!

G

" ,+ T

7

(

!J

8

T

! T 7

"

K & /+ T & /I

T

&

,+ T

&

!

$

7 = 2 20 = 2 2 n 20 = 2n 20 / 2 = n 10 = n

'

8

8

b

% & / & /< , & ] , & /< ] @

tan ø = 1/1 = 1

(

= =

0?O! =

-

= -

?5O

"

%

& x=

98 sen36º sen 29º

& x=

98sen36º sen115º

& x=

98sen36º sen 29º

& x=

98sen36º sen 65º

= a

$

x 98 = sen 36 º sen 29º 98sen 36 º x= sen 29 º

#

I @

= $

(!

"

(!

8

$ 8 =

8 T

!

/+ T 7

8

J

C 8

K

& /+ /< @ & /+ ,< 0 & ,5 & 0+ = ') '

T

!

) ?

p = 10 + p / 2 + p / 6

2p = 60

6( p = 10 + p / 2 + p / 6)

p = 60 / 2 p = 30

6 p = 60 + 3 p + p 6 p = 60 + 4 p 6 p − 4 p = 60

$

0+ 8

#

! /5 % & 0+>/< G& & >0+< G& & ' ) >, A 0+< /5& &

30 15

(1 / 4)15 (3 / 4)15

$ !

= '>

& ) /< G

'> ' >(& )

& ) 0< G "

<

) 0+ ) /5 ' >(

'>

&)

/5

b ) ,< 0 J

&)

8

8 N

b K

c (8

!

%

& : ) 0+O & : ) G+O & : ) ?+O & : ) @+O = 8

$

/@+O x + 60º +90º = 180º x + 150º = 180º x = 180º −150º x = 30º

8 & 0+O & G+O & 5+O & ?+O 60º +80º + x = 180º 140º + x = 180º x = 180º −140º x = 40º

?+O

%

V

N

!'

7

"

$

N

"

$

# '

! %

& /< ? & /< 0 & /< , & ,< 0 = V

N

,/) ,

"

/ 0,/) ? ' >V

N

$

'

& ) ,< ? ) /< 0

8 7

8 -

c (!

c

-

%

& ] /5 & /5 & ,+ & ,5 =

h 2 = 24 2 + 7 2

25 2 − 20 2 = BD 2

h = 576 + 49

BD 2 = 625 − 400

h = 625

BD = 225 BD = 15

h = 25

=

V /, d V

8

=

=

6+O!

$ 8

V

& x=

12sen60º sen50º

& x=

12sen50º sen60º

& x=

12sen 70º sen 60º

& x=

12sen60º sen70º

=

?+O %

= 7

' (

8

/,! $

6+O 3 ?+O 3 b ) /@+O /0+O 3 b ) /@+O b ) /@+O 1 /0+O b ) 5+O

x 12 12sen60º = asi : x = sen60º sen50º sen50º

/@+O!

8 &

$

&

!

%

8 =

& & '

8

8

B

%

&

c (-H

c -H

&

c (-H

c (-

&

c -H

c (-

&

c (-H ! c -H

! c (= 7 H

/+X

B

" 8 "

$

B $

=

%

"

/+ B

8 5

8

&

!

"

J

"

8

K

10 5

& 5 >/+& & /+ >I & @ >6& ? & 5e

E F /5

,

!

V = (

8

=

! &

) 0+

&

) G+

&

) 5+

&

) ?+

E F 15 +

%

n − 20 n = = 3 2

n − 20 n = 3 2 90 + 2n − 40 = 3n 6 15 +

50 + 2n = 3n 50 = 3n − 2n 50 = n

'

" $

B

7

!

8 "

8 7

5+O J

"

8

8 K

& /< G+ & ,< I & 5< /@ & /0< /@

= 8

$

/@+O A + 50º +90º = 180º A + 140º = 180º A = 180º −140º A = 40º

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& x = 142 2 + 77 2 − 2(142)77 cos 80º & x = 142 2 + 77 2 & x = 77 cos 80º & x = 142 csc 80º.

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]− 1,2]x[1,3[ ]− 1,1[ ]− 1,3[

& ]− 1,2]x[1,3[ & [1,3[x ]− 1,2] & [− 1,2[x ]1,3] & [1,3[x[− 1,2]

8 0@G!+++ d

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&

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&

) 0@G!+++

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&

) 0@G!+++

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/O

=

x = tan1º 384,000 x = 384,000 tan1º

%