The document contains examples of algebraic expressions and equations. Some expressions are set equal to numbers to form equations. Steps are shown to solve equations for unknown variables by isolating them on one side of the equal sign.

2. 3 − 6 = 2(4 − 2 ) − 12
3 + 2 = 6( − 2) + 8
+ = 10 − = 2
4 8
4 , , −
3 3

3. 2 + 3 + 4 + 5 + 7 = 21
−2 − 3 − 4 − 5 − 7 = −21
4 − 2 + 8 − 6 + 10 − 12 = 2
:
( ) ( ) ( )( )( )( ) ( )
( ) ( ) ( )( )( )( ) ( )
(3 + 2)(4 − 3) = 12 − −6
(2 − 3) = 4 − 12 + 9
(3 − 2) = 27 − 54 + 36 −8
( + ) = +2 +
2 2
( − ) = −2 +
2 2
( + ) = +3 +3 +
3 3
( − ) = −3 +3 −
3 3

4. 5( − 2) + 4 − 8 = 2( + 5) − 3( − 1) + 9
5 − 10 + 4 − 8 = 2 + 10 − 3 + 3 + 9
5 +4 −2 +3 = 10 + 3 + 9 + 10 + 8
10 = 40
40
=
10
= 4
10 6 −4
− =
+2 −2 −2
( + 2)( − 2),
10( − 2) − 6( + 2) = −4( + 2)
10 − 20 − 6 − 12 = −4 − 8
10 − 6 + 4 = −8 + 20 + 12
Por transposición de términos:
8 = 24
24
=
8
=3

5. 6 3 4 8
+ − =
−1 +1 −1 +1
( − 1) ( + 1)( − 1)
( + 1)( − 1)
6 + 3( − 1) − 4( + 1) = 8( − 1)
6 +3 −3−4 −4 =8 −8
6 + 3 − 4 − 8 = −8 + 3 + 4
9 − 12 = −8 + 7
−3 = −1
−1
=
−3
=
− + −
− =
d)
+ − −
( − ) = ( + )( − )
( − 1)( − 4) − ( + 3)( + 4) = 12 − 2
−5 +4−( + 7 + 12) = 12 − 2
−5 +4− − 7 − 12 = 12 − 2
−12 − 8 = 12 − 2
−12 + 2 = 12 + 8
−10 = 20

6. 20
=
−10
= −2
6 − 3, 4 + 2, 16 + 8
6 − 3 = 3(2 − 1)
4 + 2 = 2(2 + 1)
16 + 8 = 8(2 + 1) = 2 (2 + 1)
2 . 3(2 + 1)(2 − 1) = 24(2 + 1)(2 − 1)
6 − 12, 16 − 64, −4 +4
6 − 12 = 6( − 2) = 2.3 ( − 2)
16 − 64 = 16 ( − 4) = 2 ( + 2)( − 2)
− 4 + 4 = ( − 2)( − 2) = ( − 2)
2 . 3( − 2) ( + 2) = 48( − 2) ( + 2)
8 − 2, 32 − 32 + 8, 12 + 12 + 3

7. 8 − 2 = 2(4 − 1) = 2(2 + 1)(2 − 1)
32 − 32 + 8 = 8(4 − 4 + 1) = 2 (2 − 1)(2 − 1) = 2 (2 − 1)
12 + 12 + 3 = 3(4 + 4 + 1) = 3(2 + 1)(2 + 1) = 3(2 + 1)
2 . 3(2 + 1) (2 − 1) = 24(2 + 1) (2 − 1)
( − ) = ( + )( − )
) 16 − 9 = (4 + 3)(4 − 3)
) 36 − 25 = (6 + 5)(6 − 5)
) 81 − 16 = (9 + 4)(9 − 4)
± ±

8. ) + 5 + 6 = ( + 3)( + 2)
+3
+2
) 12 + 14 − 10 = (4 − 2)(3 + 5)
4 −2
3 +5
) 10 − 14 − 12 = (5 + 3)(2 − 4)
5 +3
2 −4
+ = ( + )( − + )
8 + 27 = (2 + 3)(4 − 6 + 9)
)6 + 8 = (4 + 2)(16 − 8 + 4)
− = ( − )( + + )
) 27 − 8 = (3 − 2)(9 + 6 + 4)
) 125 − 64 = (5 − 4)(25 + 20 + 16)
)8 − 27 = (2 − 3)(4 + 6 + 9)
) 125 − 64 = (5 − 4)(25 + 20 + 16)

9. + + =0
) −5 +6 =0
( − 3)( − 2) = 0
−3=0 −2=0
=3 =2
) 12 + 14 − 10 = 0
(4 − 2)(3 + 5) = 0
4 −2 =0 3 +5=0
2 1 5
= = =−
4 2 3
+ + = 0 >0
4
4 +4 +4 = 0
4 +4 = −4
+

10. 4 +4 + = −4
(2 + ) = −4
2 + = ± ±4
− ±√ −4
=
2
∆= −4
∆≥0
∆=0
∆<0
5+2 , 7−3 = √−1
2, 4
√−8 (8)(−1) √8 √−1 = √8
√−4 = (4)(−1) = √4 √−1 = 2

11. + +2=0
− ±√ −4
=
2
= 1, = 1, =2
−1 ± (−1) − 4(1)(2)
=
(2)(1)
−1 ± √1 − 8
=
2
−1 ± √−7
=
2
−1 ± √7
=
2
−1 + √7 −1 − √7
= =
2 2
) + =0 =0 =−
) + =0 = ± −
, + 1, +2

12. + + 1 + + 2 = 51
3 + 3 = 51
3 = 51 − 3
3 = 48
48
=
3
= 16
40 −
− (40 − ) = 400
− (1600 − 80 + ) = 400
− 1600 + 80 − = 400
80 = 400 + 1600
80 = 2000
= 25
33 −
(33 − ) = 270
33 − = 270
− 33 + 270 = 0

13. = 18 = 15
28 −
+ (28 − ) = 400
+ 784 − 56 + = 400
2 − 56 + 384 = 0
− 28 + 192 = 0
= 16 = 12
70 −
50
70 −
(50) = + (70 − )
2500 = + 4900 − 140 +
2 − 140 + 2400 = 0
= 40 = 30

14. −
+ .
− + + + = 54
+ + = 54
3 = 54
= 18
= 18
− = 18 − 6 = 12
+ = 18 + 6 = 24
+3
20 20
+3
2
1 1
+ = ( − 15)
3 9

15. 3 + = 9( − 15)
3 + = 9 − 135
3 + − 9 = −135
−5 = −135
−135
=
−5
= 27
+6
( + 6)
( + 6 + 5)( + 5) = ( + 11)( + 5)
( + 11)( + 5) − ( + 6) = 175
+ 16 + 55 − − 6 = 175
10 + 55 = 175
10 = 120
= 12
195
195
−( ) = 56

16. 38025
− = 56
− 38025 = 56
− 56 − 38025 = 0
− ±√ −4
=
2
56 ± (−56) − 4(1)(−38025)
=
2(1)
56 ± 3136 + 152100)
=
2(1)
56 ± √155236
=
2
56 ± 394
=
2
56 + 394
=
2
450
=
2
= 225
= √225
= 15
15
= 13
, −5
+ 10
3 + ( − 5) = 4( + 10) − 3

17. 3 + − 10 + 25 = 4 + 40 − 3
− 7 + 25 = 4 − 37
− 7 − 4 + 25 − 37 = 0
− 11 − 12 = 0
= 12
= −1
(30 − )
4
2(30 − )
4 + 2(30 − ) = 84
4 + 60 − 2 = 84
2 = 84 − 60
2 = 24
24
=
2
= 12
(40 + )
(10 + )
(40 + ) = 2( 10 + )

18. 40 + = 20 + 2
40 − 20 = 2 −
20 =
720
15, 225 480
90
260
40 50
117
3
9, 27, 81
√2
1
35 625
20, 15
10
8,67
300 144
5,32
152
98
5 3
2 5 3
+ =
−3 +3 2( − 1)
= 1,45 + 1,40 = 1,45 − 1,40
2 5 3
− =
+4 +4 +2 +2
= 2,24 = 0,36

20. / /
/
/
1, 2, 3, 4
/

21. N = { 1, 2, 3, 4 …}
Z = { . . .−3, −2, −1, 0, 1 , 2, 3, . . . }
= {1, 2, 3, 4,5. . } = {… − 4, −3, −2, −1}
+ =0
3 2 4 1
= , , ,
5 7 3 2
I = 1,23, 2,47, √2, √5, e, π
C = a + bi
a bi
3+2 , 5+3 , 8+ 2
= √−1

22. A B
A- B B-A
A =U−A
U
A A′
A∪B
A∩B
A B
A∩B

23. A∆B = (A − B) ∪ (B − A)
60 20 18 15
8 9
5 3
5
5
3, 8
2, de la diferencia de 7 que practican fútbol y básquet.
4, 9
20 − (3 + 5 + 4) = 8
18 − (3 + 5 + 2) = 8
15 − (4 + 5 + 2) = 4
8 + 8 + 4 = 20
60 − (20 + 4 + 5 + 3 + 2) = 60 − 34 = 26
18 6
8

24. Plata
Oro Plata
6
2
Bronce
6
10,
13,
8,
4 + 7 + 2 + 6 = 19
4 + 7 + 2 = 13
0 30 40

25. 60
30 − + + 40 − = 60
= 10
29 12 8 5

26. 12 + 8 + + 5 = 29
=4 12 + 4 = 16
38 17 19 20 7
9 6 4
24
48 20 25 8
5
45 35 13
5
2
37 20 22 18 9
11 8 5
13
50 20 25
5
10
49 23 25 27
9 11 10
4

27. 12 > 7 4<8
( , )
A = {2, 4, 7} B = {3, 5}
AxB = {(2, 3), (2, 5), (4, 3), (4, 5), (7, 3), (7, 5)}
AxB
R = {(2, 3), (4, 3), (7, 5)}
R = {(2, 5), (4, 3), (7, 3)}
dom (R ) = {2, 4, 7}
(R ) = {2, 4, 7}
ran (R ) = {3, 5}
dom (R ) = {5, 3}
{ , }
{0, 0}

28. A(2, 3).
B(−3, 4).
C(−1, −3)
(5, −2).
y
-5 -4 -3 -2 1 2 3 4 x Abscisas
y
5 x

29. A(5, 0), B(−3, 0).
C(0, 5), D(0, −3).
- -5 -4 -3 -2 -1 0 1 2 3 4 5 +
|AB| = − = 4 − (−5) = 4 + 5 = 9
|BA| = − = |−5 − 4| = |−9| = 9
A B B A
y2 – y1
CC y22
y
y1 x2 – x1
x1
x2
B En el ∆ Rectángulo A B C
AB = 2 y2 y1 2
x2 x1
y2 – y1
A C
x2 – x1

30. A (2 5) B (-6 -3)
x1 y1 x2 y2
AB = (− − ) + (− − ) = √64 + 64 = √128
.
B
= =
y2 – y1
=
A C
−
x2 – x1
=
−
A(2, −5) B(−6, −3)
−3 − (−5) −8 + 5 −3 3
m= = = =
−6 − 2 −8 −8 8
90°
90°

31. A = (−3, 4) B = (5, 1)
y −y 1−4 −3
m= = =
x −x 5 − (−3) 8
y = mx + b
3
y=− x+b
8
b y
A (−3, 4) −3
4 B
b
(−3)(−3)
4= +b
8
9
4= +b
8
b=4−8=
9 23
b
8
3 23
y=− x+
8 8
a b
+ =1
a b
b "
" "
y=− x+ =0 =
=0
3 23
y=− x+
8 8

32. 3 23
0=− x+
8 8
).
23
=
3
+ =1
23 23
3 8
A
4
23
=
8
B
1
23
=
-3
3
5
+ + =0
3 23
y=− x+
8 8
8
8y = −3x + 23

33. 3 + 8 − 23 = 0
= +3 = −2
= +2 =− −1
−
= −4 = −9
5
A(5, −3) B(2, −1)
AB = √13
A(7, −3) B(5, −2)
=−
A(2, 5)
B(7, 3)
= − + + =1 2 + 5 − 29 = 0
A(−2, 5) =3
= 3 + 11 + =1 3 − + 11 = 0
) = −1 ) =5 +3 ) =4 +4 ) =− +2
=4 −2 ) =6 −5
(3, 4), (−3, 1) (2, −1)

34. x 0 1 2 3 -1 -2 -3 1 1
y 0 1 2 3 -1 -2 -3 1 1

35. x -2 -1 0 1
y -1 1 3 5
x -2 -1 0 1 2
y 4 1 0 1 4

36. y = x
x 4 3 2 1 0
y 2 1.7 1.4 1 0
X 0 /2 3/2 2
y 0 1 0 -1 0

37. y = cos x
X 0 /2 3/2 2
y 1 0 -1 0 1
y = ex
X 0 1 2 3
y 1 2,718 7,38 20,07

38. y
y = 3
x
y = -4
=3 −6
b = −6 a = 2.
= + −6
D( f ) =< −∞, ∞ >= Reales R( f ) = [−3, ∞ >.
= 2 − 3 + 1, f (2)
= + ,
f (2) = −4 f (5) = 2
=2 = −8
( )=0 = − 7 + 12
=4 =3
( )=
( )= + 2

40. Aristóteles (384-332 a.c.)
Platón (427-347 a.c.)
Sócrates (470-399 a.c.)
Parménides
Zenón

41. -
-
-
-
-
- .

42. - Los abogados poseen conocimientos jurídicos si y solo si estudian leyes.
- Las obstetrices atienden partos o los farmacéuticos conocen de medicamentos.

44. 12 + 9 = 21 9 < 21
5 1

46. (p q) (~ p q)

48. Conclusión: Si P implica a Q su valor de verdad es una tautología

49. es la conclusi n
p
(p p ) c
{(p p ) p } c
p q
q r
p r
[(p ) (q r] (p r)

50. p q
~q
~q ] ~p
~p
[(p )
~q ~p

52. p q
q r
p
p q q r p r
p q q r p r
p q q r p r
p r
p q q r p r

53. p q
q r
p r
p q q r p
p q q r p r
p q q r p r
p r
p q q r p r

54. p v q
p q Modus Ponnes
p
q
p q Modus Tollendo Tollens
~q
~p
pvq Silogismo Disyuntivo
~p
q
p q Silogismo Hipot tico
q r
e)
p r

55. .
apagado o encendido
p q
Bit 1 Bit 0
1 1 V V
1 0 V F
0 1 F V
0 0 F F

56. 2 =8
2

57. p q r
FOCO
BATERIA

58. p
q
r
FOCO
BATERIA

59. X
X Y
Y
X W
Y
Z

60. X W
Z
Z
X Y W
X Y Z´
Y
X´
Z´
X Y Z´
Y´ U V X
Z Y´ U

61. Respuesta:
4.18.4.1 Puerta Y (and).
F=p q
Su ecuación es:
p
pq
q
F=p +q
p
p+q
q

62. F = p
p p
(p )
p ~q
q v ~r
p r

63. >6