18jonea one year ago Add/Subtract: : Simplify and state the domain.

1. 18jonea

|dw:1440004351229:dw|

2. 18jonea

@Michele_Laino

3. freckles

can't really understand this? what is going on with that 1/4 over there... And I see you have - + in between the first two fractions.

4. 18jonea

the plus is suppose to be over before the 1/4

5. freckles

$\frac{7}{x+6}-\frac{x+2}{2x+2}+\frac{1}{4}?$

6. 18jonea

yes

7. freckles

well first 2x+2 can be written as 2(x+1) $\frac{7}{x+6}-\frac{x+2}{2(x+1)}+\frac{1}{4}$ now look at the bottoms of all the fractions and ask yourself how you can find a common denominator

8. 18jonea

no?

9. freckles

what does that mean exactly

10. freckles

"no?" I don't understand the response to what I said.

11. freckles

does that mean you don't know how to find a common denominator ?

12. 18jonea

i dont know how to find the common denominator

13. freckles

oh okay

14. freckles

well if you notice the denominators are x+6 and 2(x+1) and 4

15. freckles

so a common denominator could be (x+6)*(x+1)*4

16. freckles

the first fraction... $\frac{7}{x+6}$ how can we write this as something over/[4(x+6)(x+1)]?

17. 18jonea

{4(x+6)(x+1)}

18. freckles

notice on bottom it is lacking the 4 and the (x+1) so just multiply both top and bottom by 4(x+1) that is how you can get this new bottom we speak of ... $\frac{7}{x+6} \\ \text{ multiply top and bottom by } 4(x+1) \\ \frac{7}{x+6} \cdot \frac{4(x+1)}{4(x+1)}=\frac{7 \cdot 4(x+1)}{4(x+6)(x+1)}$

19. freckles

now look at the second fraction

20. freckles

$-\frac{x+2}{2(x+1)}$ remember we want 4(x+6)(x+1) on bottom so what is this bottom lacking in order to get that?

21. 18jonea

x+6

22. freckles

and also 2 right since 2*2=4

23. 18jonea

i am so confused

24. freckles

$-\frac{x+2}{2(x+1)} \cdot \frac{2(x+6)}{2(x+6)} =-\frac{2(x+2)(x+6)}{2 \cdot 2 (x+6)(x+1)}$

25. freckles

we wanted 4(x+1)(x+6) on bottom

26. freckles

27. freckles

2(x+1)*what=4(x+1)(x+6)

28. freckles

divide both sides 2(x+1) $what=\frac{4(x+1)(x+6)}{2(x+1)}=\frac{4}{2} \frac{(x+1)}{(x+1)} \frac{x+6}{1} \\ what=2 (1)(x+6) \\ what=2(x+6)$ so 2(x+1)*2(x+6)=4(x+1)(x+6)

29. freckles

so what we needed to do to get the bottom 4(x+1)(x+6) for the second fraction was multiply the second fraction on top and bottom by 2(x+6)

30. freckles

so far we have: $\frac{7}{x+6}-\frac{x+2}{2(x+1)}+\frac{1}{4} \\ = \\ \frac{7 \cdot 4(x+1)}{4(x+6)(x+1)}-\frac{2(x+6)(x+2)}{4(x+1)(x+6)}+\frac{1}{4} \\ \text{ so we are already able to combine the first two fractions } \\ \frac{7 \cdot 4(x+1)-2(x+6)(x+2)}{4(x+1)(x+6)}+\frac{1}{4}$

31. freckles

you still need to do the same thing you did to the first two fractions to the third

32. freckles

what do we need to multiply 4 by so that we have 4(x+1)(x+6)

33. 18jonea

x+1 x+6

34. freckles

right so multiply top and bottom by (x+1)(x+6)

35. freckles

$\frac{1}{4}=\frac{1(x+1)(x+6)}{4(x+1)(x+6)} \text{ or just } =\frac{(x+1)(x+6)}{4(x+1)(x+6)}$

36. freckles

so this gives us: $\frac{7}{x+6}-\frac{x+2}{2(x+1)}+\frac{1}{4} \\ = \\ \frac{7 \cdot 4(x+1)}{4(x+6)(x+1)}-\frac{2(x+6)(x+2)}{4(x+1)(x+6)}+\frac{1}{4} \\ \text{ so we are already able to combine the first two fractions } \\ \frac{7 \cdot 4(x+1)-2(x+6)(x+2)}{4(x+1)(x+6)}+\frac{1}{4}$ $\frac{7 \cdot 4(x+1)-2(x+6)(x+2)+(x+1)(x+6)}{4(x+1)(x+6)}$

37. freckles

you should multiply everything out on top and combine any like terms

38. freckles

also notice the domain can be found from the start

39. freckles

you don't want any of the fractions to be 0 on bottom so to find the domain say all real numbers except x+1=0 or x+6=0 (I will let you solve those equations)

40. 18jonea

why is there a x+2

41. 18jonea

and x+1

42. freckles

the second fraction had x+2 in the numerator

43. freckles

and the second fraction also had 2x+2 on bottom 2x+2=2(x+1)

44. 18jonea

right

45. 18jonea

is the domain all real numbers except when x = 6 or 1

46. 18jonea

@freckles

47. 18jonea

wait it would be -6 right?

48. freckles

-6 is right 1 is incorrect this is because 1+1=0 isn't a true equation

49. freckles

-1+1=0 is true

50. freckles

x+1=0 means x=-1

51. freckles

the domain is all real numbers except x=-1 or x=-6

52. 18jonea

right i just figured that out

53. freckles

did you ever try multiplying the top and combining like terms yet?

54. 18jonea

i am trying to figure out the second equation you did... can you show me how you did it again please

55. freckles

fraction?

56. 18jonea

yes the second fraction

57. freckles

this was the second fraction $-\frac{x+2}{2(x+1)}$

58. freckles

we wanted the bottom to be 4(x+1)(x+6)

59. freckles

the bottom is missing factors 2 and (x+6)

60. freckles

so we multiply top and bottom by 2(x+6)

61. 18jonea

so how would it look

62. freckles

$-\frac{x+2}{2(x+1)} \cdot \frac{2(x+6)}{2(x+6)} \\ =-\frac{2(x+2)(x+6)}{2 \cdot 2(x+1)(x+6)} \\ = -\frac{2(x+2)(x+6)}{4(x+1)(x+6)}$

63. 18jonea

ok i am goimg to try and multiply now

64. freckles

ok

65. 18jonea

yeah i have no idea how to even start... can you help

66. 18jonea

@freckles

67. 18jonea

wouldnt the second one be 2x squared +16x + 24

68. 18jonea

@freckles

69. freckles

2(x+2)(x+6) = 2(x^2+6x+2x+12) 2(x^2+8x+12) 2x^2+16x+24 is right and then there is a negative in front of that mess -2(x+2)(x+6) so it is actually -2x^2-16x-24

70. freckles

$\frac{7 \cdot 4(x+1)-2(x+6)(x+2)+(x+1)(x+6)}{4(x+1)(x+6)} \\ \frac{28(x+1)-2x^2-16x-24+(x+1)(x+6)}{4(x+1)(x+6)}$

71. freckles

28(x+1)=28x+28 and (x+1)(x+6)=x^2+6x+1x+6 (x+1)(x+6)+x^2+7x+6

72. freckles

$\frac{28x+28-2x^2-16x-24+x^2+7x+6}{4(x+1)(x+6)}$

73. freckles

combine like terms on top

74. 18jonea

to combine all of them i got -x^2 +12x +10 All divided by 4(x+6)(x+1)

75. 18jonea

is that right?

76. freckles

$28x+28-2x^2-16x-24+x^2+7x+6 \\ \text{ reorder terms } \\ (-2x^2+x^2)+(28x-16x+7x)+(28-24+6) \\ (-2+1)x^2+(28-16+7)x+(28-24+6) \\ -1x^2+19x+10 \\ -x^2+19x+10$ is what I got for the numerator you might want to check my work

77. 18jonea

yes thats what i got

78. 18jonea

sorry i meanyt 19 x

79. 18jonea

then how would i siplify it

80. freckles

in my mind $\frac{-x^2+19x+10}{4(x+1)(x+6)}$ this is simplified

81. freckles

however your teacher may want you to multiply the bottom out

82. freckles

not sure in your teacher's mind

83. 18jonea

thank you

84. freckles

np

85. 18jonea

can you help me with one more of these problems that similar..

86. freckles

you can show me your work and I can help you fix it if it needs any fixing

87. 18jonea

ok i will

88. 18jonea

$\frac{ 1 }{ 2} +\frac{ 15 }{ 2x-14 } - \frac{ 3x-5 }{ x^2-7x }$

89. 18jonea

first one stays the same 2(x-7) x(x-7)

90. freckles

check mark so far! :)

91. 18jonea

domain is all real numbers except when x=7

92. freckles

well or x=0

93. freckles

you have x on the bottom

94. freckles

on that 3rd fraction

95. freckles

$\frac{1}{2}+\frac{15}{2(x-7)}+\frac{3x-5}{x(x-7)} \\ \text{ so we want } x-7 \neq 0 \text{ and also we want } x \neq 0$

96. 18jonea

ok

97. freckles

do you want to try to figure out what the denominators should be so we can combine the fractions? or do you want me to tell you and you try to figure out the new numerators?

98. 18jonea

you tell me and i try to figure out the numerators

99. freckles

ok notice the denominators are: 2 2(x-7) x(x-7) ---- if we look at just 2 and 2(x-7) then the denominator would be 2(x-7) but we also have x(x-7) for the third bottom so looking at 2(x-7) and x(x-7) the new denominator of all them should be 2x(x-7) this is because all the denominators given go into 2x(x-7) that I can think of something to multiply all of your original denominators to get this new denominator of 2x(x-7) for example: 2*x(x-7) is 2x(x-7) and 2(x-7)*x is 2x(x-7) and x(x-7)*2 is 2x(x-7)

100. 18jonea

$\frac{ 1 }{ 2 } * x(x-7)$

101. 18jonea

x * x(x-7)

102. 18jonea

is that right

103. freckles

well you multiply top and bottom by x(x-7)

104. freckles

$\frac{1}{2} \cdot \frac{x(x-7)}{x(x-7)}$

105. freckles

$\frac{x(x-7)}{2x(x-7)}$ but yep this right for the first fraction

106. 18jonea

ok

107. 18jonea

the next one you only need to multiply top and bottom by x

108. 18jonea

and the last one you only multiply it by 2

109. freckles

(well 2/2 but I get what you are saying) right!!! so we have: $\frac{1}{2}+\frac{15}{2(x-7)}+\frac{3x-5}{x(x-7)} \\ =\frac{x(x-7)}{2x(x-7)}+\frac{15x}{2x(x-7)}+\frac{2(3x-5)}{2x(x-7)}$

110. 18jonea

thats what i got

111. freckles

so now we have the same denominators we can now combine the fractions

112. freckles

$\frac{1}{2}+\frac{15}{2(x-7)}+\frac{3x-5}{x(x-7)} \\ =\frac{x(x-7)}{2x(x-7)}+\frac{15x}{2x(x-7)}+\frac{2(3x-5)}{2x(x-7)} \\ =\frac{x(x-7)+15x+2(3x-5)}{2x(x-7)}$

113. freckles

multiply top out and combine like terms

114. 18jonea

x^2 +2x - 10

115. 18jonea

so i can take the 2x out of the top and bottom and then i would get $\frac{ x^2-10 }{ x-7 }$

116. 18jonea

@freckles

117. freckles

sorry my internet stopped working for a few minutes there

118. freckles

and you can't cancel out common terms but you can cancel out common factors

119. freckles

so you should have: $\frac{1}{2}+\frac{15}{2(x-7)}+\frac{3x-5}{x(x-7)} \\ =\frac{x(x-7)}{2x(x-7)}+\frac{15x}{2x(x-7)}+\frac{2(3x-5)}{2x(x-7)} \\ =\frac{x(x-7)+15x+2(3x-5)}{2x(x-7)} \\ =\frac{x^2+14x-20}{2x(x-7)}$

120. freckles

@18jonea

121. 18jonea

so what would be the final equation

122. 18jonea

@freckles

123. freckles

$\frac{x^2+14x-20}{2x(x-7)}$ and you can multiply to the bottom out if you want or if your teacher wants but to me this is simplified

124. 18jonea

how did you get 14x and -20

125. 18jonea

the last fraction you are supppose to subtract

126. freckles

the problem I have is: $\frac{1}{2}+\frac{15}{2(x-7)}+\frac{3x-5}{x(x-7)}$

127. freckles

is that not the right problem?

128. 18jonea

sorry it was subtract the last fraction

129. 18jonea

thats how i got $\frac{ x^2 +2x-10 }{ 2x(x-7) }$

130. freckles

$\frac{1}{2}+\frac{15}{2(x-7)}-\frac{3x-5}{x(x-7)} \\ =\frac{x(x-7)}{2x(x-7)}+\frac{15x}{2x(x-7)}-\frac{2(3x-5)}{2x(x-7)} \\ =\frac{x(x-7)+15x-2(3x-5)}{2x(x-7)} \\ =\frac{x^2-7x+15x-6x+10}{2x(x-7)}$ $=\frac{x^2+2x+10}{2x(x-7)}$

131. 18jonea

wait it is -10

132. freckles

-2(-5)=10

133. 18jonea

why is it -2 for the rest it was +2

134. freckles

you have -2(3x-5) distributive property says a(b+c)=ab+ac so this means -2(3x-5) is -2(3x)+(-2)(-5) or -6x+10

135. 18jonea

why is it -2 cause it has +2 at bottom

136. 18jonea

i have 6x-10

137. 18jonea

hence how i got x2+2x−10 / 2x(x−7)

138. freckles

$\frac{1}{2}+\frac{15}{2(x-7)}\color{red}-\frac{3x-5}{x(x-7)} \\ =\frac{x(x-7)}{2x(x-7)}+\frac{15x}{2x(x-7)}\color{red}-\frac{2(3x-5)}{2x(x-7)} \\ =\frac{x(x-7)+15x\color{red}-2(3x-5)}{2x(x-7)} \\ =\frac{x^2-7x+15x-6x+10}{2x(x-7)} \\ =\frac{x^2+2x+10}{2x(x-7)}$

139. freckles

do you see that red subtraction symbol I just brought it down

140. 18jonea

got it thank you!