Question

6) Se consideră numerele:
$x = \frac{1}{2} - \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + ... + \frac{1}{2010} - \frac{1}{2011}$
Și
$y= 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + ... + \frac{1}{2009} - \frac{1}{2010}$
Calculează media aritmetică a numerelor x și y ( pana la 9:15-9:30 va rog)

7). Rezolvă, în mulțimea numerelor raționale, ecuațiile :
a)
$x + 17 = 12$
b)
$3x - 5 = 22$
c)
$5x + 4 = 7x + 1$
D)
$2(x - 3) + 1 = 5 - 4(x - 2)$
E)
$\frac{x}{3} - \frac{x - 3}{4} = \frac{x + 1}{6}$
F)
$\frac{2}{5} \times ( \frac{5}{4} x - \frac{5}{6} ) = 3x - \frac{1}{6}$
G)
$0,(3)x + 1,5 = 1$
H)
$3 + |2x - 1| = 5$

(ora 10 /10:30)

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Answer 1

Răspuns:

Explicație pas cu pas:

6)

$x+y = (\frac{1}{2} -\frac{1}{3} +\frac{1}{4} -\frac{1}{5} +...+\frac{1}{2010} -\frac{1}{2011}) +(1-\frac{1}{2} +\frac{1}{3}-\frac{1}{4} +\frac{1}{5}-\frac{1}{6} + ... \frac{1}{2009}-\frac{1}{2010})=$

$=1 + (\frac{1}{2} -\frac{1}{2} )+(-\frac{1}{3} +\frac{1}{3} )+(\frac{1}{4} -\frac{1}{4} )+(-\frac{1}{5} +\frac{1}{5} ) + ... + (\frac{1}{2010} -\frac{1}{2010} )-\frac{1}{2011} =$

$= 1 + 0 + 0 + 0 + 0 + ... + - \frac{1}{2011} =$

$= 1 - \frac{1}{2011} =$

$= \frac{2011}{2011} - \frac{1}{2011} =$

$= \frac{2011-1}{2011} =$

$= \frac{2010}{2011}$

Deci media aritmetica a numerelor x si y este:

$\frac{x+y}{2} = \frac{\frac{2010}{2011}}2} = \frac{1005}{2011}$

7.

a)

x + 17 = 12

x = 12 - 17

x = -5

b)

3x - 5 = 22

3x = 22 + 5

3x = 27

x = 27/3

x = 9

c)

5x + 4 = 7x + 1

5x - 7x = 1 - 4

- 2x = - 3

2x = 3

x = 3/2

d)

2(x-3) + 1 = 5 - 4(x-2)

2x - 6 + 1 = 5 - 4x + 8

2x - 5 = 13 - 4x

2x + 4x = 13 + 5

6x = 18

x = 18/6

x = 3

e)

$\frac{x}{3} -\frac{x-3}{4} =\frac{x+1}{6}$

$\frac{4x}{12} -\frac{3(x-3)}{12} =\frac{2(x+1)}{12}$

$4x -3(x-3) =2(x+1)$

$4x -3x +9 = 2x +2$

$x +9 = 2x +2$

$2x -x= 9 - 2$

$x = 7$

f)

$\frac{2}{5}*(\frac{5}{4}x -\frac{5}{6} )=3x -\frac{1}{6}$

$\frac{2}{5}*\frac{5}{4}x - \frac{2}{5}*\frac{5}{6}=3x -\frac{1}{6}$

$\frac{x}{2} - \frac{1}{3}=3x -\frac{1}{6}$

$\frac{3x}{6} - \frac{2}{6}=\frac{6*3x}{6} -\frac{1}{6}$

$3x- 2=18x -1$

$3x- 18x = -1 + 2$

$-15x = 1$

$x=-\frac{1}{15}$

g)

$0,(3)x + 1,5 = 1$

$\frac{1}{3}x + \frac{3}{2} = 1$

$\frac{2x}{6} + \frac{9}{6} = \frac{6}{6}$

$2x + 9 = 6\\2x = 6-9\\2x = -3\\x = -\frac{3}{2}$

h)

$3 + |2x - 1|=5\\|2x - 1\ = 5 - 3\\|2x - 1\ = 2\\\\\left \{{{2x-1=+2} \atop{2x-1=-2}} \right.$

$2x-1 = 2\\2x = 2 + 1 \\2x = 3\\x = \frac{3}{2}$

$2x-1 = -2\\2x = -2 + 1 \\2x = -1\\x = -\frac{1}{2}$