Giải bất phương trình sau
\(\dfrac{x+1}{2006}\) + \(\dfrac{x+2}{2005}\) +\(\dfrac{x}{2006}\) >= -6
K
Khách
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UK
13 tháng 9 2017
Sửa đề:
\(VP=\sqrt{1+2005^2+\dfrac{2005^2}{2006^2}}+\dfrac{2005}{2006}\)
Ta có: \(2005^2+1=\left(2005+1\right)^2-2.2005.1=2006^2-2.2005\)
\(\Rightarrow VP=\sqrt{2006^2-2.2005+\dfrac{2005^2}{2006^2}}+\dfrac{2005}{2006}\)
\(=\sqrt{\left(2006-\dfrac{2005}{2006}\right)^2}+\dfrac{2005}{2006}\)
\(=2006-\dfrac{2005}{2006}+\dfrac{2005}{2006}=2006\)
Phương trình đã cho tương đương
\(\sqrt{x^2-2x+1}+\sqrt{x^2-4x+4}=2006\)
\(\Leftrightarrow\sqrt{\left(x-1\right)^2}+\sqrt{\left(x-2\right)^2}=2006\)
\(\Leftrightarrow\left|x-1\right|+\left|x-2\right|=2006\)
Đến đây thì tự xét trường hợp và giải tìm nghiệm, bài này không cần điều kiện nhé
NV
Nguyễn Việt Lâm
Giáo viên
20 tháng 11 2018
\(\sqrt{1+2005^2+\dfrac{2005^2}{2006^2}}=\dfrac{1}{2006}\sqrt{2006^2+2005^2+\left(2005.2006\right)^2}\)
\(=\dfrac{1}{2006}\sqrt{\left(2006-2005\right)^2+2.2005.2006+\left(2005.2006\right)^2}\)
\(=\dfrac{1}{2006}\sqrt{1+2.2005.2006+\left(2005.2006\right)^2}\)
\(=\dfrac{1}{2006}\sqrt{\left(2005.2006+1\right)^2}=\dfrac{2005.2006+1}{2006}=2005+\dfrac{1}{2006}\)
Phương trình tương đương:
\(\sqrt{\left(x-1\right)^2}+\sqrt{\left(x-2\right)^2}=2005+\dfrac{1}{2006}+\dfrac{2005}{2006}\)
\(\Leftrightarrow\left|x-1\right|+\left|x-2\right|=2006\)
TH1: \(x\ge2\): \(x-1+x-2=2006\Rightarrow2x=2009\Rightarrow x=\dfrac{2009}{2}\)
TH2: \(x\le1\) : \(1-x+2-x=2006\Rightarrow-2x=2003\Rightarrow x=\dfrac{-2003}{2}\)
TH3: \(1< x< 2:\) \(x-1+2-x=2006\Rightarrow3=2006\) (vô nghiệm)
Vậy \(\left[{}\begin{matrix}x=\dfrac{2009}{2}\\x=\dfrac{-2003}{2}\end{matrix}\right.\)
6 tháng 7 2017
\(\dfrac{x+4}{2016}+\dfrac{x+2}{2018}\ge\dfrac{x+14}{2006}+\dfrac{x+83}{1937}\)
\(\Leftrightarrow\dfrac{x+4}{2016}+1+\dfrac{x+2}{2018}+1\ge\dfrac{x+14}{2006}+1+\dfrac{x+83}{1937}+1\)
\(\Leftrightarrow\dfrac{x+2020}{2016}+\dfrac{x+2020}{2018}-\dfrac{x+2020}{2006}-\dfrac{x+2020}{1937}\ge0\)
\(\Leftrightarrow\left(x+2020\right)\left(\dfrac{1}{2016}+\dfrac{1}{2018}-\dfrac{1}{2006}-\dfrac{1}{1937}\right)\ge0\)
\(\Leftrightarrow x+2020\ge0\Leftrightarrow x\ge-2020\)
Vậy \(x\ge-2020\)
KK
13 tháng 4 2018
\(\dfrac{x+1}{2008}+\dfrac{x+2}{2007}+\dfrac{x+3}{2006}=\dfrac{x+4}{2005}+\dfrac{x+5}{2004}+\dfrac{x+6}{2003}\)
⇔\(\dfrac{x+1}{2008}+1+\dfrac{x+2}{2007}+1+\dfrac{x+3}{2006}+1=\dfrac{x+4}{2005}+1+\dfrac{x+5}{2004}+1+\dfrac{x+6}{2003}+1\)
⇔ \(\dfrac{x+2009}{2008}+\dfrac{x+2009}{2007}+\dfrac{x+2009}{2006}=\dfrac{x+2009}{2005}+\dfrac{x+2009}{2004}+\dfrac{x+2009}{2003}\)
⇔ \(\dfrac{x+2009}{2008}+\dfrac{x+2009}{2007}+\dfrac{x+2009}{2006}-\dfrac{x+2009}{2005}-\dfrac{x+2009}{2004}-\dfrac{x+2009}{2003}=0\)
⇔ \(\left(x+2009\right)\left(\dfrac{1}{2008}+\dfrac{1}{2007}+\dfrac{1}{2006}-\dfrac{1}{2005}-\dfrac{1}{2004}-\dfrac{1}{2003}\right)=0\)
⇔ x+2009=0
⇔ x=-2009
vậy x=-2009 là nghiệm của pt
13 tháng 4 2018
a) ( x2 + x )2 + 4( x2 + x ) = 12
<=> ( x2 + x )2 + 4( x2 + x ) + 4 - 16 = 0
<=> ( x2 + x + 2)2 - 16 = 0
<=> ( x2 + x + 2 + 4)( x2 + x + 2 - 4) = 0
<=> ( x2 + x + 6 )( x2 + x - 2) = 0
Do : x2 + x + 6
= x2 + 2.\(\dfrac{1}{2}x+\dfrac{1}{4}+6-\dfrac{1}{4}=\left(x+\dfrac{1}{2}\right)^2+\dfrac{23}{4}\) ≥ \(\dfrac{23}{4}\) > 0 ∀x
=> x2 + x - 2 = 0
<=> x2 - x + 2x - 2 = 0
<=> x( x - 1) + 2( x - 1) = 0
<=> ( x - 1)( x + 2 ) = 0
<=> x = 1 hoặc : x = - 2
KL.....
b) Kuroba kaito làm rùi nhé 
24 tháng 1 2018
\(\Leftrightarrow\dfrac{x}{2005}+1+\dfrac{x-1}{2006}+1=\dfrac{x-2}{2007}+1-1+1\)
\(\Leftrightarrow\dfrac{x+2005}{2005}+\dfrac{x+2005}{2006}=\dfrac{x+2005}{2007}\)
\(\Leftrightarrow\left(x+2005\right)\left(\dfrac{1}{2005}+\dfrac{1}{2006}-\dfrac{1}{2007}\right)=0\)
\(\Leftrightarrow x+2005=0\) (vì \(\dfrac{1}{2005}+\dfrac{1}{2006}-\dfrac{1}{2007}\ne0\))
\(\Leftrightarrow x=-2005\)
24 tháng 1 2018
\(\dfrac{x}{2005}+\dfrac{x-1}{2006}=\dfrac{x-2}{2007}-1\)
\(\Leftrightarrow\dfrac{x+2005}{2005}+\dfrac{x+2005}{2006}-\dfrac{x+2005}{2007}=0\)
\(\Leftrightarrow\left(x+2005\right)\left(\dfrac{1}{2005}+\dfrac{1}{2006}-\dfrac{1}{2007}\right)=0\)
\(\Leftrightarrow x=-2005\).
BT
2
19 tháng 4 2023
\(\dfrac{x+2}{2008}+\dfrac{x+3}{2007}+\dfrac{x+5}{2005}+\dfrac{x+4}{2006}=-4\\ \Rightarrow\dfrac{x+2}{2008}+\dfrac{x+3}{2007}+\dfrac{x+5}{2005}+\dfrac{x+4}{2006}+4=0\\ \Rightarrow\dfrac{x+2}{2008}+\dfrac{x+3}{2007}+\dfrac{x+5}{2005}+\dfrac{x+4}{2006}+1+1+1+1=0\\ \Rightarrow\left(\dfrac{x+2}{2008}+1\right)+\left(\dfrac{x+3}{2007}+1\right)+\left(\dfrac{x+5}{2005}+1\right)+\left(\dfrac{x+4}{2006}+1\right)=0\\ \Rightarrow\dfrac{x+2010}{2008}+\dfrac{x+2010}{2007}+\dfrac{x+2010}{2005}+\dfrac{x+2010}{2006}=0\\ \Rightarrow\left(x+2010\right)\left(\dfrac{1}{2008}+\dfrac{1}{2007}+\dfrac{1}{2005}+\dfrac{1}{2006}\right)=0\)
mà `1/2008+1/2007+1/2005+1/2006≠ 0`
`=> x+2010=0`
`=>x=-2010`
19 tháng 4 2023
\(\Leftrightarrow\left(\dfrac{x+2}{2008}+1\right)+\left(\dfrac{x+3}{2007}+1\right)+\left(\dfrac{x+5}{2005}+1\right)+\left(\dfrac{x+4}{2006}+1\right)=0\)
=>x+2010=0
=>x=-2010
10 tháng 5 2023
a:=>3x=15
=>x=5
b: =>8-11x<52
=>-11x<44
=>x>-4
c: \(VT=\left(\dfrac{x^2-\left(x-6\right)^2}{x\left(x+6\right)\left(x-6\right)}\right)\cdot\dfrac{x\left(x+6\right)}{2x-6}+\dfrac{x}{6-x}\)
\(=\dfrac{12x-36}{2x-6}\cdot\dfrac{1}{x-6}-\dfrac{x}{x-6}=\dfrac{6}{x-6}-\dfrac{x}{x-6}=-1\)
4 tháng 8 2017
số số hạng của A là :
( 2007 - 3 ) : 3 + 1 = 669 ( số )
tổng A là :
( 2007 + 3 ) . 669 : 2 = 672345
B = \(\dfrac{\dfrac{2006}{2}+\dfrac{2006}{3}+\dfrac{2006}{4}+...+\dfrac{2006}{2007}}{\dfrac{2006}{1}+\dfrac{2005}{2}+\dfrac{2004}{3}+...+\dfrac{1}{2006}}\)
B = \(\dfrac{2006.\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2007}\right)}{\left(\dfrac{2005}{2}+1\right)+\left(\dfrac{2004}{3}+1\right)+...+\left(\dfrac{1}{2006}+1\right)+1}\)
B = \(\dfrac{2006.\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2007}\right)}{\dfrac{2007}{2}+\dfrac{2007}{3}+...+\dfrac{2007}{2006}+\dfrac{2007}{2007}}\)
B = \(\dfrac{2006.\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2007}\right)}{2007.\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2006}+\dfrac{1}{2007}\right)}\)
B = \(\dfrac{2006}{2007}\)
4 tháng 8 2017
2006/1 là 2006, tách 1 của 2006 ra 2005 phân số còn lại 1
sai đề ko bn