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a)\(x+\frac{1}{3}=\frac{3}{4}\)
\(\Rightarrow x=\frac{3}{4}-\frac{1}{3}\)
\(\Rightarrow x=\frac{5}{12}\)
b)\(x-\frac{2}{5}=\frac{5}{7}\)
\(\Rightarrow x=\frac{5}{7}+\frac{2}{5}\)
\(\Rightarrow x=1\frac{4}{35}\)
c)\(-x-\frac{2}{3}=-\frac{6}{7}\)
\(\Rightarrow-x=-\frac{6}{7}+\frac{2}{3}\)
\(\Rightarrow-x=-\frac{4}{21}\)
\(\Rightarrow x=\frac{4}{21}\)
d)\(\frac{4}{7}-x=\frac{1}{3}\)
\(x=\frac{4}{7}-\frac{1}{3}\)
\(\Rightarrow x=\frac{5}{21}\)
Lời giải:
Đặt $A=x^{2011}+x^{2010}+....+x+1$
$Ax=x^{2012}+x^{2011}+...+x^2+x$
$\Rightarrow Ax-A=x^{2012}-1$
$\Rightarrow A=\frac{x^{2012}-1}{x-1}$
$B=x^{502}+x^{501}+...+x+1$
$Bx=x^{503}+x^{502}+....+x^2+x$
$\Rightarrow Bx-B=x^{503}-1$
$\Rightarrow B=\frac{x^{503}-1}{x-1}$
Khi đó: $A:B = \frac{x^{2012}-1}{x-1}: \frac{x^{503}-1}{x-1}=\frac{x^{2012}-1}{x^{503}-1}=\frac{(x^{503})^4-1}{x^{503}-1}$
Đặt $x^{503}=a$ thì:
$A:B=\frac{a^4-1}{a-1}=a^3+a^2+a+1$
$\Rightarrow A\vdots B$
\(=\dfrac{x^3+x^2+x^2+x}{x^2+x}\\ =\dfrac{x^2\left(x+1\right)+x\left(x+1\right)}{x\left(x+1\right)}\\ =\dfrac{\left(x+1\right)\left(x^2+x\right)}{x\left(x+1\right)}\\ =\dfrac{x\left(x+1\right)\left(x+1\right)}{x\left(x+1\right)}\\ =x+1\)
\(\dfrac{\left(2x+1\right)}{2}-\dfrac{1}{3}=\dfrac{\left(x+1\right)}{6}\left(MC=12\right)\\ \Leftrightarrow\dfrac{\left(2x+1\right)2}{2.6}-\dfrac{1.4}{3.4}=\dfrac{\left(x+1\right)2}{6.2}\\ \Leftrightarrow\dfrac{4x+2}{12}-\dfrac{4}{12}=\dfrac{2x+2}{12}\\ \Leftrightarrow4x+2-4=2x+2\\ \Leftrightarrow4x-2x=2+2-4\\ \Leftrightarrow2x=0\\ \Leftrightarrow x=0\)
\(x^2+4y^2+9-4xy-6x+12y\)
\(=\left(x^2-4xy+4y^2\right)+\left(-6x+12y\right)+9\)
\(=\left(x-2y\right)^2-6\left(x-2y\right)+9\)
\(=\left(x-2y-3\right)^2\)
Ta có : \(x\left|x+2\right|\ge x^2-x-6\)
\(\Rightarrow x\left|x+2\right|-x^2+x+6\ge0\)
TH1 : \(x+2\ge0\left(x\ge-2\right)\)
BPt \(\Leftrightarrow x\left(x+2\right)-x^2+x+6\ge0\)
\(\Leftrightarrow x^2+2x-x^2+x+6\ge0\)
\(\Leftrightarrow3x+6\ge0\)
\(\Leftrightarrow x\ge-\dfrac{6}{3}=-2\)
- Kết hợp điều kiện \(\Rightarrow x\ge-2\)
TH2 : \(x+2< 0\left(x< -2\right)\)
BPt \(\Leftrightarrow-x\left(x+2\right)-x^2+x+6\ge0\)
\(\Leftrightarrow-x^2-2x-x^2+x+6\ge0\)
\(\Leftrightarrow-2x^2-x+6\ge0\)
\(\Leftrightarrow-2\le x\le\dfrac{3}{2}\)
- Kết hợp điều kiện \(\Rightarrow\)Không có x thỏa mãn .
Vậy bpt có nghiệm \(x=[-2;+\infty)\)