\(\left|x-5\right|=2x\)ĐK : x>=0 

TH1 : x - 5 = 2x <=> x = -5 ( loại )

TH2 : x - 5 = -2x <=> 3x = 5 <=> x = 5/3 ( tm )

Vậy tập nghiệm pt là S = { 5/3 } 

\(\left(x-2\right)^2+2\left(x-1\right)\le x^2+4\)

\(\Leftrightarrow x^2-4x+4+2x-2-x^2-4\le0\)

\(\Leftrightarrow-2x-2\le0\Leftrightarrow x+1\ge0\Leftrightarrow x\ge-1\)

Vậy tập nghiệm bft là S = { x | x > = -1 } 

Ta có: \(\left|x-5\right|=2x\)

\(\Leftrightarrow\left[{}\begin{matrix}x-5=2x\left(x\ge5\right)\\x-5=-2x\left(x< 5\right)\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x-2x=5\\x+2x=5\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}-x=5\\3x=5\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-5\left(loại\right)\\x=\dfrac{5}{3}\left(nhận\right)\end{matrix}\right.\)

\(\Leftrightarrow x^2-4x+4+2x^2-4x+2-x^2< =4\)

=>-8x<=-2

hay x>=1/4

\(a,f'\left(x\right)=3x^2-6x\\ f'\left(x\right)\le0\Leftrightarrow3x^2-6x\le0\\ \Leftrightarrow3x\left(x-2\right)\le0\Leftrightarrow0\le x\le2\)

Lời giải:

a. $f'(x)\leq 0$

$\Leftrightarrow 3x^2-6x\leq 0$

$\Leftrightarrow x(x-2)\leq 0$

$\Leftrightarrow 0\leq x\leq 2$

b.

$f'(x)=x^2-3x+2=0$

$\Leftrightarrow 3x^2-6x=x^2-3x+2=0$

$\Leftrightarrow 3x(x-2)=(x-1)(x-2)=0$

$\Leftrightarrow x-2=0$

$\Leftrightarrow x=2$

c.

$g(x)=f(1-2x)+x^2-x+2022$

$g'(x)=(1-2x)'f(1-2x)'_{1-2x}+2x-1$

$=-2[3(1-2x)^2-6(1-2x)]+2x-1$
$=-24x^2+2x+5$

$g'(x)\geq 0$

$\Leftrightarrow -24x^2+2x+5\geq 0$

$\Leftrightarrow (5-12x)(2x-1)\geq 0$

$\Leftrightarrow \frac{-5}{12}\leq x\leq \frac{1}{2}$

\(\frac{9}{x^2-4}=\frac{x-1}{x+2}+\frac{3}{x-2}\)

\(ĐKXĐ:x\ne\pm2\)

\(pt\Leftrightarrow\frac{9}{x^2-4}=\frac{x^2-3x+2}{x^2-4}+\frac{3x+6}{x^2-4}\)

\(\Leftrightarrow\frac{9}{x^2-4}=\frac{x^2+8}{x^2-4}\)

\(\Leftrightarrow x^2+8=9\Leftrightarrow x=\pm1\left(tm\right)\)

Vậy pt có 2 nghiệm là 1 và -1

Điều kện :  \(x+2\ne0\) và \(x-2\ne0\Leftrightarrow x=\pm2\)

( Khi đó \(x^2-4=\left(x+2\right)\left(x-2\right)\ne0\) )

\(\frac{9}{x^2-4}=\frac{x-1}{x+2}+\frac{3}{x-2}\)

\(\Leftrightarrow\frac{9}{\left(x-2\right)\left(x+2\right)}=\frac{\left(x-1\right)\left(x-2\right)+3\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}\)

\(\Rightarrow x^2-3x+2+3x+6=9\Leftrightarrow x^2=1\Leftrightarrow x=\pm1\)

Vậy tập nghiệm của PT là: \(S=\left\{-1;1\right\}\)

Chúc bạn học tốt !!!

`|5x| = - 3x + 2`

Nếu `5x>=0<=> x>=0` thì phương trình trên trở thành :

`5x =-3x+2`

`<=> 5x +3x=2`

`<=> 8x=2`

`<=> x= 2/8=1/4` ( thỏa mãn )

Nếu `5x<0<=>x<0` thì phương trình trên trở thành :

`-5x = -3x+2`

`<=>-5x+3x=2`

`<=> 2x=2`

`<=>x=1` ( không thỏa mãn ) 

Vậy pt đã cho có nghiệm `x=1/4`

__

`6x-2<5x+3`

`<=> 6x-5x<3+2`

`<=>x<5`

Vậy bpt đã cho có tập nghiệm `x<5`

Đk: \(x\ge1\)

BPT \(\Leftrightarrow2\sqrt{x-1}-\sqrt{x+2}-\left(x-2\right)>0\)

Đặt \(a=\sqrt{x-1}\left(a\ge0\right)\)

\(\Rightarrow\left\{{}\begin{matrix}a^2+3=x+2\\a^2-1=x-2\end{matrix}\right.\)

Bpttt: \(2a-\sqrt{a^2+3}-\left(a^2-1\right)>0\)

\(\Leftrightarrow2a-a^2+1>\sqrt{a^2+3}\)

\(\Leftrightarrow\left\{{}\begin{matrix}2a-a^2+1>0\\\left(2a-a^2+1\right)^2>a^2+3\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2a-a^2+1>0\\a^4-4a^3+a^2+4a-2>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(a-1-\sqrt{2}\right)\left(1-\sqrt{2}-a\right)>0\\\left(a-1\right)\left(a+1\right)\left(a^2-4a+2\right)>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}1-\sqrt{2}< a< 1+\sqrt{2}\left(1\right)\\\left(a-1\right)\left(a+1\right)\left(a-2-\sqrt{2}\right)\left(a-2+\sqrt{2}\right)>0\left(2\right)\end{matrix}\right.\)

Kết hợp \(a\ge0\) và (1) 

\(\Rightarrow\left\{{}\begin{matrix}a+1>0\\a-2-\sqrt{2}< 1+\sqrt{2}-2-\sqrt{2}< 0\end{matrix}\right.\) \(\Rightarrow\left(a+1\right)\left(a-2-\sqrt{2}\right)< 0\)

Chia cả hai vế của (2) cho \(\Rightarrow\left(a+1\right)\left(a-2-\sqrt{2}\right)< 0\) ta được:

\(\left(a-1\right)\left(a-2+\sqrt{2}\right)< 0\)

\(\Leftrightarrow2-\sqrt{2}< a< 1\)

\(\Leftrightarrow2-\sqrt{2}< \sqrt{x-1}< 1\)

\(\Leftrightarrow7-4\sqrt{2}< x< 2\)

Vậy...(Lol, dài ha)

\(BPT\Leftrightarrow\dfrac{\left(x-2\right)\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}\ge\dfrac{\left(3x+2\right)\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}-\dfrac{2\left(x+1\right)\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}\)

\(\Rightarrow x^2-x-2x+2-3x^2-3x-2x-2-2x^2-2\ge0\)

\(\Leftrightarrow-4x^2-8x-2\ge0\)

\(\Leftrightarrow x^2+2x+\dfrac{1}{2}\ge0\)

\(\Leftrightarrow\left(x+1\right)^2-\dfrac{1}{2}\ge0\)

Vậy bất phương trình luôn đúng \(\forall x\).

ĐKXĐ: \(x\ne1,-1\)

Ta có: \(\dfrac{x-2}{x+1}\ge\dfrac{3x+2}{x-1}-2\)

\(\dfrac{x-2}{x+1}\ge\dfrac{3x+2-2\left(x-1\right)}{x-1}\)

\(\dfrac{x-2}{x+1}-\dfrac{3x+2-2x+2}{x-1}\ge0\)

\(\dfrac{x-2}{x+1}-\dfrac{x+4}{x-1}\ge0\)

\(\dfrac{\left(x-2\right)\left(x-1\right)-\left(x-4\right)\left(x+1\right)}{x^2-1}\ge0\)

\(\dfrac{x^2-3x+2-x^2+3x+4}{x^2-1}\ge0\)

\(\dfrac{6}{x^2-1}\ge0\)

\(\Rightarrow x^2-1>0\Leftrightarrow x^2>1\Leftrightarrow\left\{{}\begin{matrix}x< -1\\x>1\end{matrix}\right.\)(TM)

Biểu thức vế trái có nghĩa khi 

 \(x\ne-2;x\ne1\\ \dfrac{x-2}{x+2}+\dfrac{x+1}{x-1}\Leftrightarrow\dfrac{x-2}{x+2}+\dfrac{x+1}{x-1}-2>0\\ \dfrac{\left(x-2\right)\left(x-1\right)+\left(x+2\right)\left(x+1\right)-2\left(x+2\right)\left(x-1\right)}{\left(x+2\right)\left(x-1\right)}\\ >0\\ \Leftrightarrow\dfrac{8-2x}{\left(x+2\right)\left(x-1\right)}>0\\ \Leftrightarrow\dfrac{4-x}{\left(x+2\right)\left(x-1\right)}>0\)  

\(\Leftrightarrow\left(x+2\right)\left(x-1\right)\left(x-4\right)< 0\) 

Lập bảng xét dấu

Vậy nghiệm của bất pt là 

\(x< -2.hay.1< x< 4\) 

ĐK: \(x\ne\dfrac{1\pm\sqrt{5}}{2}\)

TH1: \(x^2-x-1>0\Leftrightarrow\left[{}\begin{matrix}x>\dfrac{1+\sqrt{5}}{2}\\x< \dfrac{1-\sqrt{5}}{2}\end{matrix}\right.\)

\(\dfrac{\left|x^2-x\right|-2}{x^2-x-1}\ge0\)

\(\Leftrightarrow\left|x^2-x\right|-2\ge0\)

\(\Leftrightarrow\left|x^2-x\right|\ge2\)

\(\Leftrightarrow\left(\left|x^2-x\right|\right)^2\ge4\)

\(\Leftrightarrow x^4-2x^3+x^2-4\ge0\)

\(\Leftrightarrow\left(x-2\right)\left(x+1\right)\left(x^2-x+2\right)\ge0\)

\(\Leftrightarrow\left(x-2\right)\left(x+1\right)\ge0\)

\(\Leftrightarrow\left[{}\begin{matrix}x\ge2\\x\le-1\end{matrix}\right.\)

TH2: \(x^2-x-1< 0\Leftrightarrow\dfrac{1-\sqrt{5}}{2}< x< \dfrac{1+\sqrt{5}}{2}\)

\(\dfrac{\left|x^2-x\right|-2}{x^2-x-1}\ge0\)

\(\Leftrightarrow\left|x^2-x\right|\le2\)

\(\Leftrightarrow\left(x-2\right)\left(x+1\right)\left(x^2-x+2\right)\le0\)

\(\Leftrightarrow\left(x-2\right)\left(x+1\right)\le0\)

\(\Leftrightarrow-1\le x\le2\)

\(\Rightarrow\dfrac{1-\sqrt{5}}{2}< x< \dfrac{1+\sqrt{5}}{2}\)

Vậy \(S=[2;+\infty)\cup(-\infty;-1]\cup\left(\dfrac{1-\sqrt{5}}{2};\dfrac{1+\sqrt{5}}{2}\right)\)