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

`<=> 2x-6 >= 0`

`<=> 2x >=6`

`<=> x>=3`

Vật bpt đã cho có tập nghiệm \(S=\left\{x|x\ge3\right\}\)

=>2x-6>=0 hoặc x+1<0

=>x>=3 hoặc x<-1

Ta có: \(\sqrt{2x+7}-6=x\)

\(\Leftrightarrow\sqrt{2x+7}=x+6\)

\(\Leftrightarrow x^2+12x+36-2x-7=0\)

\(\Leftrightarrow x^2+10x+29=0\)(Vô lý)

Vậy: \(S=\varnothing\)

Điều kiện : x ≥ 0

\(\sqrt{2x+97}-6=x\text{⇔}\sqrt{2x+97}=x+6\\ \text{⇔}2x+97=x^2+12x+36\text{⇔}x^2+10x-61=0\\ \text{⇔}\left[{}\begin{matrix}x=-5+\sqrt{86}\\x=-5-\sqrt{86}\end{matrix}\right.\)

\(\sqrt{2x+97}-6=x\)

\(\Leftrightarrow\sqrt{2x+97}=x+6\)

\(\Leftrightarrow\left\{{}\begin{matrix}x+6\ge0\\2x+97=\left(x+6\right)^2\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-6\\x^2+10x-61=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-6\\\left[{}\begin{matrix}x=-5+\sqrt{86}\\x=-5-\sqrt{86}\end{matrix}\right.\end{matrix}\right.\)\(\Rightarrow x=-5+\sqrt{86}\)

Vậy..,

a) => (12x-4)+(6x+15)=16

=>12x-48+6x+15=16

=>(12x+6x)-(15-48)=16

=>18x-(-33)=16

=>18x=16+(-33)

=>18x=-17

=>x=-17/18

KL

b) =>(-20x+(-40))-(12-30x)=48

=>-20x+(-40)-12+30x=48

=>(-20x+30x)+(-40-12)=48

=>10x+(-52)=48

=>10x=48-(-52)

=>10x=4

=>x=4/10

KL

a) 12( x - 4 ) + 3( 2x + 5 ) = 16

<=> 12x - 48 + 6x + 15 = 16

<=> 18x - 33 = 16

<=> 18x = 49

<=> x = 49/18

b) -10( 2x + 4 ) - 2( 6 - 15x ) = 48

<=> -20x - 40 - 12 + 30x = 48

<=> 10x - 52 = 48

<=> 10x = 100

<=> x = 10

`|2x+1|-3=x+4`

`<=>|2x+1|=x+4+3=x+7(x>=-7)`

`**2x+1=x+7`

`<=>x=7-1=6(tm)`

`**2x+1=-x-7`

`<=>3x=-6`

`<=>x=-2(tm)`

`|3x-5|=1-3x(x<=1/3)`

`**3x-5=1-3x`

`<=>6x=6`

`<=>x=1(l)`

`**3x-5=3x-1`

`<=>-5=-1` vô lý

`|2x+2|+|x-1|=10`

Nếu `x>=1`

`pt<=>2x+2+x-1=10`

`<=>3x+1=10`

`<=>3x=9`

`<=>x=3(tm)`

Nếu `x<=-1`

`pt<=>-2x-2+1-x=10`

`<=>-1-3x=10`

`<=>-11=3x`

`<=>x=-11/3(tm)`

Nếu `-1<=x<=1`

`pt<=>2x+2+1-x=10`

`<=>x+3=10`

`<=>x=7(l)`

Vậy `S={3,-11/3}`

d) 

+) Với \(x< -4\), PT \(\Rightarrow3-x-x-4-2x-6=10\) \(\Leftrightarrow x=-\dfrac{17}{4}\)  (Nhận)

+) Với \(-4\le x\le-3\), PT \(\Rightarrow3-x+x+4-2x-6=10\) \(\Leftrightarrow x=-\dfrac{9}{2}\) (Loại)

+) Với \(-3< x\le3\), PT \(\Rightarrow3-x+x+4+2x+6=10\) \(\Leftrightarrow x=-\dfrac{3}{2}\) (Nhận)

+) Với \(x>3\), PT \(\Rightarrow x+3+x+4+2x+6=10\) \(\Leftrightarrow x=-\dfrac{3}{4}\) (Loại)

  Vậy \(x\in\left\{-\dfrac{3}{2};-\dfrac{17}{4}\right\}\) 

5x+2x=6 mũ 2 - 5 mũ 0 

5x + 2x = 6² - 5⁰

=> 7x = 36 - 1

=> 7x = 35

=> x = 35 : 7 = 5

\(5x+2x=6^2-5^0\)

\(\left(5+2\right)x=36-\left(5^1\div5^1\right)\)

\(7x=36-1\)

\(7x=35\)

\(x=5\)

\(5x+2x=6^2-5^0\)

<=> \(7x=35\)

<=> \(x=5\)

vậy........

Lời giải:
ĐKXĐ: $x\neq -1$

$F=\frac{2x}{x^2+2x+1}$

$F-\frac{1}{2}=\frac{2x}{x^2+2x+1}-\frac{1}{2}=\frac{4x-x^2-2x-1}{2(x^2+2x+1)}$

$=\frac{-(x^2-2x+1)}{2(x^2+2x+1)}=\frac{-(x-1)^2}{2(x+1)^2}\leq 0$ với mọi $x\neq -1$

$\Rightarrow F\leq \frac{1}{2}$
Vậy gtln của $F$ là $\frac{1}{2}$ khi $x-1=0\Leftrightarrow x=1$

\(F=\dfrac{2x}{\left(x+1\right)^2}=\dfrac{2\left(x+1\right)-2}{\left(x+1\right)^2}=\dfrac{2}{x+1}-\dfrac{2}{\left(x+1\right)^2}\)

Đặt x + 1 = y => F = \(\dfrac{2}{y}-\dfrac{2}{y^2}\)

Đặt \(\dfrac{1}{y}=t\Rightarrow F=2t-2t^2=-2\left(t^2-t\right)=-2\left(t^2-2.t.\dfrac{1}{2}+\dfrac{1}{4}-\dfrac{1}{4}\right)=-2\left(t-\dfrac{1}{2}\right)^2+\dfrac{1}{2}\)

\(\Rightarrow F\le\dfrac{1}{2}\).Dấu "=" xảy ra khi: \(t-\dfrac{1}{2}=0\Leftrightarrow t=\dfrac{1}{2}\Leftrightarrow\dfrac{1}{y}=\dfrac{1}{2}\Leftrightarrow y=2\Leftrightarrow x+1=2\Leftrightarrow x=1\)