\(\dfrac{x-1}{1}+\dfrac{x-1}{2}=\dfrac{x}{3}+\dfrac{x}{4}-\dfrac{7}{12}\\ =>x-1+\dfrac{x}{2}-\dfrac{1}{2}=\dfrac{x}{3}+\dfrac{x}{4}-\dfrac{7}{12}\\ =>\left(x+\dfrac{x}{2}\right)+\left(-1-\dfrac{1}{2}\right)=\left(\dfrac{x}{3}+\dfrac{x}{4}\right)-\dfrac{7}{12}\\ =>\dfrac{3}{2}x-\dfrac{3}{2}=\dfrac{7x}{12}-\dfrac{7}{12}\\ =>\dfrac{3}{2}x-\dfrac{7}{12}x=-\dfrac{7}{12}+\dfrac{3}{2}\\ =>\dfrac{11}{12}x=\dfrac{11}{12}=\\ =>x=\dfrac{11}{12}:\dfrac{11}{12}\\ =>x=1\)

Ta có:

\(\left|x-9\right|+\left|2-x\right|\ge\left|x-9+2-x\right|=\left|-7\right|=7\)

Dấu "=" xảy ra:

\(\left(x-9\right)\left(2-x\right)\ge0\\ \Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x-9\ge0\\2-x\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}x-9\le0\\2-x\le0\end{matrix}\right.\end{matrix}\right.\\ \Leftrightarrow2\le x\le9\)

a: Trên cùng một nửa mặt phẳng bờ chứa tia Ox, ta có: \(\widehat{xOt}< \widehat{xOy}\left(50^0< 100^0\right)\)

nên tia Ot nằm giữa hai tia Ox và Oy

=>\(\widehat{xOt}+\widehat{tOy}=\widehat{xOy}\)

=>\(\widehat{tOy}=100^0-50^0=50^0\)

b: 

Vì tia Ot nằm giữa hai tia Ox và Oy

và \(\widehat{xOt}=\widehat{yOt}\left(=50^0\right)\)

nên Ot là phân giác của góc xOy
 

\(\dfrac{4}{15}< \dfrac{x}{30}< \dfrac{1}{3}\)

=>\(\dfrac{8}{30}< \dfrac{x}{30}< \dfrac{10}{30}\)

=>8<x<10

=>x=9

\(\dfrac{4}{15}< \dfrac{x}{30}< \dfrac{1}{3}\\ =>\dfrac{8}{30}< \dfrac{x}{30}< \dfrac{10}{30}\\ =>8< x< 10\)

\(2x^2+4x+3\)

\(=2\left(x^2+2x+\dfrac{3}{2}\right)\)

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

\(=2\left(x+1\right)^2+1>=1>0\forall x\)

Ta có:

\(2x^2+4x+3\\ =\left(2x^2+4x+2\right)+1\\ =2\left(x^2+2x+1\right)+1\\ =2\left(x+1\right)^2+1\ge1>0\)

=> Bt luôn dương 

\(\dfrac{\left(\dfrac{2}{3}\right)^3\cdot\left(\dfrac{3}{4}\right)^2\cdot\left(-1\right)^5}{\left(\dfrac{2}{5}\right)\cdot\left(-\dfrac{5}{12}\right)^2}=\dfrac{\dfrac{2^3}{3^3}\cdot\dfrac{3^2}{4^2}\cdot\left(-1\right)}{\dfrac{2}{5}\cdot\dfrac{25}{144}}\)

\(=\dfrac{\dfrac{1}{2\cdot3}\cdot\left(-1\right)}{\dfrac{5}{72}}=-\dfrac{1}{6}\cdot\dfrac{72}{5}=-\dfrac{12}{5}\)

\(\dfrac{\left(-3\right)^{10}\cdot15^5}{25^3\cdot\left(-9\right)^7}=\dfrac{3^{10}\cdot3^5\cdot5^5}{5^6\cdot\left(-1\right)\cdot3^{14}}=-\dfrac{3}{5}\)

\(\dfrac{\left(-3\right)^{10}.15^5}{25^3.\left(-9\right)^7}\)

\(=\dfrac{\left(-3\right)^{10}.\left(3.5\right)^5}{\left(5^2\right)^3.\left(-3^2\right)^7}\)

\(=\dfrac{\left(-3\right)^{10}.3^5.5^5}{5^6.\left(-3\right)^{14}}\)

\(=\dfrac{1.3^5.1}{5.3^4}\)

\(=\dfrac{3}{5.1}\)

\(=\dfrac{3}{5}\)

\(N=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{49}-\dfrac{1}{50}\)

\(=1+\dfrac{1}{3}+...+\dfrac{1}{49}+\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{50}-2\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{50}\right)\)

\(=1+\dfrac{1}{2}+...+\dfrac{1}{50}-1-\dfrac{1}{2}-...-\dfrac{1}{25}\)

\(=\dfrac{1}{26}+\dfrac{1}{27}+...+\dfrac{1}{50}\)

=>M=N

\(A=\left(3x+1\right)^3-\left(y-2\right)^2+\left(y-1\right)^3+\left(x+y\right)^2\)

Thay x=-1/3;y=3 vào A, ta được:

\(A=\left[3\cdot\dfrac{-1}{3}+1\right]^3-\left(3-2\right)^2+\left(3-1\right)^3+\left(-\dfrac{1}{3}+3\right)^2\)

\(=-1^2+2^3+\left(\dfrac{8}{3}\right)^2\)

\(=\dfrac{64}{9}+7=\dfrac{127}{9}\)

\(A=\left(3x+1\right).3-\left(y-2\right).2+\left(y-1\right).3+\left(x+y\right).2\\ \Leftrightarrow A=3.\left(3x+1+y-1\right)+2.\left(x+y-y+2\right)\\ \Leftrightarrow A=3.\left(3x+y\right)+2.\left(x+2\right)\)
Thay \(x=-\dfrac{1}{3};y=-3\) được:
\(A=3.\left[3.\left(-\dfrac{1}{3}\right)+\left(-3\right)\right]+2.\left[\left(-\dfrac{1}{3}\right)+2\right]\\ \Leftrightarrow A=3.\left(-1-3\right)+2.\dfrac{5}{3}\\ \Leftrightarrow A=3.\left(-4\right)+2.\dfrac{5}{3}\\ \Leftrightarrow A=-12+\dfrac{10}{3}\\ \Leftrightarrow A=-\dfrac{26}{3}\)
Vậy \(A=-\dfrac{26}{3}\) tại \(x=-\dfrac{1}{3};y=-3\)