\(\left(4+3x\right)^2=\left(4+3x\right)^4\)

\(\Leftrightarrow1=\left(4x+3\right)^2\)

\(\Leftrightarrow\left(4+3x\right)^2=1\)

\(\Leftrightarrow\orbr{\begin{cases}4+3x=1\\4+3x=-1\end{cases}\Leftrightarrow}\orbr{\begin{cases}3x=-3\\3x=-5\end{cases}\Leftrightarrow}\orbr{\begin{cases}x=-1\\x=-\frac{5}{3}\end{cases}}\)

Vậy \(x=\left\{-1,\frac{-5}{3}\right\}\)

Gọi ƯCLN(2x + 3 ; 6x + 11) = d

=> \(\hept{\begin{cases}2x+3⋮d\\6x+11⋮d\end{cases}\Rightarrow\hept{\begin{cases}3\left(2x+3\right)⋮d\\6x+11⋮d\end{cases}\Rightarrow}\hept{\begin{cases}6x+9⋮d\\6x+11⋮d\end{cases}\Rightarrow}\left(6x+11\right)-\left(6x+9\right)⋮d}\)

=> \(2⋮d\Rightarrow d\inƯ\left(2\right)\Rightarrow d\in\left\{1;2\right\}\)

Vì \(\hept{\begin{cases}2x+3\in2k+1\\6x+11\in2k+1\end{cases}\Leftrightarrow\hept{\begin{cases}\left(2x+3\right):2\text{ dư 1}\\\left(6x+11\right):2\text{ dư 1}\end{cases}}}\)

=> d = 1

=> \(\frac{2x+3}{6x+11}\)là phân số tối giản với mọi số nguyên x

\(\frac{2x+3}{6x+11}\)

Gọi d là ƯC(2x + 3 ; 6x + 11)

\(\Rightarrow\hept{\begin{cases}2x+3⋮d\\6x+11⋮d\end{cases}}\Rightarrow\hept{\begin{cases}3\left(2x+3\right)⋮d\\6x+11⋮d\end{cases}}\Rightarrow\hept{\begin{cases}6x+9⋮d\\6x+11⋮d\end{cases}}\)

=> ( 6x + 11 ) - ( 6x + 9 ) chia hết cho d

=> 6x + 11 - 6x - 9 chia hết cho d

=> 2 chia hết cho d

=> d = 1 hoặc d = 2

* d = 2 => \(\hept{\begin{cases}2x+3⋮̸2\\6x+11⋮̸2\end{cases}}\) vì \(\hept{\begin{cases}3⋮̸2\\11⋮̸2\end{cases}}\)

=> d = 1

=> ƯCLN(2x + 3 ; 6x + 11) = 1

=> \(\frac{2x+3}{6x+11}\)tối giản với mọi số nguyên x ( đpcm ) 

Bài làm:

\(7\left|\frac{3}{5}x-40\%\right|-\frac{3}{5}:\frac{1}{2}=0,2\)

\(\Leftrightarrow7\left|\frac{3}{5}x-\frac{2}{5}\right|-\frac{3}{5}.2=\frac{1}{5}\)

\(\Leftrightarrow7\left|\frac{3}{5}x-\frac{2}{5}\right|=\frac{1}{5}+\frac{6}{5}\)

\(\Leftrightarrow\frac{7}{5}\left|3x-2\right|=\frac{7}{5}\)

\(\Leftrightarrow\left|3x-2\right|=1\)

\(\Leftrightarrow\orbr{\begin{cases}3x-2=1\\3x-2=-1\end{cases}\Leftrightarrow\orbr{\begin{cases}3x=3\\3x=1\end{cases}\Leftrightarrow}\orbr{\begin{cases}x=1\\x=\frac{1}{3}\end{cases}}}\)

Vậy \(x=1\)hoặc \(x=\frac{1}{3}\)

Học tốt!!!!

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