a, \(2^{x-1}+5.2^{x-2}=\frac{7}{32}\)
=>\(2^{x-1}+\frac{5}{2}.2^{x-1}=\frac{7}{32}\)
=>\(2^{x-1}\left(1+\frac{5}{2}\right)=\frac{7}{32}\)
=>\(2^{x-1}\cdot\frac{7}{2}=\frac{7}{32}\)
=>\(2^{x-1}=\frac{1}{16}=\frac{1}{2^4}=2^{-4}\)
=>x-1=-4
=>x=-5
b, |x - 4| + |x - 10| + |x + 101| + |x + 990| + |x + 1000| = |4-x|+|10-x|+|x+101|+|x+990|+|x+1000|
Ta có: \(\left|4-x\right|\ge4-x;\left|10-x\right|\ge10-x;\left|x+990\right|\ge x+990;\left|x+1000\right|\ge x+1000\)
\(\Rightarrow\left|4-x\right|+\left|10-x\right|+\left|x+990\right|+\left|x+1000\right|\ge4-x+10-x+x+990+x+1000\)
\(\Rightarrow\left|4-x\right|+\left|10-x\right|+\left|x+101\right|+\left|x+990\right|+\left|x+1000\right|\ge2004+\left|x+101\right|\)
\(\Rightarrow2005\ge2004+\left|x+101\right|\)
\(\Rightarrow\left|x+1\right|\le1\)
\(\Rightarrow-1\le x+101\le1\)
\(\Rightarrow-102\le x\le-100\)
Vì \(x\in Z\)
\(\Rightarrow x\in\left\{-102;-101;-100\right\}\)
