Question

[Trần Thị Hảo]

Tìm x biết: \(\left|x+\dfrac{1}{2}\right|+\left|x+\dfrac{1}{6}\right|+\left|x+\dfrac{1}{12}\right|+\left|x+\dfrac{1}{20}\right|+...+\left|x+\dfrac{1}{110}\right|=11x\)

Answer 1

[Nguyễn Thanh Hằng]

Với \(\forall x\) ta có :

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

+) \(\left|x+\dfrac{1}{6}\right|\ge0\)

..........................

+) \(\left|x+\dfrac{1}{110}\right|\ge0\)

\(\Leftrightarrow\left|x+\dfrac{1}{2}\right|+\left|x+\dfrac{1}{6}\right|+.........+\left|x+\dfrac{1}{110}\right|\ge0\)

Mà \(\left|x+\dfrac{1}{2}\right|+\left|x+\dfrac{1}{6}\right|+........+\left|x+\dfrac{1}{110}\right|=11x\)

\(\Leftrightarrow11x\ge0\)

\(\Leftrightarrow x\ge0\)

Với \(x\ge0\) thì :

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

+) \(\left|x+\dfrac{1}{6}\right|=x+\dfrac{1}{6}\)

.....................................

+) \(\left|x+\dfrac{1}{110}\right|=x+\dfrac{1}{110}\)

\(\Leftrightarrow x+\dfrac{1}{2}+x+\dfrac{1}{6}+......+x+\dfrac{1}{110}=11x\)

\(\Leftrightarrow11x+\left(\dfrac{1}{2}+\dfrac{1}{6}+........+\dfrac{1}{110}\right)=11x\)

\(\Leftrightarrow0x=\dfrac{1}{2}+\dfrac{1}{6}+....+\dfrac{1}{110}\) (vô lí)

\(\Leftrightarrow x\in\varnothing\)