Nghiệm của phương trình \(\left| {x - \dfrac{1}{{2020}}} \right| + \left| {x - \dfrac{2}{{2020}}} \right| + \left| {x - \dfrac{3}{{2020}}} \right|\)\( + ... + \left| {x - \dfrac{{2019}}{{2020}}} \right| = 2020x - 2020\) là:

NX: \(VT \ge 0\) nên \(VP = 2020x - 2020 \ge 0 \Leftrightarrow x \ge 1\)

Khi đó \(x - \dfrac{1}{{2020}} > 0\), \(x - \dfrac{2}{{2020}} > 0\),…,\(x - \dfrac{{2019}}{{2020}} > 0\)

Phương trình trở thành:

\(x - \dfrac{1}{{2020}} + x - \dfrac{2}{{2020}} + x - \dfrac{3}{{2020}} + ... + x - \dfrac{{2019}}{{2020}} = 2020x - 2020\)

\( \Leftrightarrow 2019x - \left( {\dfrac{1}{{2020}} + \dfrac{2}{{2020}} + ... + \dfrac{{2019}}{{2020}}} \right) = 2020x - 2020\)

\( \Leftrightarrow 2019x - \dfrac{{1 + 2 + 3 + ... + 2019}}{{2020}} = 2020x - 2020\)

\( \Leftrightarrow 2019x - \dfrac{{\left( {1 + 2019} \right).2019}}{{2.2020}} = 2020x - 2020\)

\( \Leftrightarrow 2019x - \dfrac{{2019}}{2} = 2020x - 2020\)

\( \Leftrightarrow 2020 - \dfrac{{2019}}{2} = 2020x - 2019x\)

\( \Leftrightarrow x = \dfrac{{2021}}{2}\left( {TM} \right)\)

Vậy phương trình có nghiệm \(x = \dfrac{{2021}}{2}\).