\(\Leftrightarrow\left(\dfrac{x+1}{2022}+1\right)+\left(\dfrac{x+3}{2020}+1\right)+\left(\dfrac{x+5}{2018}+1\right)+\left(\dfrac{x+7}{2016}+1\right)=0\)

=>x+2023=0

=>x=-2023

\(\dfrac{1}{x+1}\)-\(\dfrac{5}{x-2}\)=\(\dfrac{15}{\left(x+1\right)\left(x-2\right)}\)

\(\Leftrightarrow\)\(\dfrac{x-2}{\left(x+1\right)\left(x-2\right)}\)-\(\dfrac{5\left(x+1\right)}{\left(x+1\right)\left(x-2\right)}\)=\(\dfrac{15}{\left(x+1\right)\left(x-2\right)}\)

\(\Leftrightarrow\)x-2-5(x+1)=15

\(\Leftrightarrow\) x-2-5x-5=15

\(\Leftrightarrow\)x-5x=15+2+5

\(\Leftrightarrow\)-4x=22

\(\Leftrightarrow\)x=-\(\dfrac{11}{2}\)

vậy

nhớ like nha

\(\frac{x-2}{2017}+\frac{x-3}{2018}=\frac{x-4}{2019}+\frac{x-5}{2020}\)

<=> \(\frac{x-2}{2017}+1+\frac{x-3}{2018}+1=\frac{x-4}{2019}+1+\frac{x-5}{2020}+1\)

<=> \(\frac{x+2015}{2017}+\frac{x+2015}{2018}-\frac{x+2015}{2019}-\frac{x+2015}{2020}=0\)

<=> \(\left(x+2015\right)\left(\frac{1}{2017}+\frac{1}{2018}-\frac{1}{2019}-\frac{1}{2020}\right)=0\)

<=> x + 2015 = 0  ( vì \(\frac{1}{2017}+\frac{1}{2018}-\frac{1}{2019}-\frac{1}{2020}\ne0\)) 

<=> x = - 2015 

Vậy x = -2015.

Giải phương trình :

\(\frac{x-2}{2017}+\frac{x-3}{2018}=\frac{x-4}{2019}+\frac{x-5}{2020}\)

\(\Rightarrow\frac{x-2}{2017}+1+\frac{x-3}{2018}+1=\frac{x-4}{2019}+1+\frac{x-5}{2020}+1\)

\(\Rightarrow\frac{x+2015}{2017}+\frac{x+2015}{2018}-\frac{x+2015}{2019}-\frac{x+2015}{2020}=0\)

\(\Rightarrow\left(x+2015\right)\left(\frac{1}{2017}+\frac{1}{2018}-\frac{1}{2019}-\frac{1}{2020}\right)=0\)

Mà \(\left(\frac{1}{2017}+\frac{1}{2018}-\frac{1}{2019}-\frac{1}{2020}\right)>0\)

\(\Rightarrow x+2015=0\)

\(\Rightarrow x=-2015\)

Ta có:\(\frac{x-2}{2017}+1+\frac{x-3}{2018}+1=\frac{x-4}{2019}+1+\frac{x-5}{2020}+1\)

\(\Rightarrow\frac{x+2015}{2017}+\frac{x+2015}{2018}-\frac{x+2015}{2019}-\frac{x+2015}{2020}=0\)

\(\Rightarrow\left(x+2015\right)\left(\frac{1}{2017}+\frac{1}{2018}-\frac{1}{2019}-\frac{1}{2020}\right)=0\)

Mà \(\frac{1}{2017}+\frac{1}{2018}-\frac{1}{2019}-\frac{1}{2020}>0\)

\(\Rightarrow x+2015=0\Rightarrow x=-2015\)

\(S=\left\{-2015\right\}\)

gợi ý 

2017-x-2=2018-3-x=2019-4-x=2020-5-x

Tham khảo :

Nhận thấy vế trái luôn dương nên \(x-2020\ge0\Leftrightarrow x\ge2020\)

Với \(x\ge2020\Rightarrow\left\{{}\begin{matrix}x-2017\ge0\\2x-2018\ge0\\3x-2019\ge0\end{matrix}\right.\)

PT trở thành: \(x-2017+2x-2018+3x-2019=x-2020\)

Hay kết hợp với điều kiện \(x=\dfrac{4034}{5}\) suy ra PT đã cho vô nghiệm 

helpppppppppppppppp

Bài làm:

Pt <=> \(\left(\frac{x-1}{2020}-1\right)+\left(\frac{x-3}{2018}-1\right)+\left(\frac{x-5}{2016}-1\right)+\left(\frac{x-7}{2014}-1\right)=4-4\)

\(\Leftrightarrow\frac{x-2021}{2020}+\frac{x-2021}{2018}+\frac{x-2021}{2016}+\frac{x-2021}{2014}=0\)

\(\Rightarrow x-2021=0\Rightarrow x=2021\)

Với x = 2023 

<=> x + 1 = 2024

Khi đó P(2023) = x²⁰²³ - (x + 1).x²⁰²² + ... + (x + 1).x - 1

= x²⁰²³ - x²⁰²³ - x²⁰²² + .. + x² + x - 1

= x - 1 = 2023 - 1 = 2022

\(\left(x-2018\right)^3+\left(x-2020\right)^3=\left(2x-4038\right)\)

\(\Leftrightarrow\left(x-2018\right)^3+\left(x-2020\right)^3+\left(4038-2x\right)^3=0^{^{\left(1\right)}}\)

Áp dụng bđt \(a+b+c=0\Leftrightarrow a^3+b^3+c^3=3abc\)

Ta có \(\left(x-2018\right)+\left(x-2020\right)+\left(4038-2x\right)=0\)

\(\Leftrightarrow\left(x-2018\right)^3+\left(x-2020\right)^3+\left(4038-2x\right)^3=3\left(x-2018\right)\left(x-2020\right)\left(4038-2x\right)\)

Do đó (1) \(\Leftrightarrow3\left(x-2018\right)\left(x-2020\right)\left(4038-x\right)=0\)

<=> x-2018 =0 hoặc x-2020 = 0 hoặc 4038 -2x =0 

<=> x=2018 hoặc x=2020 hoặc x=2019

Vậy phương trình đã cho có nghiệm S={2018;2020;2019}

\(\left(x-2018\right)^3+\left(x-2020\right)^3=\left(2x-4038\right)^3\)

\(\Leftrightarrow\left(x-2018\right)^3+\left(x-2020\right)^3+\left(4038-2x\right)^3=0\)

ta có \(\left(x-2018\right)+\left(x-2020\right)+\left(4038-2x\right)=0\)

nên đặt \(\left(x-2018\right)=a;\left(x-2020\right)=b;\left(4038-2x\right)=c\Leftrightarrow a+b+c=0\)

Khi đó a³ + b³+c³ = 0 ( 1) 

mà a+b+c=0 \(\Leftrightarrow\)a+b=-c

\(\Leftrightarrow\)(a+b)³ = -c³

\(\Leftrightarrow\)a³+b³+c³ = 3abc (2)

Từ (1) và (2) \(\Leftrightarrow\)abc=0

\(\Leftrightarrow\)\(\left(x-2018\right)=0hoặc\left(x-2020\right)=0hoặc\left(4038-2x\right)=0\)

\(\Leftrightarrow\)\(x=2018hoặcx=2020hoặcx=2019\)

Vậy tập nghiệm của PT là S={2018;2019;2020}