Question

So sánh \(\frac{a}{b}\)và \(\frac{a+2019}{b+2019}\)với a,b thuộc Z, b>0

MK làm rồi mà k bt đúng hay sai. Giúp mk với

Answer 1

Vì b > 0 => b + 2019 > 0

Ta có: \(\frac{a}{b}=\frac{a.\left(b+2019\right)}{b.\left(b+2019\right)}=\frac{a.b+a.2019}{b.\left(b+2019\right)}=\frac{a+2019}{b+2019}=\)

\(\frac{b.\left(a+2019\right)}{b.\left(b+2019\right)}=\frac{a.b+b.2019}{b.\left(b+2019\right)}\)

TH1: Nếu a < b => \(\frac{a.b+a.2019}{b.\left(b+2019\right)}< \frac{a.b+b.2019}{b.\left(b+2019\right)}\)

                       hay \(\frac{a}{b}< \frac{a+2019}{b+2019}\)

TH2: Nếu a = b => \(\frac{a.b+a.2019}{b.\left(b+2019\right)}=\frac{a.b+b.2019}{b.\left(b+2019\right)}\)

                       hay \(\frac{a}{b}=\frac{a+2019}{b+2019}\)

TH3: Nếu a > b => \(\frac{a.b+a.2019}{b.\left(b+2019\right)}>\frac{a.b+b.2019}{b.\left(b+2019\right)}\)

                       hay \(\frac{a}{b}=\frac{a+2019}{b+2019}\)

Answer 2

Xét tích : \(a(b+2019)=ab+2019a\)

\(b(a+2019)=ab+2019b\)

Vì b > 0 nên b + 2019 > 0

Nếu a > b thì \(ab+2019a>ab+2019b\)

\(a(b+2019)>b(a+2019)\)

\(\Rightarrow\frac{a}{b}>\frac{a+2019}{b+2019}\)

Nếu a < b thì \(ab+2019a< ab+2019b\)

\(a(b+2019)< b(a+2019)\)

\(\Rightarrow\frac{a}{b}< \frac{a+2019}{b+2019}\)

Nếu a = b thì rõ ràng \(\frac{a}{b}=\frac{a+2019}{b+2019}\)