Question

[hoang thi my nga]

Cho A>1,B>0,C>0 so sanh 
a/b va a+c/b+c

Answer 1

[kudo shinichi]

Xét hiệu:

\(\frac{a}{b}-\frac{a+c}{b+c}=\frac{a\left(b+c\right)-b\left(a+c\right)}{b.\left(b+c\right)}=\frac{ab+ac-ab-bc}{b.\left(b+c\right)}=\frac{c\left(a-b\right)}{b.\left(b+c\right)}\)

Ta có: \(b.\left(b+c\right)>0\)

Với \(a=b\Rightarrow a-b=0\Rightarrow\frac{c.\left(a-b\right)}{b.\left(b+c\right)}=\frac{c.0}{b+\left(b+c\right)}=0\Rightarrow\frac{a}{b}=\frac{a+c}{b+c}\)

Với \(a>b\Rightarrow a-b>0\Rightarrow\frac{c.\left(a-b\right)}{b.\left(b+c\right)}>0\Rightarrow\frac{a}{b}>\frac{a+c}{b+c}\)

Với \(a< b\Rightarrow a-b< 0\Rightarrow\frac{c.\left(a-b\right)}{b.\left(b+c\right)}< 0\Rightarrow\frac{a}{b}< \frac{a+c}{b+c}\)

Vậy với \(a=b th\text{ì} \frac{a}{b}=\frac{a+c}{b+c}\)

           \(a>bth\text{ì}\frac{a}{b}>\frac{a+c}{b+c}\)

          \(a< th\text{ì}\frac{a}{b}< \frac{a+c}{b+c}\) 

Tham khảo nhé~