Question

chứng tỏ rằng nếu a/b < c/d ( b>0,d>0) thì a/b < a+c/b+d<c/d

Answer 1

Ta có : \(\frac{a}{b}< \frac{c}{d}\Rightarrow ad< bc\)

\(\Rightarrow ab+ad< ab+bc\)

\(\Rightarrow a.\left(b+d\right)< b.\left(a+c\right)\)

\(\Rightarrow\frac{a}{b}< \frac{a+c}{b+d}\left(1\right)\)

Ta lại có : \(ad< bc\Rightarrow ad+cd< bc+cd\)

\(\Rightarrow d.\left(a+c\right)< c.\left(b+d\right)\)

\(\Rightarrow\frac{a+c}{b+d}< \frac{c}{d}\left(2\right)\)

Từ (1) và (2), suy ra nếu :\(\frac{a}{b}< \frac{c}{d}\)

thì : \(\frac{a}{b}< \frac{a+c}{b+d}< \frac{c}{d}\)