Question

1. Cho 2 số hữu tỉ \(\dfrac{a}{b}\) và \(\dfrac{c}{d}\) ( b > 0, d > 0 ). Chứng tỏ rằng:

a) Nếu \(\dfrac{a}{b}< \dfrac{c}{d}\) thì ad < bc

b) Nếu ad < bc thì \(\dfrac{a}{b}< \dfrac{c}{d}\)

2. Chứng tỏ rằng nếu \(\dfrac{a}{b}< \dfrac{c}{d}\) ( b > 0, d > 0 ) thì \(\dfrac{a}{b}< \dfrac{a+c}{b+d}< \dfrac{c}{d}\)

Answer 1

1. Ta có: \(\dfrac{a}{b}=\dfrac{ab}{cd},\dfrac{c}{d}=\dfrac{bc}{bd}\)

a) Mẫu chung bd > 0 ( do b > 0, d > 0 ) nên nếu \(\dfrac{ad}{bd}< \dfrac{bc}{bd}\) thì ad < bc

b) Ngược lại, Nếu ad < bc thì \(\dfrac{ad}{bd}< \dfrac{bc}{bd}.\Rightarrow\dfrac{a}{b}< \dfrac{c}{d}\)

Ta có thể viết: \(\dfrac{a}{b}< \dfrac{c}{d}\Leftrightarrow ad< bc\)

Answer 2

2. a) Ta có: \(\dfrac{a}{b}< \dfrac{c}{d}\Rightarrow ad< bc\) ( 1 )

Thêm ab vào 2 vế của (1): \(ad+ab< bc+ab\)

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

Thêm cd vào 2 vế của (1): \(ad+cd< bc+cd\)

\(d\left(a+c\right)< c\left(b+d\right)\Rightarrow\dfrac{a+c}{b+d}< \dfrac{c}{d}\) ( 3 )

Từ (2) và (3) ta có: \(\dfrac{a}{b}< \dfrac{a+c}{b+d}< \dfrac{c}{d}\)

Answer 3

1.

a) Ta có: \(\dfrac{a}{b}< \dfrac{c}{d}\)

\(\Rightarrow\dfrac{ad}{bd}< \dfrac{bc}{bd}\)

\(\Rightarrow ad< bc\left(đpcm\right)\)

Vậy ad < bc

Answer 4

1. Ta có :

\(\dfrac{a}{b}\) và \(\dfrac{c}{d}\left(b>0;d>0\right)\)

a, +) \(\dfrac{a}{b}=\dfrac{c}{d}\Leftrightarrow ad=bc\)

\(\Leftrightarrow\) Nếu \(\dfrac{a}{b}< \dfrac{c}{d}\Leftrightarrow ad< bc\)

b, +) \(ad=bc\Leftrightarrow\dfrac{a}{b}=\dfrac{c}{d}\)

\(\Leftrightarrow\) Nếu \(ad< bc\Leftrightarrow\dfrac{a}{b}< \dfrac{c}{d}\)

2. Ta có :

\(\dfrac{a}{b}< \dfrac{c}{d}\left(b>0;d>0\right)\)

\(\Leftrightarrow ad< bc\)

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

\(\Leftrightarrow a\left(d+b\right)< b\left(a+c\right)\)

\(\Leftrightarrow\dfrac{a}{b}< \dfrac{a+c}{b+d}\left(1\right)\)

\(\Leftrightarrow ad+cd< bc+cd\)

\(\Leftrightarrow d\left(a+c\right)< c\left(b+d\right)\)

\(\Leftrightarrow\dfrac{a+c}{b+d}< \dfrac{c}{d}\left(2\right)\)

Từ \(\left(1\right)+\left(2\right)\left(\Leftrightarrowđpcm\right)\)

Answer 5

b)nếu ad < bd\(\Rightarrow\dfrac{ad}{bd}< \dfrac{cb}{bd}\)

\(\Rightarrow\dfrac{a}{b}< \dfrac{c}{d}\) đpcm