Question

Giả sử \(x=\frac{a}{m}\), \(y=\frac{b}{m}\)\(\left(a,b,m\in Z,m>0\right)\)và \(x< y\). Chứng tỏ rằng nếu chọn \(z=\frac{a+b}{2m}\)thì ta có \(x< z< y\)

Answer 1

Do x < y

=> a/m < b/m

=> a/m + a/m < a/m + b/m < b/m + b/m

=> 2a/m < a+b/m < 2b/m

=> a/m < a+b/m : 2 < b/m

=> a/m < a+b/m × 1/2 < b/m

=> a/m < a+b/2m < b/m

=> x < z < y

Answer 2

=> am<bm

=>am+am<am+bm =>a.2m<m.(a+b)

=>a/m<a+b/2m         (1)

=>am+bm<bm+bm=>m(a+b)<b.2m

=>a+b/2m<b/m      (2)

tu (1) va (2)

=>a/m<a+b/m2<b/m