\(\text{Cách 1}:\text{Vì }x< y\)
\(\Rightarrow\frac{a}{m}< \frac{b}{m}\)
\(\Rightarrow a< b\)
\(\text{Ta có : }x=\frac{a}{m}=\frac{2.a}{2.m}=\frac{a+a}{2m}< \frac{a+b}{2m}=z\)
\(\Rightarrow x< z\left(1\right)\)
\(\text{Lại có : }z=\frac{a+b}{2m}< \frac{b+b}{2m}=\frac{2b}{2m}=\frac{b}{m}\)
\(\Rightarrow z< y\left(2\right)\)
\(\text{Từ }\left(1\right)\text{và }\left(2\right)\Rightarrow x< z< y\left(\text{đpcm}\right)\)
\(\text{Cách }2:\text{Vì }x< y\)
\(\Rightarrow\frac{a}{m}< \frac{b}{m}\)
\(\Rightarrow\frac{a}{m}\times\frac{1}{2}< \frac{b}{m}\times\frac{1}{2}\)
\(\Rightarrow\frac{a}{2m}< \frac{b}{2m}\)
\(\Rightarrow\frac{a}{2m}+\frac{a}{2m}< \frac{b}{2m}+\frac{a}{2m}\)
\(\Rightarrow\frac{2a}{2m}< \frac{a+b}{2m}\)
\(\Rightarrow\frac{a}{m}< \frac{a+b}{2m}\)
\(\Rightarrow x< z\left(1\right)\)
\(\text{Lại có : }\)\(\frac{a}{2m}< \frac{b}{2m}\)
\(\Rightarrow\frac{a}{2m}+\frac{b}{2m}< \frac{b}{2m}+\frac{b}{2m}\)
\(\Rightarrow\frac{a+b}{2m}< \frac{2b}{2m}\)
\(\Rightarrow\frac{a+b}{2m}< \frac{b}{m}\)
\(\Rightarrow z< y\left(2\right)\)
\(\text{Từ }\left(1\right)\text{và}\left(2\right)\Rightarrow x< z< y\left(\text{đpcm}\right)\)