Giải bài 125 trang 60

Đề bài

Tìm các số nguyên x sao cho:

a) $\frac{-2}{5}<\frac{x}{15}<\frac{1}{6}$

b) $\frac{1}{2}+\frac{1}{3}-\frac{5}{12}<2x<\frac{-12}{31}+\frac{-136}{-31}$

Lời giải chi tiết

a)

Ta có:

$\begin{array}{c}\frac{-2}{5}=\frac{-2.18}{5.18}=\frac{-36}{90}\\ \frac{x}{15}=\frac{x.6}{15.6}=\frac{6x}{90}\\ \frac{1}{6}=\frac{1.15}{6.15}=\frac{15}{90}\end{array}$

Do đó:

$\begin{array}{c}\frac{-2}{5}<\frac{x}{15}<\frac{1}{6}\iff \frac{-36}{90}<\frac{6x}{90}<\frac{15}{90}\\ \Rightarrow -36<6x<15\Rightarrow -6<x<2,5\\ \Rightarrow x=\left\{-5;-4;-3;-2;-1;0;1;2\right\}\end{array}$

b) Ta có:

 $\begin{array}{c}\frac{1}{2}+\frac{1}{3}-\frac{5}{12}=\frac{1.6}{2.6}+\frac{1.4}{3.4}-\frac{5}{12}=\frac{6}{12}+\frac{4}{12}-\frac{5}{12}=\frac{6+4-5}{12}=\frac{5}{12}\\ \frac{-12}{31}+\frac{-136}{-31}=\frac{-12}{31}+\frac{136}{31}=\frac{-12+136}{31}=\frac{124}{31}=4\end{array}$

Do đó:

$\begin{array}{c}\frac{1}{2}+\frac{1}{3}-\frac{5}{12}<2x<\frac{-12}{31}+\frac{-136}{-31}\\ \iff \frac{5}{12}<2x<4\iff \frac{5}{24}<x<2\\ \iff x=1\end{array}$