Question

8．已知实数a，b，c满足：|a-b|=5，|b-c|=7，则|a-c|=2或12．

Answer 1

分析 根据绝对值性质得出a-b=5或a-b=-5，b-c=7或b-c=-7，从而罗列出所有情况：$\left\{\begin{array}{l}{a-b=5}\\{b-c=7}\end{array}\right.$或$\left\{\begin{array}{l}{a-b=5}\\{b-c=-7}\end{array}\right.$或$\left\{\begin{array}{l}{a-b=-5}\\{b-c=7}\end{array}\right.$或$\left\{\begin{array}{l}{a-b=-5}\\{b-c=-7}\end{array}\right.$，上下两式相加可得a-c，进而得出答案．

解答 解：∵|a-b|=5，|b-c|=7，
∴a-b=5或a-b=-5，b-c=7或b-c=-7，
则$\left\{\begin{array}{l}{a-b=5}\\{b-c=7}\end{array}\right.$或$\left\{\begin{array}{l}{a-b=5}\\{b-c=-7}\end{array}\right.$或$\left\{\begin{array}{l}{a-b=-5}\\{b-c=7}\end{array}\right.$或$\left\{\begin{array}{l}{a-b=-5}\\{b-c=-7}\end{array}\right.$，
上下两式相加可得a-c=12或a-c=-2或a-c=2或a-c=-12，
即|a-c|=2或12，
故答案为：2或12．

点评 本题主要考查绝对值，熟练掌握①当a是正有理数时，a的绝对值是它本身a；②当a是负有理数时，a的绝对值是它的相反数-a；③当a是零时，a的绝对值是零是解题的关键．