Question

8．已知實數(shù)a，b，c滿足：|a-b|=5，|b-c|=7，則|a-c|=2或12．

Answer 1

分析 根據(jù)絕對值性質(zhì)得出a-b=5或a-b=-5，b-c=7或b-c=-7，從而羅列出所有情況：$\left\{\begin{array}{l}{a-b=5}\\{b-c=7}\end{array}\right.$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$\left\{\begin{array}{l}{a-b=-5}\\{b-c=-7}\end{array}\right.$，上下兩式相加可得a-c，進(jìn)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，
則$\left\{\begin{array}{l}{a-b=5}\\{b-c=7}\end{array}\right.$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$\left\{\begin{array}{l}{a-b=-5}\\{b-c=-7}\end{array}\right.$，
上下兩式相加可得a-c=12或a-c=-2或a-c=2或a-c=-12，
即|a-c|=2或12，
故答案為：2或12．

點評 本題主要考查絕對值，熟練掌握①當(dāng)a是正有理數(shù)時，a的絕對值是它本身a；②當(dāng)a是負(fù)有理數(shù)時，a的絕對值是它的相反數(shù)-a；③當(dāng)a是零時，a的絕對值是零是解題的關(guān)鍵．