Question

設(shè)不等式|x+1|-|x-2|＞k的解集為R,求實(shí)數(shù)k的取值范圍.

Answer 1

解法一：根據(jù)絕對(duì)值的幾何意義,|x+1|可以看作數(shù)軸上點(diǎn)P(x)到點(diǎn)A(-1)的距離|PA|,|x-2|可以看作是數(shù)軸上點(diǎn)P(x)到點(diǎn)B(2)的距離|PB|,則|x+1|-|x-2|=|PA|-|PB|.如圖所示:

當(dāng)點(diǎn)P在線段AB上時(shí),-3≤|PA|-|PB|≤3,

當(dāng)P在A點(diǎn)左側(cè)時(shí),|PA|-|PB|=-3,

當(dāng)P在B點(diǎn)右側(cè)時(shí),|PA|-|PB|=3,

則不等式-3≤|x+1|-|x-2|≤3恒成立.

故使原不等式的解集為R的實(shí)數(shù)k的取值范圍是k＜-3.

解法二:令y=|x+1|-|x-2|

=

在直角坐標(biāo)系中,作出函數(shù)圖象如圖.

要使不等式|x+1|-|x-2|＞k對(duì)一切實(shí)數(shù)成立,則函數(shù)圖象全部都落在直線y=k的上方,則k的取值范圍為k＜

-3.