我國(guó)著名數(shù)學(xué)家華羅庚說(shuō)過(guò)“數(shù)缺形時(shí)少直觀,形少數(shù)時(shí)難入微”,數(shù)形結(jié)合是解決數(shù)學(xué)問(wèn)題的重要思想方法.例如,代數(shù)式|x-2|的幾何意義是數(shù)軸上x所對(duì)應(yīng)的點(diǎn)與2所對(duì)應(yīng)的點(diǎn)之間的距離;因?yàn)閨x+1|=|x-(-1)|.所以|x+1|的幾何意義就是數(shù)軸上x所對(duì)應(yīng)的點(diǎn)與-1所對(duì)應(yīng)的點(diǎn)之間的距離.
發(fā)現(xiàn)問(wèn)題:代數(shù)式|x+1|+|x-2|的最小值是多少?
探究問(wèn)題:如圖,點(diǎn)A,B,P分別表示的是-1,2,x,AB=3.
∵|x+1|+|x-2|的幾何意義是線段PA與PB的長(zhǎng)度之和.
∴當(dāng)點(diǎn)P在線段AB上時(shí),PA+PB=3;當(dāng)點(diǎn)P在點(diǎn)A的左側(cè)或點(diǎn)B的右側(cè)時(shí),PA+PB>3,
∴|x+1|+|x-2|的最小值是3.
(1)解決問(wèn)題,|5-(-3)|的值是
8
8
.
(2)|x-4|+|x+2|的最小值是
6
6
.
(3)當(dāng)a為何值時(shí),代數(shù)式|x+a|+|x-3|的最小值是2.