Solve y/(y-2) <= 2
The solution says that it divides into 2 cases.
Case 1, y>2
then the inequality becomes
y <= 2(y-2)
Case 2, y<2
then the inequality becomes
y >= 2(y-2)
Why it is needed to do so? why the inequality sign becomes <= in Case 1
and becomes >= in Case 2?
Amath - Inequality...
2008-02-28 10:27 pm
回答 (2)
2008-02-28 11:19 pm
✔ 最佳答案
Case 1, for y>2 , (y-2)>0-----and the inequality sign remains unchangedCase 2, y<2 ,(y-2)<0 and the inequality sign changes
Why we need to divid into 2 case is to ensure the denominator is either positive or negative and hence to determine the inequality sign !!!
[Note , if a>b then -a<-b]
2008-03-03 4:58 am
The key is, if you have multiply a negative number on both sides, the inequality sign change direction.
One obvious example is.
1 ＜ 2 and -2 ＞ -4
A more clever way to solve this is:
y/(y-2) ＜= 2
y(y-2) ＜= 2(y-2)^2
y^2 - 2y ＜= 2y^2 - 8y 8
y^2 - 6y 8 ＞= 0
(y-2)(y-4) ＞= 0
Therefore y ＜= -2 or y ＞= 4 (This can be done by drawing a graph or analyzing ＋ x ＋ and － x －)
One obvious example is.
1 ＜ 2 and -2 ＞ -4
A more clever way to solve this is:
y/(y-2) ＜= 2
y(y-2) ＜= 2(y-2)^2
y^2 - 2y ＜= 2y^2 - 8y 8
y^2 - 6y 8 ＞= 0
(y-2)(y-4) ＞= 0
Therefore y ＜= -2 or y ＞= 4 (This can be done by drawing a graph or analyzing ＋ x ＋ and － x －)
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