Hello Can some one please check if i did this question right? Thank you.
-(x-4)^2 (x-1)^2 < 0
so, -(x-4)^2 = 0 and (x-1)^2 =0
so, x=4 and x =1
there fore 1<x<4
Is this correct?
Let's take a step back and think about the problem. $\displaystyle (x-4)^2(x-1)^2>0 $ is always greater than zero since everything is being squared. So the answer to this problem is ALL REAL VALUES except $\displaystyle x=4 $ and $\displaystyle x=1 $.
Now if the question is $\displaystyle (x-4)^2(x-1)^2 \geq 0 $, then the answer will be ALL REAL VALUES ONLY. Can you see the point here?
I don't see your point. The problem starts with: -(x-4)^2 (x-1)^2 < 0
It's pretty obvious there's a catch because when you first look at it you see a minus sign and you don't think of taking the square root. They're telling you to multiply by -1 to make it positive and then you can take the square root, which is positive by convention, and that gives the quadratic inequality which is the topic of the thread.
As is, it is trivial.
Like you said, first multiple both side by -1 to obtain $\displaystyle (x-4)^2 (x-1)^2 > 0$. The point here is WE DON'T HAVE TO TAKE SQUARE ROOT, because it is obvious that anything square will always be greater than zero, except for zero itself. So the answer is ALL REAL VALUE EXCEPT 4 & 1. Beside taking a square on both side of the inequality is a dangerous maneuver. For example, would you take the square root on both side when solving this inequality: $\displaystyle (x+2)^2<1$?
Why do extra work to get the answer? Just simply think about the problem, analyze it, extract the most out of it.
Hartlw. For the example $\displaystyle (x+2)^2<1$, would it still be true if x=-2.5? The right answer is $\displaystyle -3<x<-1$. The point is that it is a fatal mistake to take square root both sides when solving an inequality. The best way to solve this problem is to expand them, bring everything to one side, factor and solve. See below.
$\displaystyle (x+2)^2<1~ \Leftrightarrow~ x^2+4x+4<1~ \Leftrightarrow~ x^2+4x+3<0~ \Leftrightarrow~ (x+1)(x+3)<0$
This is a proper way of solving inequality that ensures no lost of solution.
From your solution, when x=-2 the inequality still holds.