How do you solve the inequality (x-1)(x+4)(x-3)>0?

Dec 12, 2016

Explanation:

There can be three situations for the inequality to hold good:

I.All terms are positive that means $x > 1 , x > - 4 \mathmr{and} x > 3$. This simply means that x>3

II $x - 1 > 0 , x + 4 < 0 \mathmr{and} x - 3 < 0$. This means x>1, but x< -4 and x<3, that is x>1 and x<-4 ( if x< -4, it would automatically be less than 3)

III$x + 4 > 0 , x - 1 < 0 \mathmr{and} x - 3 < 0$. This means x>-4, x<1 and x<3. This implies that x>-4 and x<1 (if x<1, it would automatically be less than 3)

IV $x + 3 > 0 , x - 1 < 0 \mathmr{and} x + 4 < 0$. This would mean x> -3, x<1 and x<-4. This implies that x> -3 and x<-4 (if x<-4, it would automatically be less than 1)

Analyzing the above options, it would be clear that II is not possible, because x cannot be greater than 1 and less than -4 at the same time.
Like wise IV is also not possible because x can not be greater than -3 and less than -4 at the same time

Hence I and III are the only solutions for the Inequality.