inequality problem

[surabi]

If z² - 4z > 5 then which of the following is always true

A) z > -5

B) z < 5

C) z > -1

D) z < 1

E) z < -1

The equation can be written as z² - 4z - 5 > 0. . . (z - 5)(z + 1) > 0. If we set it equal to zero we find z > 5, z > -1. right?....
But the explanation given to this says z>5 and Z<-1 ..how is this?..I think i have some problem with solving these inequalities. Some one please help me.
The aswer is E

[ash]

Hi Surabi,
The answer is right. It is indeed E. First of all, don't try to solve inequalities as if they are equalities as you have done. Now coming to the question,
z² - 4z > 5 i.e. z(z-4)>5. Therefore either both z & z-4 are +ve or both are -ve(to be greater than 5) and the minimum value of z has to be at least 4 if z is +ve. z>4 is not an answer. Now if both are -ves, then the value of z must be less than -1 so that the product becomes more than 5.
If you find yourself in a bind, always try to plug in some value & see the result. e.g. in this case if you take z>-1, suppose z=2. The result is 2(2-4) i.e.-4 which can't be greater than 5.I rest my case...
HTH,
ash

[surabi]

Hi ashi
I got it
Thank you very much

[dkpbus]

Originally posted by ash

Hi Surabi,
The answer is right. It is indeed E. First of all, don't try to solve inequalities as if they are equalities as you have done. Now coming to the question,
z² - 4z > 5 i.e. z(z-4)>5. Therefore either both z & z-4 are +ve or both are -ve(to be greater than 5) and the minimum value of z has to be at least 4 if z is +ve. z>4 is not an answer. Now if both are -ves, then the value of z must be less than -1 so that the product becomes more than 5.
If you find yourself in a bind, always try to plug in some value & see the result. e.g. in this case if you take z>-1, suppose z=2. The result is 2(2-4) i.e.-4 which can't be greater than 5.I rest my case...
HTH,
ash

There is a mathematical solution to such type of problems which is much easier to understand

See z² - 4z > 5
=> z² - 4z - 5 > 0
=> (Z-5)(Z+1) > 0
This condition will hold true in two cases
1st case
z-5 > 0 and Z + 1 > 0
=> z > 5 and z > -1
means if z > 5 then z > -1 also so 1st case gives z > 5

2nd case both are less than 0
i.e.
z-5 < 0 and z+1 < 0
i.e.
z < 5 and z < -1
so if z < -1 it will be less than 5 also

hence z² - 4z > 5 will be true if z > 5 or z < -1
And thats the answer

DKP [^]