GmatKiss wrote: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
Try to show that the answers DON'T have to be true.
Plug z=10 into z² - 4z > 5:
10² - 40 > 5
60 > 5.
This works.
Since it's possible that z=10, eliminate B, D and E.
Plug z= -10 into z² - 4z > 5:
(-10)² - 4(-10) > 5
100+40 > 5
140 > 5.
This works.
Since it's possible that z=-10, eliminate A and C.
None of the answers must be true.
Algebraically:
z² - 4z - 5 > 0
(z+1)(z-5) > 0.
Thus, z=-1 and z=5 are the critical points: the values where z² - 4z - 5 = 0.
When z is ANY other value, z² - 4z - 5 < 0 or z² - 4z - 5 > 0.
To determine the range of z, test values to the left and right of each critical point.
Case 1: z < -1.
Let z = -2.
(-2)² - 4(-2) - 5 > 0
7>0. This works.
z<-1 is part of the range.
Case 2: -1<z<5.
Let z=0.
(0)² - 4(0) - 5 > 0
-5>0.
Doesn't work.
-1<z<5 is not part of the range.
Case 3: z>5.
Let z=6.
(-6)² - 4(6) - 5 > 0.
7>0.
This works.
z>5 is part of the range.
Thus, z<-1 OR z>5 .
Since it's possible that z<-1 or that z>5, none of the answers MUST be true.
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