If z^2 – 4z > 5, which of the following is always true?

Hi, answer must be E.

As z^2 - 4z > 5
=> z^2 - 4z - 5 > 0
(z - 5)(z + 1) > 0

when -1 < z < 5, (z-5)(z+1) < 0 always
when z = -1 or 5, (z-5)(z+1) = 0
when z < -1 or z > 5, then (z-5)(z+1) > 0 always

since none of the options match z < -1 or z > 5, answer must be E

My doubt is, since any value Z>5 satisfy the inequality, why A is wrong?

We’ll treat it as an equation first:

z^2 – 4z = 5

z^2 – 4z – 5 = 0

(z - 5)(z + 1) = 0

z = 5 or z = -1

The two solutions above divide the number line into three intervals:

1) z < -1

2) -1 < z < 5

3) z > 5

For each of these intervals, if one number from the interval satisfies the inequality, then the whole interval will satisfy the inequality.

1) z < -1

Let’s pick z = -2:

(-2)^2 - 4(-2) > 5?

4 + 8 > 5? (Yes)

2) -1 < z < 5

Let’s pick z = 0:

0^2 - 4(0) > 5?

0 - 0 > 5? (No)

3) z > 5

Let’s pick z = 6:

6^2 - 4(6) > 5?

36 - 24 > 5? (Yes)

We see that the solution to the inequality is z < -1 or z > 5. Note that the question is asking for a statement that is ALWAYS true; for any of the given answer choices, we can always find a value of z that does not satisfy the given inequality. Thus, none of the provided answer choices are always true.

Answer: E

I think the answer should be A. It looks same as the Question which asks "must be true". There is in value of z>5 which would not satisfy the given question condition.
Please let me know where i am wrong.

Hi, the answer is actually E. The question asks which MUST BE TRUE.

if you solve the equation, it looks like (z+1)(z-5)>0.
This signifies that either (a) z<-1 or (b) z>5
Therefore it z>5 is NOT A MUST because z can be less than -1. Does this make sense? let me know if you need further explanation.

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