vitorpteixeira wrote:

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

(A) z > 5

(B) z < 5

(C) z > -1

(D) z < 1

(E) none of above

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

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