HOT Competition 4 Sep/8PM: If x^4 > 50, then which of the following mu

If x^4>50, then which of the following must be true?


A. |x|>3
Not necessarily. Because 3^4 = 81
We need only x^4> 50. So, some number close to 3, but just less that 3 (for eg: 2.999999999999999) will also satisfy the inequation. So, A is wrong

B. x>2.5
Not necessary. x = -3 also satisfies the inequation.

C. x<2.5
Not necessary. x = 3 also satisfies the inequation

Now, we are left with only D and E. We need value of x such that x^4 > 50
=> |x| > something
=> 1/|x| < something
So, D has to be the answer. E can't be the answer

D. 1/|x|<0.41
Right answer

E. 1/|x|>0.4
Wrong because of reason mentioned above

Answer: D

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VeritasKarishma, ScottTargetTestPrep, Bunuel

Could you please explain how does option D satisfy the "MUST BE TRUE" condition?

According to option D, |x| > 2.5

If I take x = 2.51 (x > 2.5) or x = -2.51 (x < 2.5), I get (2.51)^4 = 39.69. In fact any value of x between x = 2.5 and 2.66 (exclusive) will not satisfy the given equation x^4 > 50.

Notice that "must be true" condition means that for any value of x satisfying x^4 > 50, x MUST satisfy the other inequality. The value x = 2.51 is not a valid counter example because it doesn't satisfy x^4 > 50; in other words, we can't even pick x = 2.51 since (2.51)^4 is not greater than 50. In order to prove that 1/|x| < 0.4 is not necessarily true, you have to come up with a value of x such that x^4 > 50 is satsified, but 1/|x| < 0.4 is not satisfied.

To see why 1/|x| must be true, notice that fourth root of x^4 is |x|. Notice also that the fourth root of 50 is the square root of the square root of 50, i.e. √(√50). It is roughly equal to 2.66, so the fourth root of 50 is greater than 2.5. We have:

x^4 > 50
(x^4)^(1/4) > 50^(1/4) > 2.5
|x| > 2.5
1/|x| < 1/2.5 = 0.4

If you would like to solve the question by finding counter examples, you can actually find counter examples for each answer choice besides D. For instance, choice A cannot be correct because if we pick x = 2.7, then (2.7)^4 is greater than 50 (so x satisfies x^4 > 50) but x = 2.7 does not satisfy |x| > 3. Picking x = -2.7 and x = 2.7 show that the inequalities x > 2.5 and x < 2.5 are not necessarily true, respectively. So, B and C are also eliminated. Finally, picking x = 2.7 once more show that 1/|x| is not necessarily greater than 0.4, since 1/|2.7| is around 3.7. That eliminates E and we're left with D.

You should check out this post on my blog:

https://anaprep.com/algebra-must-be-tru ... questions/

A. |x|>3
What about 2.9?

B. x>2.5
What about x=-3?

C. x<2.5
What about 3?

D. \(\frac{1}{|x|}<0.4\)
Yes!! This means |x|>2.5

E. 1|x|>0.4
No this means |x|<2.5

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