If x is an integer and x not = 0, is x^4 < 90?

Given: \(x=integer\neq{0}\). Question: is \(x^4<90\)?

(1) \(x^{\sqrt{4}}<10\) --> \(x^2<10\). Now, since \(x\) is an integer, then max value of \(x\) is 3 --> \(3^4=81<90\). Sufficient. (Side note: possible values of \(x\) from this statement are: -3, -2, -1, 1, 2, and 3).

(2) \(\frac{2}{x^4} >0.2\) --> \(\frac{1}{x^4} >\frac{1}{10}\) --> \(x^4<10\). Sufficient. (Side note: the possible values of \(x\) from this statement are: -1 and 1).

Answer: D.

As for your doubt (red part):

Square root function cannot give negative result.

Any nonnegative real number has a unique non-negative square root called the principal square root and unless otherwise specified, the square root is generally taken to mean the principal square root.

When the GMAT provides the square root sign for an even root, such as \(\sqrt{x}\) or \(\sqrt[4]{x}\), then the only accepted answer is the positive root.

That is, \(\sqrt{25}=5\), NOT +5 or -5. In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5. Even roots have only non-negative value on the GMAT.

Odd roots will have the same sign as the base of the root. For example, \(\sqrt[3]{125} =5\) and \(\sqrt[3]{-64} =-4\).

Hope it helps.
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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Since x is an integer, \(4^4 = 256 > 90\) and \(3^4 = 81\), \(x^4 < 90\) is equivalent \(-3 ≤ x ≤ 3\).

Since we have 1 variable (\(x\)) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
\(x^\sqrt{4} = x^2 < 10\) tells \(-3 ≤ x ≤ 3\), which is equivalent to the question.
Condition 1) is sufficient.

Condition 2)
\(\frac{2}{x^4} > 0.2\)
\(⇔ \frac{x^4}{2} < \frac{1}{0.2} = 5\)
\(⇔ x^4 < 10\)
\(⇔ -1 ≤ x ≤ 1\), since \(x\) is an integer and \(2^4 = 16\).

In inequality questions, the law “Question is King” tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient

Condition 2) is sufficient

Therefore, D is the answer.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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