Data Sufficiency

[Math Revolution GMAT math practice question]

[x] is the greatest integer less than or equal to x. < x > is the least integer greater than or equal to x. What is the value of x?

1) [x] = 2
2) < x > = 2

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

[x] is the greatest integer less than or equal to x. < x > is the least integer greater than or equal to x. What is the value of x?

1) [x] = 2
2) < x > = 2

$$? = x$$
$$\left( 1 \right)\,\,\,\,\left[ x \right]\,\, = \,\,2\,\,\,\,\, \Leftrightarrow \,\,\,\,\,2 \le x < 3\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,x = 2\,\,\, \hfill \cr
\,{\rm{Take}}\,\,x = 2.5 \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\,\left\langle x \right\rangle = 2\,\,\,\,\, \Leftrightarrow \,\,\,\,\,1 < x \le 2\,\,\,\,\left\{ \matrix{
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,x = 2\,\,\, \hfill \cr
\,{\rm{Take}}\,\,x = 1.5 \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,\left\{ \matrix{
\,\,2 \le x < 3 \hfill \cr
\,\,1 < x \le 2 \hfill \cr} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,x = 2\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

[ x ] is analyzed as follows.
If n ≤ x < n + 1 for some integer n, then [ x ] = n.
< x > is analyzed as follows.
If n - 1 < x ≤ n for some integer n, then < x > = n.

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 ] = 2
=> 2 ≤ x < 3
Thus, condition 1) is not sufficient, since it does not yield a unique solution.

Condition 2)
< x > = 2
=> 1 < x ≤ 2
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Conditions 1) & 2)
Only x = 2 satisfies both conditions.
Since the answer is unique, both conditions together are sufficient.

Therefore, C is the answer.
Answer: C

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.