Asad wrote:
GMATGuruNY wrote:
Asad wrote:
If \(a ≠ 0\), is \(a + a^{−1} > 2\)?
(1) \(a > 0\)
(2) \(a < 1\)
CRITICAL POINTS occur when the two sides of the inequality are EQUAL or when the inequality is UNDEFINED.
Given \(a + a^{−1} > 2\):
The two sides are equal when a=1, since \(1 + 1^{−1} = 1 + \frac{1}{1^1} = 2\).
The inequality is undefined when a=0, since \(0^{-1} = \frac{1}{0^1} =\) undefined.
To determine which ranges satisfy the inequality, test one value to the left and one value to the right of each critical point.
Here, we must test a<0, 0<a<1 and a>1.If we test a=-1, a=1/2 and a=2, only a=1/2 and a=2 satisfy \(a + a^{−1} > 2\), implying that the valid ranges are 0<a<1 and a>1.
Question stem, rephrased:
Is \(a\) a positive value
other than 1?
Statement 1:
If a=2, the answer to the rephrased question stem is YES.
If a=1, the answer to the rephrased question stem is NO.
INSUFFICIENT.
Statement 2:
If a=1/2, the answer to the rephrased question stem is YES.
If a=-1, the answer to the rephrased question stem is NO.
INSUFFICIENT.
Statements combined:
Since 0<a<1, the answer to the rephrased question stem is YES.
SUFFICIENT.
GMATGuruNYThank you so much for your response. Sir, could explain a bit the highlighted part? i mean: how do someone convinced that we
must test those values?
Also, why it is
other than 1Thanks__
The critical points, plotted on a number line:
<------0
-------1------>0 and 1 are the only values where the \(a + a^{−1}\) is undefined or where \(a + a^{−1} = 2\).
Implication:
In each of the colored ranges, \(a + a^{−1} < 2\) or \(a + a^{−1} > 2\).
To determine in which ranges \(a + a^{−1} > 2\), we must test only one value in each of the colored ranges.
a=-1 is in the red range.
When a=-1, \(a + a^{−1} < 2\).
Thus, the red range (a<0) is not valid.
a=1/2 is in the green range.
When a=1/2, \(a + a^{−1} > 2\).
Thus, the green range (0<a<1) is valid.
a=2 is in the blue range.
When a=2, \(a + a^{−1} > 2\).
Thus, the blue range (a>1) is valid.
Result:
\(a + a^{−1} > 2\) when \(a\) is in the green range (0<a<1) or in the blue range (a>1).
Together, the green and blue ranges include every positive value except for 1.
Thus, the answer to the question stem is YES if \(a\) is ANY POSITIVE VALUE OTHER THAN 1.
Question stem. rephrased:
Is \(a\) a positive value other than 1?