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Problem solving: Difficulty level: 700, Source: Jamboree

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by Brent@GMATPrepNow » Sun Oct 07, 2018 7:13 am
aishwaryav12 wrote:If x is an integer, how many possible values of x exist for x² + 5|x| + 6 = 0?
A. 4
B. 2
C. 3
D. 1
E. 0
Take: x² + 5|x| + 6 = 0
Subtract 6 from both sides to get: x² + 5|x| = -6

KEY CONCEPT: x² ≥ 0 and |x| ≥ 0 for all values of x
In other words, x² will always be greater than or equal to 0
And |x| will always be greater than or equal to 0, which means 5|x| will always be greater than or equal to 0

So, we can take our equation, x² + 5|x| = -6, and rewrite it as follows:
(some number that's greater than or equal to zero) + (some number that's greater than or equal to zero) = -6
As we can see, it's impossible for the left side of the equation to equal a NEGATIVE value.
As such, there can be no solution.

Answer: E

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by fskilnik@GMATH » Sun Oct 07, 2018 7:20 am
aishwaryav12 wrote:If x is an integer, how many possible values of x exist for x2+5|x|+6=0?
A. 4
B. 2
C. 3
D. 1
E. 0
$$?\,\,\,:\,\,\,\# \,\,\,{\mathop{\rm int}} \,\,\,{\rm{roots}}\,\,\,{\rm{for}}\,\,\,\,{x^2} + 5\left| x \right| + 6 = 0$$
$$\left. \matrix{
{x^2} \ge 0 \hfill \cr
\left| x \right|\,\, \ge 0\,\, \hfill \cr} \right\}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,?\,\,\,\,:\,\,\,{x^2} + 5\left| x \right|\, + 6\,\,\, \ge 6\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 0$$

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

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Fabio.
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by [email protected] » Sun Oct 07, 2018 2:58 pm
Hi aishwaryav12,

We're told that X is an integer. We're asked for the number of possible solutions for the equation X^2+5|X|+6=0. This question can approached in a couple of different ways, but you can actually solve it with Number Properties and avoid doing any calculations whatsoever.

To start, since X is an INTEGER, we know that X^2 will either be 0 or a positive integer.
We also know that 5|X| will also either be 0 or a positive integer.
Since "+6" is a positive integer, we're going to be adding up 3 numbers that will have a sum that is a positive integer (regardless of what X actually is). By extension, the given equation (which is set equal to 0) has NO solutions.

Final Answer: E

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