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Inequality Question

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Inequality Question

by sajibfin06 » Wed Jul 29, 2015 2:21 am
How many of the integers that satisfy the inequality (x+2)(x+3)/(x-2)≥0 are less than 5 ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Can you also tell me whether (x+2)(x+3)/(x-2)≥0 and (x+2)(x+3)≥0 mean the same thing ?

If question were asked as what the possible range of value of (x+2)(x+3)/(x-2)≥0 , how do we solve them , please explain in detail .
Thnaks in advance .
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by nikhilgmat31 » Wed Jul 29, 2015 2:48 am
sajibfin06 wrote:How many of the integers that satisfy the inequality (x+2)(x+3)/(x-2)≥0 are less than 5 ?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Can you also tell me whether (x+2)(x+3)/(x-2)≥0 and (x+2)(x+3)≥0 mean the same thing ?

If question were asked as what the possible range of value of (x+2)(x+3)/(x-2)≥0 , how do we solve them , please explain in detail .
Thnaks in advance .
x + 2 >= 0 OR x + 3 >= 0

x >=-2 or x = -2 or x > -3 or x =-3

x = -1,1,2 doesn't fit

By plugin integer values below 5

possible values of x are 4,3,-2 ,-3


Answer is D = 4
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by Spencer@Prep4GMAT » Wed Jul 29, 2015 6:24 am
Hi sajibfin06,

To answer your question about "the possible range of values":

(1) Note that x can't be 2; otherwise we have zero in the denominator.
(2) For the fraction to be equal to zero, x could be -2 or -3.
(3) For the fraction to be positive (> 0), the numerator and denominator must have the same sign. Let's use a number line:

x <--------(-3)---(-2)---(-1)---(0)---(1)---(2)--->
x+2 ---------------0+++++++++++++++++++
x+3 --------0++++++++++++++++++++++++
(x+2)(x+3) ++0-----0+++++++++++++++++++
x-2 -------------------------------------0+++

So we see two ranges where (x+2)(x+3) and (x-2) have the same sign: -2 > x > -3, and x > 2.

Putting this together, the possible ranges for x are -2 ≥ x ≥ -3, and x > 2.

And the integer values less than 5 where the inequality is true are x = -3, -2, 3, and 4 (as nikhilgmat31 has said correctly).

Also, as you can see from the number line exercise, (x+2)(x+3)/(x-2) ≥ 0 and (x+2)(x+3) ≥ 0 are not the same thing. For (x+2)(x+3) ≥ 0, either -3 ≥ x or x ≥ -2. After all, since we multiply both sides of the inequality by (x-2), we'd have to flip the inequality sign if x-2 < 0.

Let me know if that answers your questions or not!
Spencer
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by sajibfin06 » Wed Jul 29, 2015 8:14 am
Thank you -Spencer@Prep4GMAT- very much . I get my problem cleared.
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by GMATGuruNY » Wed Jul 29, 2015 8:22 am
To determine the valid solutions for x, we can use the CRITICAL POINTS approach:
https://www.beatthegmat.com/inequalities ... 76123.html
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by [email protected] » Wed Jul 29, 2015 9:41 am
Hi sajibfin06,

The solution that Mitch linked to is great; it emphasizes a particular part of the process that is so important for GMAT questions of all types: you have to take notes and do work in an organized way.

In this prompt, we're asked to focus on integer solutions that are less than 5. From the answers, we know that there is at least one solution, but no more than five solutions. This means that there aren't that many options and they shouldn't be too hard to find.

If you were "stuck" on this question, then you could just start plugging in integers until you've "found" all of the ones that "fit." Start with the number 4, then 3, then 2, etc. You'd be amazed how often you can use what's called "brute force" against a Quant question; plug in numbers and pound on the question until you've found the solution.

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