Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized for You
we will pick new questions that match your level based on your Timer History
Track Your Progress
every week, we’ll send you an estimated GMAT score based on your performance
Practice Pays
we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:
Join the live F2F session with the Admission Directors from Rotman School of Management and Ivey Business School to find out why Canada is increasingly becoming the go-to destination for several MBA aspirants, and gain critical insights....
A live session in which Greg Guglielmo, Haas MBA & Founder of Avanti Prep, analyses resumes of MBA applicants, and suggests ways of improvement in one on one talk with concerned applicants.
Leadership is one of the most important characteristics schools look for in applicants. This session will help define exactly what leadership is and how you can best showcase it in your essays.
Start by taking a free, full-length practice test—at your convenience. Receive a detailed score analysis to understand your strengths and see where you need to improve.
GMAT tests your ability to think critically by presenting "tricky" arguments that require you to notice flaws or vulnerabilities in the construct. e-GMAT is conducting a free webinar in which you can learn the art to decode difficult CR questions.
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
09 Jun 2012, 01:51
40
2
95
macjas 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
\(\frac{(x+2)(x+3)}{x-2}\geq{0}\) --> the roots are -3, -2, and 2 (equate the expressions to zero to get the roots and list them in ascending order), this gives us 4 ranges: \(x<-3\), \(-3\leq{x}\leq{-2}\), \(-2<x<2\) and \(x>2\) (notice that we have \(\geq\) sign, so, we should include -3 and -2 in the ranges but not 2, since if \(x=2\) then the denominator becomes zero and we cannot divide by zero).
Now, test some extreme value: for example if \(x\) is very large number then all three terms will be positive which gives the positive result for the whole expression, so when \(x>2\) the expression is positive. Now the trick: as in the 4th range expression is positive then in 3rd it'll be negative, in 2nd it'l be positive again and finally in 1st it'll be negative: - + - +. So, the ranges when the expression is positive are: \(-3\leq{x}\leq{-2}\), (2nd range) and \(x>2\) (4th range).
\(-3\leq{x}\leq{-2}\) and \(x>2\) means that only 4 integers that are less than 5 satisfy given inequality: -3, -2, 3, and 4.
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
09 Jun 2012, 02:20
3
2
Bunuel wrote:
macjas 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
\(\frac{(x+2)(x+3)}{x-2}\geq{0}\) --> the roots are -3, -2, and 2 (equate the expressions to zero to get the roots and list them in ascending order), this gives us 4 ranges: \(x<-3\), \(-3\leq{x}\leq{-2}\), \(-2<x<2\) and \(x>2\) (notice that we have \(\geq\) sign, so, we should include -3 and -2 in the ranges but not 2, since if \(x=2\) then the denominator becomes zero and we cannot divide by zero).
Now, test some extreme value: for example if \(x\) is very large number then all three terms will be positive which gives the positive result for the whole expression, so when \(x>2\) the expression is positive. Now the trick: as in the 4th range expression is positive then in 3rd it'll be negative, in 2nd it'l be positive again and finally in 1st it'll be negative: - + - +. So, the ranges when the expression is positive are: \(-3\leq{x}\leq{-2}\), (2nd range) and \(x>2\) (4th range).
\(-3\leq{x}\leq{-2}\) and \(x>2\) means that only 4 integers that are less than 5 satisfy given inequality: -3, -2, 3, and 4.
Thanks Bunuel, while I could easily solve this one using numbers, I couldn't get the algebraic approach. You explanation with the graphical approach is bang on. Thanks!
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
02 Dec 2012, 05:13
1
Bunuel wrote:
macjas 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
\(\frac{(x+2)(x+3)}{x-2}\geq{0}\) --> the roots are -3, -2, and 2 (equate the expressions to zero to get the roots and list them in ascending order), this gives us 4 ranges: \(x<-3\), \(-3\leq{x}\leq{-2}\), \(-2<x<2\) and \(x>2\) (notice that we have \(\geq\) sign, so, we should include -3 and -2 in the ranges but not 2, since if \(x=2\) then the denominator becomes zero and we cannot divide by zero).
Now, test some extreme value: for example if \(x\) is very large number then all three terms will be positive which gives the positive result for the whole expression, so when \(x>2\) the expression is positive. Now the trick: as in the 4th range expression is positive then in 3rd it'll be negative, in 2nd it'l be positive again and finally in 1st it'll be negative: - + - +. So, the ranges when the expression is positive are: \(-3\leq{x}\leq{-2}\), (2nd range) and \(x>2\) (4th range).
\(-3\leq{x}\leq{-2}\) and \(x>2\) means that only 4 integers that are less than 5 satisfy given inequality: -3, -2, 3, and 4.
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
02 Dec 2012, 05:15
2
3
eaakbari wrote:
Bunuel wrote:
macjas 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
\(\frac{(x+2)(x+3)}{x-2}\geq{0}\) --> the roots are -3, -2, and 2 (equate the expressions to zero to get the roots and list them in ascending order), this gives us 4 ranges: \(x<-3\), \(-3\leq{x}\leq{-2}\), \(-2<x<2\) and \(x>2\) (notice that we have \(\geq\) sign, so, we should include -3 and -2 in the ranges but not 2, since if \(x=2\) then the denominator becomes zero and we cannot divide by zero).
Now, test some extreme value: for example if \(x\) is very large number then all three terms will be positive which gives the positive result for the whole expression, so when \(x>2\) the expression is positive. Now the trick: as in the 4th range expression is positive then in 3rd it'll be negative, in 2nd it'l be positive again and finally in 1st it'll be negative: - + - +. So, the ranges when the expression is positive are: \(-3\leq{x}\leq{-2}\), (2nd range) and \(x>2\) (4th range).
\(-3\leq{x}\leq{-2}\) and \(x>2\) means that only 4 integers that are less than 5 satisfy given inequality: -3, -2, 3, and 4.
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
30 Mar 2013, 05:36
7
1
3
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
We can analize the numerator >=0 \((x+2)(x+3)=0\) \(x+2=0, x=-2\) \(x+3=0, x=-3\) Since we have a ">=" we take the external values \(x>=-2\) and \(x<=-3\) Then we analyze the denominator >0 (it can't be =0) \(x-2>0, x>2\)
~~~~~~~(-3)~~~~~(-2)~~~~~~~(+2) negative, negative,negative|positive For the D positive | negative| positive , positive For the N You sum up the sign of the values and obtain: negative | positive | negative | positive
We are looking for >=0 value, so we keep the positive intervals and discard the negative ones. -3>=x>=-2 (in this we have also the =) and x>2 ( no = here) The values less than 5 are : -3,-2,3,4
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
01 Apr 2013, 15:06
1
Bunuel wrote:
rakeshd347 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
I am not really good with inequalities to be honest. I have solved this question and found the answer but It took me 4minutes. Is there any short approach please.
Merging similar topics. Please refer to the solutions above.
I still don't understand how -2 and -3 are solutions. Don't they make the numerator = to 0? I kind of understand the theory, but i'm having trouble reconciling the number picking strategy with the theory.
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
Updated on: 09 Jun 2013, 07:00
6
1
mp2469 wrote:
Bunuel wrote:
rakeshd347 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
I am not really good with inequalities to be honest. I have solved this question and found the answer but It took me 4minutes. Is there any short approach please.
Merging similar topics. Please refer to the solutions above.
I still don't understand how -2 and -3 are solutions. Don't they make the numerator = to 0? I kind of understand the theory, but i'm having trouble reconciling the number picking strategy with the theory.
We are given (x+2)(x+3)/(x-2)>=0
Now we can not cross multiply (x-2) as we don't about its sign. All we know from the problem is that x can not be equal to 2 as because that will make the expression undefined.
Now, as know that \((x-2)^2\) is a positive quantity. Safely multiply it on both sides, thus we get, (x-2)(x+2)(x+3)>=0. AS because there is an equality sign in the given inequality, we can say that x=-2 and x=-3 are two valid solutions, for which the expression assumes the value of zero. X can't be equal to 2, as stated before.
_________________
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
09 Apr 2013, 06:32
Zarrolou 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
We can analize the numerator >=0 \((x+2)(x+3)=0\) \(x+2=0, x=-2\) \(x+3=0, x=-3\) Since we have a ">=" we take the external values \(x>=-2\) and \(x<=-3\)
Sorry to bump this old thread, but I have a question. How is the solution for \((x+2)(x+3) >= 0\) \(x>=-2\) and \(x<=-3\) and not \(x>=-2\) and \(x>=-3\)
I guess inputting numbers [-4, -5 etc] will make the inequality true but when solving practice questions, instinctively, I am missing this range. Is this something I can get good at only by practice? any tips?
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
09 Apr 2013, 06:46
2
4
bcrawl wrote:
Sorry to bump this old thread, but I have a question. How is the solution for \((x+2)(x+3) >= 0\) \(x>=-2\) and \(x<=-3\) and not \(x>=-2\) and \(x>=-3\)
I guess inputting numbers [-4, -5 etc] will make the inequality true but when solving practice questions, instinctively, I am missing this range. Is this something I can get good at only by practice? any tips?
To solve this : \((x+2)(x+3) \geq{0}\), we can use an old method. Think it this way \((x+2)(x+3) = 0\) the solutions are x=-2 and x=-3; now I use an old trick: if the sign of \(x^2\) and the operator are "the same" ie (+,>) or (-,<) we take the external values : \(x\leq{-3}\) and \(x\geq{-2}\). In the other two cases (+,<) (-,>) we take the internal values. If the sign was < (\((x+2)(x+3) \leq{0}\)) the solution would be \(-3\leq{x}\leq{-2}\).
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
25 Aug 2013, 10:05
Disclaimer: The below is not going to be helpful for your GMAT
Strictly speaking when x takes the value of 2, the value of the expression leads to infinity. I know GMAT is way too scared of infinity but the question asks if the expression leads to equal to or greater than zero, and since numerator is positive, the infinity is in the positive direction which indeed meets the inequality criteria.
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
25 Aug 2013, 10:08
nave81 wrote:
Disclaimer: The below is not going to be helpful for your GMAT
Strictly speaking when x takes the value of 2, the value of the expression leads to infinity. I know GMAT is way too scared of infinity but the question asks if the expression leads to equal to or greater than zero, and since numerator is positive, the infinity is in the positive direction which indeed meets the inequality criteria.
<apologies for the pedantry>
\(\frac{(x+2)(x+3)}{x-2}\geq{0}\) holds true for \(-3\leq{x}\leq{-2}\) and \(x>2\). Thus four integers that are less than 5 satisfy given inequality: -3, -2, 3, and 4. Notice that 2 is not among these four integers.
_________________
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
29 Nov 2015, 03:55
For these inequality questions, the best way is to represent numbers on a number line. By representing on a number line, we will be easily able to compare the sign of the equations and will result in faster solving of inequalities.
Representing (x+2) on number line
x+2 is +ve (when x>-2) x+2 is -ve (when x<-2)
Attachment:
1.png [ 2.87 KiB | Viewed 25310 times ]
Representing (x+3) on number line
x+3 is +ve (when x>-3) x+3 is -ve (when x<-3)
Attachment:
2.png [ 2.92 KiB | Viewed 25314 times ]
Representing (x-2) on number line
x-2 is +ve (when x>2) x-2 is -ve (when x<2)
Attachment:
3.png [ 2.79 KiB | Viewed 25291 times ]
Combining all these equations. The representation of these equations on the number line is shown in the figure attached.
Attachment:
4.png [ 1.8 KiB | Viewed 25287 times ]
From the figure, we can easily deduce that the given equation \((x+2)(x+3)/(x-2) >= 0\) will be satisfied when 1) All three equations are positive 2) Two equations are negative and one equation is positive. 3) Equations give the result as zero.
Such a condition is satisfied for the following integers.
At integers 3 and 4 all the equations are positive. Hence satisfy the inequality At integers -2 and -3 the value of the equation is zero, hence satisfies the inequality.
So overall 4 integers satisfy the inequality. So option (D) is correct.
Re: How many of the integers that satisfy the inequality (x+2)(x
[#permalink]
Show Tags
29 Nov 2015, 10:09
1
Hi All,
While this question looks a bit complex, it's not as difficult as it might appear. This question 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 here's how you can go about solving it quickly - 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.