GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 04 Aug 2020, 03:55 ### GMAT Club Daily Prep

#### 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

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.  # How many of the integers that satisfy the inequality (x+2)(x

Author Message
TAGS:

### Hide Tags

Senior Manager  Affiliations: UWC
Joined: 09 May 2012
Posts: 328
GMAT 1: 620 Q42 V33
GMAT 2: 680 Q44 V38
GPA: 3.43
WE: Engineering (Entertainment and Sports)
How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

22
334 00:00

Difficulty:   95% (hard)

Question Stats: 46% (01:59) correct 54% (02:00) wrong based on 4084 sessions

### HideShow timer Statistics

How many of the integers that satisfy the inequality $$\frac{(x+2)(x+3)}{(x-2)} \geq 0$$ are less than 5?

A. 1
B. 2
C. 3
D. 4
E. 5

OG 2019 PS14203
Math Expert V
Joined: 02 Sep 2009
Posts: 65785
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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.

Solving inequalities:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Hope it helps.
_________________
Senior Manager  B
Joined: 28 Mar 2012
Posts: 286
Location: India
GMAT 1: 640 Q50 V26
GMAT 2: 660 Q50 V28
GMAT 3: 730 Q50 V38
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

40
1
28
Hi,

General method:

$${(x+2)(x+3)}/(x-2) \geq 0$$

if we plot it on number line, we have,
$$-3 \leq x \leq -2$$
& $$x > 2$$, since $$x-2 \neq 0$$ (no equality).

Also, it is given$$x < 5$$
Thus integral solutions would be x = -3, -2, 3, 4

Regards,
Attachments nline.jpg [ 4.21 KiB | Viewed 179561 times ]

Originally posted by cyberjadugar on 09 Jun 2012, 00:03.
Last edited by cyberjadugar on 19 Jun 2012, 04:56, edited 1 time in total.
##### General Discussion
Current Student Joined: 08 Jan 2009
Posts: 276
GMAT 1: 770 Q50 V46
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

18
8
How many of the integers that satisfy the inequality ((x+2)(x+3)) / (x-2) >= 0 are less than 5?

Just start testing numbers:
4,3,2,1,0,-1,-2,-3,-4 etc

4 - yep
3 - yep
2 - no
1 - no
0 - no
-1 - no
-2 - yes
-3 - yes
-4 and below - no

4,3,-2,-3, so D.
Senior Manager  Joined: 13 Jan 2012
Posts: 272
Weight: 170lbs
GMAT 1: 740 Q48 V42
GMAT 2: 760 Q50 V42
WE: Analyst (Other)
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

9
3
Yeah, you could test numbers.

Alternatively, just find the solutions to the inequality:

This solves to x>2 and -3<=x<=-2

So X can be 3, 4, ... or -2 or -3.

So 4 integers.

Senior Manager  Affiliations: UWC
Joined: 09 May 2012
Posts: 328
GMAT 1: 620 Q42 V33
GMAT 2: 680 Q44 V38
GPA: 3.43
WE: Engineering (Entertainment and Sports)
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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.

Solving inequalities:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Hope it helps.

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!
Manager  Joined: 24 Mar 2010
Posts: 58
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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.

Solving inequalities:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Hope it helps.

Bunuel,

Could you explain this graphical method you use or direct me to a post which does the same.

Math Expert V
Joined: 02 Sep 2009
Posts: 65785
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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.

Solving inequalities:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Hope it helps.

Bunuel,

Could you explain this graphical method you use or direct me to a post which does the same.

Solving inequalities:
x2-4x-94661.html#p731476 (check this one first)
inequalities-trick-91482.html
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Hope it helps.
_________________
Director  Status: Far, far away!
Joined: 02 Sep 2012
Posts: 991
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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

Is it clear?
Intern  Joined: 12 Dec 2012
Posts: 3
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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.
Verbal Forum Moderator B
Joined: 10 Oct 2012
Posts: 562
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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.
_________________

Originally posted by mau5 on 01 Apr 2013, 20:33.
Last edited by mau5 on 09 Jun 2013, 07:00, edited 1 time in total.
Intern  Joined: 07 Mar 2013
Posts: 11
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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 $$x>=-3$$

Like: $$(x+2) >= 0$$ => $$x>= -2$$
and $$(x+3) >=0$$ => $$x>= -3$$

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?
Director  Status: Far, far away!
Joined: 02 Sep 2012
Posts: 991
Location: Italy
Concentration: Finance, Entrepreneurship
GPA: 3.8
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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 $$x>=-3$$

Like: $$(x+2) >= 0$$ => $$x>= -2$$
and $$(x+3) >=0$$ => $$x>= -3$$

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 $$-3\leq{x}\leq{-2}$$.

Let me know if it's clear now
Manager  Joined: 08 Dec 2012
Posts: 60
Location: United Kingdom
WE: Engineering (Consulting)
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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>
Math Expert V
Joined: 02 Sep 2009
Posts: 65785
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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.
_________________
Intern  Joined: 24 Apr 2014
Posts: 1
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

I don't see why 2 isn't a solution? Maybe someone can clarify!
Math Expert V
Joined: 02 Sep 2009
Posts: 65785
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

sunchild wrote:
I don't see why 2 isn't a solution? Maybe someone can clarify!

If x=2, then $$\frac{(x+2)(x+3)}{x-2}$$ would be undefined and not $$\geq{0}$$ because the denominator would become zero. We cannot divide by zero.

Does this make sense?
_________________
Intern  Joined: 18 Aug 2012
Posts: 9
GMAT 1: 730 Q50 V39
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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.
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 17272
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

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.

GMAT assassins aren't born, they're made,
Rich
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 65785
Re: How many of the integers that satisfy the inequality (x+2)(x  [#permalink]

### Show Tags

5
1
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

MATH REVOLUTION VIDEO SOLUTION:

_________________ Re: How many of the integers that satisfy the inequality (x+2)(x   [#permalink] 22 Jan 2016, 09:01

Go to page    1   2    Next  [ 39 posts ]

# How many of the integers that satisfy the inequality (x+2)(x  