How many of the integers that satisfy the inequality (x+2)(x : Problem Solving (PS)

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

14 May 2023, 04:06

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Ross7it wrote:

EMPOWERgmatRichC wrote:

Hi Ross7it,

With certain questions, you have to be careful about confusing the information that you've been given with the question that you are asked to answer. Your solution is correct - but I think you lost track of the question that you attempting to answer.

Your work shows that there are 4 VALUES FOR X that fit the given inequality. When you plug in each of those values for X, the resulting calculation IS greater than 0. The question is NOT asking for the end calculation though; it's asking about the possible values of X. Each of those 4 values for X ARE less than 5, so the answer to the question is 4 values.

GMAT assassins aren't born, they're made,
Rich

Dear Rich,
Thanks for answering, but it still not clear.

So, how the testing method works here? Could explain me better, please? Some guys on the first page of the topic tested numbers from -4 to 4, but it was not the same to substitute those numbers for X. Am I correct?

Many thanks
Ross

P.S. I am refering to this one:

"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."

Hi,
Solved using the shortcut method-
link- https://gmatclub.com/forum/inequalities ... 91482.html
1. Identify the number of roots- 3 roots in this qs: 2,-2,-3. Arrange in ascending order: -3,-2,2
2. Plot the roots in the diagram at the points the curve cuts the line
The rightmost will be positive infinity, then plot the roots in a descending way.
(See image)
Take any value>2 to test the sign for the rightmost area(over the line). Check what will be the test value. Here it is positive, thus the region will be positive. The points in this region will be positive. At the roots it will be 0. The signs in the regions to the left will be alternate.
(the logic is explained in the link above)
Alternatively negative values will be observed in the regions marked '-'
3. Suitable values under 5: 3,4,-2,-3
Since at x=2, denominator will become 0 and thus not possible

Voila!

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Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

25 Aug 2013, 11:05

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.

:arrow:

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Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

25 Apr 2014, 01:55

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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?

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Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

31 Mar 2017, 09:35

if I take x=2 then >=0 !
why not its satisfied ?

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Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

01 Apr 2017, 05:53

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mkumar26 wrote:

if I take x=2 then >=0 !
why not its satisfied ?

x = 2 does not satisfy (x+2)(x+3)/(x-2) >= 0. In this case the denominator (x-2) becomes 0 and the whole expression (x+2)(x+3)/(x-2) becomes undefined - recall that we cannot divide by 0.

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

17 Nov 2018, 06:43

pike wrote:

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
4,3,-2,-3, so D.

Hello
Please help
Why when I test 4,3, -3,-4 I receive wrong answer? I mean

(4+2) (4+3) / (4-2) = 21
(3+2) (3+3) / (3-2) = 30

21 and 30 are bigger than 5

Where is my mistake???
Thank you

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

17 Nov 2018, 16:24

EMPOWERgmatRichC wrote:

Hi Ross7it,

With certain questions, you have to be careful about confusing the information that you've been given with the question that you are asked to answer. Your solution is correct - but I think you lost track of the question that you attempting to answer.

Your work shows that there are 4 VALUES FOR X that fit the given inequality. When you plug in each of those values for X, the resulting calculation IS greater than 0. The question is NOT asking for the end calculation though; it's asking about the possible values of X. Each of those 4 values for X ARE less than 5, so the answer to the question is 4 values.

GMAT assassins aren't born, they're made,
Rich

Dear Rich,
Thanks for answering, but it still not clear.

So, how the testing method works here? Could explain me better, please? Some guys on the first page of the topic tested numbers from -4 to 4, but it was not the same to substitute those numbers for X. Am I correct?

Many thanks
Ross

P.S. I am refering to this one:

"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."

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Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

09 Aug 2020, 08:42

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macjas wrote:

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

Solution:

We see that if x = -3 or -2, then the value of the expression (on the left hand side of the inequality) will be 0, which is greater than or EQUAL to 0.

If x is -4 or less, then the numerator of the expression is positive (because the two factors are negative) and the denominator is negative. However, the value of the entire expression will be negative, which does not satisfy the inequality.

If x = -1, 0, or 1, then the numerator of the expression is positive (because the two factors are positive) and the denominator is negative. However, the value of the expression will be negative, which does not satisfy the inequality.

We see that x can’t be 2 since the denominator will then be 0 and we can’t divide by 0.

If x is 3 or greater, then the numerator of the expression is positive (because the two factors are negative) and the denominator is also positive. Therefore, the value of the expression will be positive, which does satisfy the inequality.

However, since we are looking for the values of x that are less than 5, so there are only 4 such values (namely, -3, -2, 3, and 4).

Answer: D

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Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

07 Apr 2021, 05:18

Top Contributor

Solution:

The question stem can be rephrased as

Is (x+2)(x+3)(x-2)>=0 ? (Multiply (x-2)^2 on both sides)

The roots of the quadratic are -3,-2 and 2

Using the wavy line concept, with a greater than sign the range would be

-3<=x<=-2

=>Integral solutions would be x = -3, -2, 3, 4 (option d)

Devmitra Sen
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How many of the integers that satisfy the inequality (x+2)(x [#permalink]

15 Oct 2021, 13:42

EMPOWERgmatRichC wrote:

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

Yes, just test numbers

x < 5
x=4 Yes
x=3 Yes
x=2 (no)(Undefined)
x= 1 (no)
x = 0 (no)
x = -1 (No)
x = -2 (Yes)
x= -3 (Yes)
x = -4 (No)
x = -5 (No)
x = -6 (No)
Pattern shows us that we're done.

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Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

15 Oct 2021, 14:50

Expert Reply

TooLong150 wrote:

EMPOWERgmatRichC wrote:

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

Yes, just test numbers

x < 5
x=4 Yes
x=3 Yes
x=2 (no)(Undefined)
x= 1 (no)
x = 0 (no)
x = -1 (No)
x = -2 (Yes)
x= -3 (Yes)
x = -4 (No)
x = -5 (No)
x = -6 (No)
Pattern shows us that we're done.

Hi TooLong150,

Yes - that's perfect! Once you get down to X = -4, you might recognize that a Number Property pattern emerges: the numerator will be the product of two negative numbers, so that product will STAY POSITIVE as X gets smaller AND the denominator will STAY NEGATIVE as X gets smaller. Dividing a positive number by a negative number ALWAYS yields a negative result, meaning that there's no reason to keep looking at integer values for X that are smaller than -4 (because they will ALL lead to total numbers that are negative numbers - which is NOT what the question is asking us for).

GMAT assassins aren't born, they're made,
Rich

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Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

10 Oct 2022, 11:32

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.

Answer: D.

Solving inequalities:
https://gmatclub.com/forum/x2-4x-94661.html#p731476
https://gmatclub.com/forum/inequalities-trick-91482.html
https://gmatclub.com/forum/everything-is ... me#p868863
https://gmatclub.com/forum/xy-plane-7149 ... ic#p841486

Hope it helps.

Bunuel Can you please help me understand as to how we can directly equate this to zero initially to find roots? This is an inequality right? Either 2 of them have to be negative or all of them have to be positive for the numbers >0?

Also after finding roots, can we solve this type by the wavy line method? Or is it limited to only basic quadratic equations/inequations?

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Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

02 Feb 2023, 20:26

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Inequality with a very different technique used by mbahouse

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Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

03 Feb 2023, 00:57

(x+2)(x+3)>=0 gives -2 and -3 as the roots. Therefore, between -2 and -3 the value is negative and for x<-3 and x>-2, the values are positive.
Now, (x-2) is positive for x>2 and negative for x<2.
We know, for the whole equation to be positive both the numerator and denominator has to be both positive or negative.
The above condition is only possible for x>2 and x lies between -2 and -3.

The question says integers less than 5, so that definitely loops 3 and 4 in since both are greater than 2 but less than 5.
Now there's no integer between -2 and -3. Therefore 2 is supposed to be the answer.

The trick we often miss is that is says : equation>=0. So we have to loop in the integers which yields 0 too. So x=-2,-3 are those 2 integers which makes the equation 0.

Therefore the total no. of integers is 4 : -2,-3,3,4

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Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

06 Feb 2023, 20:38

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Want more help? Here you go: http://linktr.ee/thegmatstrategy

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Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

03 Feb 2024, 10:20

cyberjadugar wrote:

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

Answer is (D)

Regards,

Where did you find the -3 and -2 inequality?

gmatclubot

Re: How many of the integers that satisfy the inequality (x+2)(x [#permalink]

03 Feb 2024, 10:20

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