The number of integers of n satisfying -n+2≥0 and 2n≤4 is

\(-n+2\geq{0}................-n\geq{-2}..................n\leq{2}\)....
Also \(2n\leq{5}.............n\leq{\frac{5}{2}}\)......
ans is \(n\leq{2}\)..... infinite possible solution...
E

Even if n was between 1 and 2, there will b einfinite solutions , as it is not given that n is an integer
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hi..

there are two answers we are getting \(n\leq{2}\) and not \(n\leq{\frac{5}{2}}\)
. \(n\leq{2}\) is a subset of \(n\leq{\frac{5}{2}}\) , so overlapping region is \(n\leq{2}\)

But it is given - The number of n satisfying -n+2≥0 and 2n≤5 is
n between 2 and 5/2 does not satisfy -n+2≥0...


say it is given n<10 and n>5, so n can be anything between 5 and 10 as that is overlapping region..
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hi
it would depend on question and most of the time you will be looking at the below rule when you have inequality and a negative sign and a positive sign on either side ..
example ... -x>2 but you have to find x, so x<-2 and other such situations


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Hi dave13

I am still having difficulty in understanding your exact query. Nonetheless, Pls note below points regarding inequality -

1. If you multiply or divide both sides of the inequality by a NEGATIVE number, then the sign of inequality changes

for e.g if \(x>3\) and you multiply it by \(-1\), then it will become \(-x<-3\)

if \(-2x>5\), then \(x<\frac{5}{-2}\), here you are dividing by \(-2\), hence sign changes.

2. If you are adding or subtracting something from both sides of the inequality then there is NO EFFECT in the sign of inequality

for e.g if \(x>3\) add 1 to both sides you will get \(x+1>3+1=>x+1>4\)

if \(x>3\), subtract \(-1\) from both sides you will get \(x-1>3-1=>x-1>2\)

Also you can find GMAT Quant book here -
https://gmatclub.com/forum/gmat-math-bo ... 30609.html

Find the range of solutions for
-n + 2 ≥ 0
Subtract 2 from both sides
-n ≥ -2
Divide by -1 and flip the sign
n ≤ 2
The range of solutions for n ≤ 2:

<------------0--------1--------2

Find the range of solutions for
2n ≤ 5
Divide both sides by 2
n ≤ \(\frac{5}{2} (2.5)\)
The range of solutions for n ≤ 2.5:

<--------------0--------1--------2-----2.5
Compare with n ≤ 2
<--------------0--------1--------2

The entire solution range for n ≤ 2.5 does not hold true for n ≤ 2
For the two inequalities, the range for n ≤ 2.5 includes, but goes beyond, the range for n ≤ 2

The more restrictive range, in this case, controls the answer to the question.

Every answer that satisfies n ≤ 2 , with its smaller (more restrictive) range will satisfy the inequality n ≤ 2.5, with its nearly-identical but larger range.

The reverse is not true. Some answers that satisfy the inequality n ≤ 2.5 will NOT satisfy the inequality n ≤ 2

Irrespective of infinite solutions, we cannot say, for example, that n = 2.3 satisfies the first inequality n ≤ 2
2.3 is greater than 2. Not allowed.

<--------------0--------1--------2-----2.5

Number of solutions?

Any number less than or equal to 2 satisfies n ≤ 2
n could be 2, 1.5, -8, -47 . . .-2,666,666 . . .etc

The number of \(n\) numbers that satisfy n ≤ 2 is infinite.

Answer E, none of the above
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Hi dave13

I hope you understand that it is not only "1" that you can add or subtract from both sides of the inequality but you can do so with any number.

take the example of this question only, if you were asked to find the range of n from below inequalities, then how would you do it?

-n+2≥0 and 2n≤5

If you are still facing difficulty with inequality, I would suggest grab your high school level Maths book and revise the basic properties of inequality.

Hi dave13 ,

I wonder whether the real issue here is a concept that gets muddied by the prompt.
The prompt says: The number of n satisfying -n+2≥0 and 2n≤5 is[?]

Let me try a rewritten prompt; if it does not help, no problem.
We could call the numbers we are looking for n. We could call them n and N. We could call them x and y.

The variables, no matter what they are, mean "integers, decimals, and fractions."

There are two inequalities.
Only some numbers, denoted by \(x\) and \(y\) (including integers, decimals, and fractions, but not imaginary numbers) will follow the rule each inequality "defines."
How many numbers \(x\) and \(y\) follow the rule set out by the inequality -x+2≥0, and follow the rule set out by the inequality 2y≤5?

First inequality's rule

\(-x+2≥0\)
We need x by itself. Subtract 2 from both sides:

\(-x ≥ -2\), which equals

\((-1)*(x) ≥ (-1)*(2)\)

We need x to be positive. Divide both sides by (-1).
When we divide (or multiply) an inequality by a negative number, we must flip the direction of the sign. (Long story. Trust me?) Thus:

\(\frac{(-1)*(x)}{(-1)}\)≤ \(\frac{(-1)(2)}{(-1)}\) , which equals

\(x ≤ 2\)

How many integers, decimals, and fractions are RULE -> less than or equal to 2?
Here are just a few samples. Every example follows the rule:
2 is equal to 2
These numbers are all less than 2, so they are okay, they follow the rule: 0, -\(\frac{3}{4}\), - 10.7, -2,801 . . . I can't finish this list. It is endless.

One part of the answer to, "How many numbers, \(x\), satisfy the inequality \(x ≤ 2\)?"
An infinite number of numbers.

Second inequality's rule

How many numbers, \(y\), follow the rule set out by \(2y ≤ 5\)
No pesky negative values to worry about. Just get \(y\) by itself. Divide both sides by 2

\(\frac{2y}{2} ≤ \frac{5}{2}\)

\(y ≤ 2.5\)

How many numbers, \(y,\) follow the rule that the number, \(y\), is less than or equal to 2.5?
2.5 equals 2.5
2.499 is less than 2.5
2.4 is less than 2.5
2\(\frac{1}{4}\) is less than 2.5
2.15679
2.00001 is less than 2.5
2 is less than 2.5
These numbers also all follow the rule for \(y ≤ 2.5\):
0, -\(\frac{3}{4}\), -10.79, -2,801 . . . endless

Compare number lines. Blue = "allowed, follows the rule," Red = "now allowed, breaks the rule"

\(x ≤ 2\)
<<-------0-----1------2-----2.5----3--->>

\(y ≤ 2.5\)

<<-------0-----1------2-----2.5-----3--->>


We have a problem. ALL of the numbers that are "okay" for the first inequality are okay for the second inequality.
Every number that falls on the blue parts is fine, follows both inequality rules.
BUT the reverse is not true. Some numbers that satisfy the second inequality DO NOT satisfy the first inequality.
Between 2 and 2.5, one line says RED (no) and one line says BLUE (yes)

The numbers in green, above? They do NOT follow \(x ≤ 2\)
Because of THIS more restrictive inequality,\(x ≤ 2\),
the numbers that were green above now have to be red: 2.00001, 2.15679, 2\(\frac{1}{4}\), 2.4, 2.499, 2.5
Those numbers satisfy \(y ≤ 2.5\) but not \(x ≤ 2\):

So we have to follow the stricter rule. In a way, that means \(y ≤ 2.5\) is irrelevant.

Those numbers that were green and are now red are all to the right of 2, greater than 2. Not allowed.

So we "chop off" part of the number line that we thought would answer the question properly. That part does not. It violates \(x ≤ 2\).
<<______0________________2____2.5

However, what range of answers still works? This one:
<<_-2,801___-10.7__\(-\frac{3}{4}\)__0____2

How many numbers are less than or equal to 2? An infinite number of numbers.

No answer choices says "infinite."
The answer is E.

I hope that helps.
Sometimes, when I feel as if I'm in quicksand while reviewing a topic, I go away from it for a couple of days.
I admire your tenacity. I hope this helps.
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Hi chetan2u, Hi niks18

I have two questions:

after this \(-n\geq{-2}\) how did you get this \(n\leq{2}\)


Did you use Multiplicative Inverse rule?


When a and b are both positive or both negative:

* If a < b then 1/a > 1/b
* If a > b then 1/a < 1/b

However when either a or b is negative (but not both) the direction stays the same:

* If a < b then 1/a < 1/b
* If a > b then 1/a > 1/b


or Additive Inverse rule ?

If a < b then −a > −b
If a > b then −a < −b

In which cases do we need to use additive inverse in inequilities? And in which cases do we need to use Multiplicative Inverse rule in inequilities ? I am confused now.

Now regarding the answer

Why the answer is \(n\leq{2}\) and not \(n\leq{\frac{5}{2}}\) ? we got both answers right ? so how should I choose which one is the answer \(n\leq{2}\) or \(n\leq{\frac{5}{2}}\) ?

And why correct answer is E and not C ?

Many thanks for explanation

Hi chetan2u,

many thanks for taking time to explain. I am now trying to digest what you said

ok so you give it as an example "example ... -x>2 but you have to find x, so x < -2. Are you multiplying by -1 ? I mean we imply by -x = -1 ??

And in this case if x<3 then following this additive rule a < b then −a > −b i get ---> \(-x>-3\) ? what is the point of doing - I mean changing signs, it already is an answer no ?

if \(x >3\) then following another additive rule a > b then −a < - b I get --- > \(−x < −3\) ? is it correct ??

Also what does subset mean ?

thank you very much

Hi niks18 perhaps you help me to understand the above? thank you!

Hi dave13,

Sorry but I could not understand your query. May be it is a formatting issue. Pls list down your queries properly. Also if you are facing any conceptual problems then I would suggest you to revisit Quant Book available in the forum. To know about "Subset", pls revise Set Theory.

Hi niks18, i formatted it. unfortunately i could not find this in GMAT club math book, or perhaps it is explained there not for dummies

niks18, many thanks for your explanation. I downloaded the GMAT math book long time ago but I couldn't find such as information as:

you say one can subtract 1 from both parts of inequalities or add 1 to both parts of inequalities - my question is when do I need to perform these operations and which cases should I apply subtraction, addition and multiplication ? I would appreciate if you could explain with examples. many thanks!

Greetings generis,

could you change in this question some numbers "The number of n satisfying -n+2≥0 and 2n≤5 " in which answer would 2, or 3 or any number and explain solution so as I compare with the question to which answer E please

i just wanna cement inequility in my mind cause i feel a bit of running sands in inequlity

have an terrific day

Hi Generis writing the answer to your post second time lol something went wrong with the font collors,:) so i will stick to classic black & white

ok first of all thank you for you detailed explanation but i still have some questions....i will understand if you dont reply

Question # 1) why did you take first equation as benchmark ? I meant why not start counting from 2.5 ?

Question #2) i want to change values in both equations and see some light in the logic

so we have first equation

\(2n ≥ 7\) divide by 2

\(n ≥ 3.5\)

second equation

\(2n ≤ 27\)

divide by 2

so i get \(n≤ 13.5\) now is getting really interesting

so how many solutions/ numbers are there in this range ?


so we have equation ONE

\(n ≥ 3.5\)

with range

--------------------0----1----2-----3--3.5---4------------------------------------------------------>

and equation number TWO

with range \(n≤ 13.5\)

-------------------0----1----2-----3---4-----5------6------7------8------9------10-----11-----12-----13--13.5---14--------------------------------->

ok now the most exciting moment now how many numbers / solutions does n satisfy ? please answer this question <----- Again infinity ? should i calclate all decimals, fractions etc like 13.5 12.9, 12.8 .... etc ?

many thanks for your valuable gmat insights

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