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Inequalities: Tips and hints : Quantitative Questions
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Re: Inequalities: Tips and hints [#permalink]
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Is this part of GMAT Math Book? And, if not; can it be included in the book?
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Re: Inequalities: Tips and hints [#permalink]
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Bunuel VeritasPrepKarishma chetan2u

2. You can only apply subtraction when their signs are in the opposite directions:

Quote:
If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) --> \(a-c>b-d\) (take the sign of the inequality you subtract from).
Example: \(3<4\) and \(5>1\) --> \(3-5<4-1\).


Any alternative way to memorize highlighted text under time crunch other than picking numbers?
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Re: Inequalities: Tips and hints [#permalink]
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adkikani wrote:
Bunuel VeritasPrepKarishma chetan2u

2. You can only apply subtraction when their signs are in the opposite directions:

Quote:
If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) --> \(a-c>b-d\) (take the sign of the inequality you subtract from).
Example: \(3<4\) and \(5>1\) --> \(3-5<4-1\).




Any alternative way to memorize highlighted text under time crunch other than picking numbers?



Just remember that you can add INEQUALITIES by adding the terms on same side of INEQUALITY..
So if a>b and c<d...c<d is same as d>c..
So we have a>b and d>c...
Add the same sides of INEQUALITY..
a+d>b+c.......a>b+c-d.....a-c>b-d...
Same as what you are trying to remember about SUBTRACTION
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Re: Inequalities: Tips and hints [#permalink]
Bunuel chetan2u VeritasPrepKarishma niks18

Let us say, I am given a SINGLE inequality:

a - b > a + b

Given: a and b are integers.

Can I add / subtract an integer with unknown sign (ie positive or negative)
to both sides of inequality WITHOUT knowing existing sign of another variable?

Eg. Here, can I subtract a from both sides, without knowing sign of b?
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Re: Inequalities: Tips and hints [#permalink]
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adkikani wrote:
Bunuel chetan2u VeritasPrepKarishma niks18

Let us say, I am given a SINGLE inequality:

a - b > a + b

Given: a and b are integers.

Can I add / subtract an integer with unknown sign (ie positive or negative)
to both sides of inequality WITHOUT knowing existing sign of another variable?

Eg. Here, can I subtract a from both sides, without knowing sign of b?


Yes. We are concerned about the sign of a variable when multiplying/dividing an inequality by it. However we can safely add/subtract a variable from both sides of an inequality regardless of its sign.
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Re: Inequalities: Tips and hints [#permalink]
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adkikani wrote:
Bunuel chetan2u VeritasPrepKarishma niks18

Let us say, I am given a SINGLE inequality:

a - b > a + b

Given: a and b are integers.

Can I add / subtract an integer with unknown sign (ie positive or negative)
to both sides of inequality WITHOUT knowing existing sign of another variable?

Eg. Here, can I subtract a from both sides, without knowing sign of b?

adkikani,

Inequality presearves under following operations:

- addition or subtraction of a number from both sides.

- Multiplication or division from both sides by a positive number.

Quote:
Can I add / subtract an integer with unknown sign (ie positive or negative) to both sides of inequality WITHOUT knowing existing sign of another variable?


Yes, we can add or subtract any number (NOT just integer) from both sides without knowing the existing sign.

Now, let's consider example provided by you.
Given inequality,
\(A - B > A + B\)
Assume A = 3 , B = -5. These values will satisfy the above inequality.

Case1: Add a positive value both side i.e. add A both side:

\(2A - B > 2A + B\) . You can verify that this inequality still holds true.

Case2: Add a negative value both side i.e add B both side:

\(A > A + 2B\) . Still, the inequality holds true.

I hope this helps.

Thanks.
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Re: Inequalities: Tips and hints [#permalink]
Bunuel chetan2u Gladiator59 VeritasKarishma

The post says "We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality)."

To me, the bold part means : "We can take even root of both parts of an inequality if we know that both parts of the inequality are non-negative"

However, this does not seem to hold true for the below example, can you please clarify?

Let's say : x^2 > y^4 (given)
So according to the above rule (see bold part of the excerpt), since both sides of the inequality are non-negative(as anything raised to even power is non negative), we can say:
x > y^2 (taking square root on both sides of the inequality)
But that's not necessarily true.
Consider the example :
Case 1 : X = 300, Y = 2
Case 2 : X = -300 , Y = 2
In both cases x^2 > y^4, but for case 1 : x > y^2, whereas for case 2 : x < y^2
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Re: Inequalities: Tips and hints [#permalink]
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Debo1988 wrote:
Bunuel chetan2u Gladiator59 VeritasKarishma

The post says "We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality)."

To me, the bold part means : "We can take even root of both parts of an inequality if we know that both parts of the inequality are non-negative"

However, this does not seem to hold true for the below example, can you please clarify?

Let's say : x^2 > y^4 (given)
So according to the above rule (see bold part of the excerpt), since both sides of the inequality are non-negative(as anything raised to even power is non negative), we can say:
x > y^2 (taking square root on both sides of the inequality)
But that's not necessarily true.
Consider the example :
Case 1 : X = 300, Y = 2
Case 2 : X = -300 , Y = 2
In both cases x^2 > y^4, but for case 1 : x > y^2, whereas for case 2 : x < y^2


The point is if you take the square root from x^2 > y^4, you get |x| > y^2, not x > y^2 (recall that \(\sqrt{x^2}=|x|\)).
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Re: Inequalities: Tips and hints [#permalink]
chetan2u wrote:
adkikani wrote:
Bunuel VeritasPrepKarishma chetan2u

2. You can only apply subtraction when their signs are in the opposite directions:

Quote:
If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) --> \(a-c>b-d\) (take the sign of the inequality you subtract from).
Example: \(3<4\) and \(5>1\) --> \(3-5<4-1\).




Any alternative way to memorize highlighted text under time crunch other than picking numbers?



Just remember that you can add INEQUALITIES by adding the terms on same side of INEQUALITY..
So if a>b and c<d...c<d is same as d>c..
So we have a>b and d>c...
Add the same sides of INEQUALITY..
a+d>b+c.......a>b+c-d.....a-c>b-d...
Same as what you are trying to remember about SUBTRACTION


Hi chetan2u / Bunuel

Does this concept also work in multiplication.

Like highlighted above, we can multiple inequalities only when both sides of both inequalities are positive and the inequalities have the same sign.
Say if the signs are not the same ; can we multiply the inequality with -1 to make the sign same & then multiply ?

Like
if x<a and y>b ; then (-1)y<-b
hence on multiplying : x*(-y) < a(-b) ? will the signs cancel ; nullifying the approach or multiplication of 2 inequalities with opposite signs just can't happen ?
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Re: Inequalities: Tips and hints [#permalink]
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sheldoncooper wrote:
chetan2u wrote:
adkikani wrote:
Bunuel VeritasPrepKarishma chetan2u

2. You can only apply subtraction when their signs are in the opposite directions:

Quote:
If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) --> \(a-c>b-d\) (take the sign of the inequality you subtract from).
Example: \(3<4\) and \(5>1\) --> \(3-5<4-1\).




Any alternative way to memorize highlighted text under time crunch other than picking numbers?



Just remember that you can add INEQUALITIES by adding the terms on same side of INEQUALITY..
So if a>b and c<d...c<d is same as d>c..
So we have a>b and d>c...
Add the same sides of INEQUALITY..
a+d>b+c.......a>b+c-d.....a-c>b-d...
Same as what you are trying to remember about SUBTRACTION


Hi chetan2u / Bunuel

Does this concept also work in multiplication.

Like highlighted above, we can multiple inequalities only when both sides of both inequalities are positive and the inequalities have the same sign.
Say if the signs are not the same ; can we multiply the inequality with -1 to make the sign same & then multiply ?

Like
if x<a and y>b ; then (-1)y<-b
hence on multiplying : x*(-y) < a(-b) ? will the signs cancel ; nullifying the approach or multiplication of 2 inequalities with opposite signs just can't happen ?


No that will not be correct until you know the value of the variables.
For example.
1<10 and 3>2 or -3<-2......1*-3<-2*10.....NO
3>2 and 1<10 or -1>-10....1*-3>2*-10.....YES

So same numbers but different answers depending on which inequality you are multiplying by -1.
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Re: Inequalities: Tips and hints [#permalink]
the hints and notes about inequalities are really helpful, thanks!

I have a question.

MULTIPLYING/DIVIDING AN INEQUALITY BY A NUMBER
3. Never multiply (or reduce) an inequality by a variable (or the expression with a variable) if you don't know the sign of it or are not certain that variable (or the expression with a variable) doesn't equal to zero.
⬆️

does "reduce" here mean divide or subtract? I am a little bit confuse.
as I understand, we could not multiply or divide a variable whose sign is unknown.
but we could add or subtract a variable whose sign is unknown without changing the sign of INEQUALITY.
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Re: Inequalities: Tips and hints [#permalink]
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irene727008 wrote:
the hints and notes about inequalities are really helpful, thanks!

I have a question.

MULTIPLYING/DIVIDING AN INEQUALITY BY A NUMBER
3. Never multiply (or reduce) an inequality by a variable (or the expression with a variable) if you don't know the sign of it or are not certain that variable (or the expression with a variable) doesn't equal to zero.
⬆️

does "reduce" here mean divide or subtract? I am a little bit confuse.
as I understand, we could not multiply or divide a variable whose sign is unknown.
but we could add or subtract a variable whose sign is unknown without changing the sign of INEQUALITY.



Reduce here means division.
You are correct that you can add or subtract any term to both sides without changing inequality sign.
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Re: Inequalities: Tips and hints [#permalink]
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