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# How to Solve: Inequalities

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Director
Status: Tutor - BrushMyQuant
Joined: 05 Apr 2011
Posts: 610
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE: Information Technology (Computer Software)

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Updated on: 19 Dec 2014, 04:44
15
30
How to Solve: Inequalities

Hi All,

I have learned a lot from gmatclub and am done with my gmat too. So, i have decided to contribute back.
As part of this i have decided to share the knowledge i have regarding various topics related to gmat quant.
Hope it will be useful. This post is about how to solve "Inequalities"

Theory

There are mainly four types of inequality problems which you would need to solve:--

TYPE 1
x*y > 0
When xy > 0 then we know that both x and y can be either positive or both can be negative
i.e. both x and y have the same sign
so, we have
x>0, y>0 or x<0,y<0

Example Problem
(x-1)*(x-2) > 0
So, we have two cases
Case 1
both (x-1) and (x-2) are positive
so, x-1 > 0 => x > 1
and x-2 > 0 => x > 2
Intersection of the two cases is x >2

Case 2
both (x-1) and (x-2) are negative
so, x-1 < 0 => x < 1
and x-2 < 0 => x < 2
Intersection of the two cases is x < 1

So, Solution to the question is x < 1 or x > 2

TYPE 2
x/y > 0
When x/y > 0 then we know that both x and y can be either positive or both can be negative
i.e. both x and y have the same sign
so, we have
x>0, y>0 or x<0,y<0

Example Problem
$$\frac{(x-3)}{(x-4)}$$ > 0
So, we have two cases
Case 1
Both (x-3) and (x-4) are positive
=> x-3 > 0 => x>3
And x-4 > 0 => x>4
Intersection of the two cases is x > 4

Case 2
Both (x-3) and (x-4) are negative
=> x-3 < 0 => x < 3
and x-4 < 0 => x < 4
Intersection of the two cases is x < 3

So, solution to the question is x < 3 or x > 4

TYPE 3
x*y < 0
When x*y < 0 then we know that that
(x can be positive and y will be negative) or (x can be negative and y will be positive)
i.e. x and y have opposite signs
so, we have
x>0, y<0 or x<0,y>0

Example Problem
(x+1)(x-1) < 0
So, we will have two cases
Case 1
(x+1) is positive and (x-1) is negative
=> x + 1 > 0 => x > -1
And x - 1 < 0 => x < 1
Intersection of the two cases is
-1 < x < 1

Case 2
(x+1) is negative and (x-1) is positive
=> x+1 < 0 => x < -1
And x-1 > 0 => x > 1
The two cases have no intersection. So, no solution from this case

So, solution of the problem is -1 < x < 1

TYPE 4
x/y < 0
When x/y < 0 then we know that that
(x can be positive and y will be negative) or (x can be negative and y will be positive)
i.e. x and y have opposite signs
so, we have
x>0, y<0 or x<0,y>0

Example Problem
$$\frac{(x-2)}{(x+3)}$$ < 0
So, we will have two cases
Case 1
(x-2) is positive and (x+3) is negative
=> x-2 > 0 => x > 2
And x+3 < 0 or x < -3
There is no intersection of the two cases. So, no solution from this case

Case 2
(x-2) is negative and (x+3) is positive
=> x-2 < 0 => x < 2
And x+3 > 0 => x > -3
Intersection of the two cases is -3 < x < 2

So, Solution of the question is -3 < x < 2

SUGGESTION: Try solving inequalities, they are not tough after all!

Problems:

1. x(x-1) > 0. Then value of x will be?

A. x > 0 and x > 1
B. x < 0 and x > 1
C. x < 0 and x < 1
D. x > 0 and x < 1

Solution:
x*(x-1) > 0
this is of the form xy>0 i.e. x and y have the same sign
so,
(1) either, x > 0 and x-1 >0
i.e. x >0 or x>1
taking intersection of the two possibilities we have x >1

(2)or x <0, and x-1 < 0
i.e. x <0 or x<1
taking intersection of the two possibilities we have x < 0

x-x-189656.html

2. Which of the following describes all the values of y for which y < y^2 ?

A. 1 < y
B. −1 < y < 0
C. y < −1
D. 1/y < 1
E. 0 < y < 1

Solution:
The question can be written as
y^2 - y > 0
s=> y*(y-1) > 0
It is of the form xy > 0
So, we will have two cases
Case 1
Both y and y-1 are positive
=> y > 0
And y-1 > 0 => y > 1
Intersection of the two cases is y > 1

Case 2
Both y and y-1 are negative
=>y < 0
And y -1 < 0 => y < 1
Intersection of the two cases is y <0

So, solution to the problem is y < 0 or y > 1

(As option D can be written as
1/y - 1 < 0
or, (1-y)/y < 0
or (y-1)/y > 0
And solution to this will be same as that of y*(y-1) > 0)

which-of-the-following-describes-all-the-values-of-y-for-whi-161602.html

3. Which of the following describes all values of x for which 1–x^2 >= 0?

(A) x >= 1
(B) x <= –1
(C) 0 <= x <= 1
(D) x <= –1 or x >= 1
(E) –1 <= x <= 1

Solution:
Question can be written as
x^2 - 1 <=0
=> (x+1)*(x-1) <=0
Case 1
x+1 is positive or 0 and x-1 is negative or 0
=> x+1 >= 0 => x >= -1
And x-1 <= 0 => x <= 1
Intersection is -1 <= x <= 1

Case 2
x+1 is negative or 0 and x-1 is positive or 0
x+1 <=0 => x <= -1
And x-1 >= 0 => x >= 1
No intersection in this case

So, solution to the problem is -1 <= x <= 1

which-of-the-following-describes-all-values-of-x-for-which-144461.html

4. If y>0>x, and (3+5y)/(x−1) < −7, then which of the following must be true?

A. 5y−7x+4 < 0
B. 5y+7x−4 > 0
C. 7x−5y−4 < 0
D. 4+5y+7x > 0
E. 7x−5y+4 > 0

Solution:
(3+5y)/(x−1) < −7
=> (3+5y)/(x−1) + 7 < 0
=> ((3+5y) + 7*(x-1) )/ (x-1) < 0
=> ( 7x + 5y -4 )/ (x-1) < 0
Now, we know that x < 0 so, x- 1 < 0
in (7x + 5y -4 )/ (x-1) < 0
we know that x - 1 < 0
=> (7x + 5y -4 ) > 0

if-y-0-x-and-3-5y-x-1-7-then-which-of-the-following-155220.html

5. Is k^2 + k - 2 > 0 ?

(1) k < 1
(2) k < -2

Solution:
k^2 + k - 2 > 0
=> (k+2)*(k-1) > 0
So, we will have two cases
Case 1
Both k+2 and k -1 positive
k+2 > 0 and k-1 > 0
=> k > -2 and k > 1
Intersection is k > 1

Case 2
Both k+2 and k-1 negative
k+2 < 0 and k -1 < 0
=> k < - 2 and k < 1
intersection is k < -2
So, Solution to the problem is k> 1 or k < -2

So, STAT1 is not SUFFICIENT
STAT2 is SUFFICIENT

is-k-2-k-147133.html

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Hope it helps!
Good Luck!
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How to Solve :
Statistics || Reflection of a line || Remainder Problems || Inequalities

Originally posted by BrushMyQuant on 07 Dec 2014, 07:22.
Last edited by BrushMyQuant on 19 Dec 2014, 04:44, edited 3 times in total.
Formatting.
Math Expert
Joined: 02 Sep 2009
Posts: 51121
Re: How to Solve: Inequalities  [#permalink]

### Show Tags

08 Dec 2014, 03:12
nktdotgupta wrote:
How to Solve: Inequalities

Hi All,

I have learned a lot from gmatclub and am done with my gmat too. So, i have decided to contribute back.
As part of this i have decided to share the knowledge i have regarding various topics related to gmat quant.
Hope it will be useful. This post is about how to solve "Inequalities"

Theory

There are mainly four types of inequality problems which you would need to solve:--

TYPE 1
x*y > 0
When xy > 0 then we know that both x and y can be either positive or both can be negative
i.e. both x and y have the same sign
so, we have
x>0, y>0 or x<0,y<0

Example Problem
(x-1)*(x-2) > 0
So, we have two cases
Case 1
both (x-1) and (x-2) are positive
so, x-1 > 0 => x > 1
and x-2 > 0 => x > 2
Intersection of the two cases is x >2

Case 2
both (x-1) and (x-2) are negative
so, x-1 < 0 => x < 1
and x-2 < 0 => x < 2
Intersection of the two cases is x < 1

So, Solution to the question is x < 1 or x > 2

TYPE 2
x/y > 0
When x/y > 0 then we know that both x and y can be either positive or both can be negative
i.e. both x and y have the same sign
so, we have
x>0, y>0 or x<0,y<0

Example Problem
$$\frac{(x-3)}{(x-4)}$$ > 0
So, we have two cases
Case 1
Both (x-3) and (x-4) are positive
=> x-3 > 0 => x>3
And x-4 > 0 => x>4
Intersection of the two cases is x > 4

Case 2
Both (x-3) and (x-4) are negative
=> x-3 < 0 => x < 3
and x-4 < 0 => x < 4
Intersection of the two cases is x < 3

So, solution to the question is x < 3 or x > 4

TYPE 3
x*y < 0
When x*y < 0 then we know that that
(x can be positive and y will be negative) or (x can be negative and y will be positive)
i.e. x and y have opposite signs
so, we have
x>0, y<0 or x<0,y>0

Example Problem
(x+1)(x-1) < 0
So, we will have two cases
Case 1
(x+1) is positive and (x-1) is negative
=> x + 1 > 0 => x > -1
And x - 1 < 0 => x < 1
Intersection of the two cases is
-1 < x < 1

Case 2
(x+1) is negative and (x-1) is positive
=> x+1 < 0 => x < -1
And x-1 > 0 => x > 1
The two cases have no intersection. So, no solution from this case

So, solution of the problem is -1 < x < 1

TYPE 4
x/y < 0
When x/y < 0 then we know that that
(x can be positive and y will be negative) or (x can be negative and y will be positive)
i.e. x and y have opposite signs
so, we have
x>0, y<0 or x<0,y>0

Example Problem
$$\frac{(x-2)}{(x+3)}$$ < 0
So, we will have two cases
Case 1
(x-2) is positive and (x+3) is negative
=> x-2 > 0 => x > 2
And x+3 < 0 or x < -3
There is no intersection of the two cases. So, no solution from this case

Case 2
(x-2) is negative and (x+3) is positive
=> x-2 < 0 => x < 2
And x+3 > 0 => x > -3
Intersection of the two cases is -3 < x < 2

So, Solution of the question is -3 < x < 2

SUGGESTION: Try solving inequalities, they are not tough after all!

Problems:

1. x(x-1) > 0. Then value of x will be?

A. x > 0 and x > 1
B. x < 0 and x > 1
C. x < 0 and x < 1
D. x > 0 and x < 1

Solution:
x*(x-1) > 0
this is of the form xy>0 i.e. x and y have the same sign
so,
(1) either, x > 0 and x-1 >0
i.e. x >0 or x>1
taking intersection of the two possibilities we have x >1

(2)or x <0, and x-1 < 0
i.e. x <0 or x<1
taking intersection of the two possibilities we have x < 0

x-x-189656.html

2. Which of the following describes all the values of y for which y < y^2 ?

A. 1 < y
B. −1 < y < 0
C. y < −1
D. 1/y < 1
E. 0 < y < 1

Solution:
The question can be written as
y^2 - y > 0
s=> y*(y-1) > 0
It is of the form xy > 0
So, we will have two cases
Case 1
Both y and y-1 are positive
=> y > 0
And y-1 > 0 => y > 1
Intersection of the two cases is y > 1

Case 2
Both y and y-1 are negative
=>y < 0
And y -1 < 0 => y < 1
Intersection of the two cases is y <0

So, solution to the problem is y < 0 or y > 1

(As option D can be written as
1/y - 1 < 0
or, (1-y)/y < 0
or (y-1)/y > 0
And solution to this will be same as that of y*(y-1) > 0)

which-of-the-following-describes-all-the-values-of-y-for-whi-161602.html

3. Which of the following describes all values of x for which 1–x^2 >= 0?

(A) x >= 1
(B) x <= –1
(C) 0 <= x <= 1
(D) x <= –1 or x >= 1
(E) –1 <= x <= 1

Solution:
Question can be written as
x^2 - 1 <=0
=> (x+1)*(x-1) <=0
Case 1
x+1 is positive or 0 and x-1 is negative or 0
=> x+1 >= 0 => x >= -1
And x-1 <= 0 => x <= 1
Intersection is -1 <= x <= 1

Case 2
x+1 is negative or 0 and x-1 is positive or 0
x+1 <=0 => x <= -1
And x-1 >= 0 => x >= 1
No intersection in this case

So, solution to the problem is -1 <= x <= 1

which-of-the-following-describes-all-values-of-x-for-which-144461.html

4. If y>0>x, and (3+5y)/(x−1) < −7, then which of the following must be true?

A. 5y−7x+4 < 0
B. 5y+7x−4 > 0
C. 7x−5y−4 < 0
D. 4+5y+7x > 0
E. 7x−5y+4 > 0

Solution:
(3+5y)/(x−1) < −7
=> (3+5y)/(x−1) + 7 < 0
=> ((3+5y) + 7*(x-1) )/ (x-1) < 0
=> ( 7x + 5y -4 )/ (x-1) < 0
Now, we know that x < 0 so, x- 1 < 0
in (7x + 5y -4 )/ (x-1) < 0
we know that x - 1 < 0
=> (7x + 5y -4 ) > 0

if-y-0-x-and-3-5y-x-1-7-then-which-of-the-following-155220.html

5. Is k^2 + k - 2 > 0 ?

(1) k < 1
(2) k < -2

Solution:
k^2 + k - 2 > 0
=> (k+2)*(k-1) > 0
So, we will have two cases
Case 1
Both k+2 and k -1 positive
k+2 > 0 and k-1 > 0
=> k > -2 and k > 1
Intersection is k > 1

Case 2
Both k+2 and k-1 negative
k+2 < 0 and k -1 < 0
=> k < - 2 and k < 1
intersection is k < -2
So, Solution to the problem is k> 1 or k < -2

So, STAT1 is not SUFFICIENT
STAT2 is SUFFICIENT

is-k-2-k-147133.html

Hope it helps!
Good Luck!

Good job! Thank you!
_________________
Intern
Joined: 04 Mar 2011
Posts: 13
Re: How to Solve: Inequalities  [#permalink]

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16 Dec 2014, 13:33
absolutely fantastic!!
Intern
Joined: 24 Mar 2013
Posts: 24
Re: How to Solve: Inequalities  [#permalink]

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05 May 2015, 19:12
Current Student
Joined: 14 Oct 2013
Posts: 45
Re: How to Solve: Inequalities  [#permalink]

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08 May 2015, 07:27
In number 4 - why did you multiply 7 by (x-1)? I multiplied the whole left hand side of the expression by (x-1) to get rid of (x-1) in the denominator but it seems as if you kept the (x-1) in the denominator when you did so.
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Re: How to Solve: Inequalities  [#permalink]

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08 May 2015, 09:52
Hi All,

GMAT Quant questions are typically designed in a way that allows the Test Taker more than one way to solve. In this article, the solutions are essentially all based on "doing algebra", but each of these questions can ALSO be solved by TESTing VALUES.

In many cases, TESTing VALUES is an easier and faster way to get to the correct answer, so beyond learning how to "do the math", you should also put in the proper time to learn (and practice) tactics. You'll find that it's easier to score at a higher level when you have more approaches to choose from.

GMAT assassins aren't born, they're made,
Rich
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Re: How to Solve: Inequalities  [#permalink]

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01 Oct 2018, 01:38
Hello from the GMAT Club BumpBot!

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Re: How to Solve: Inequalities &nbs [#permalink] 01 Oct 2018, 01:38
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# How to Solve: Inequalities

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