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Combining inequalities - Adding inequalities

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Combining inequalities - Adding inequalities

by gmatrant » Wed Feb 29, 2012 2:09 pm
I was reading the Manhattan "Equations and Inequalities" book where in it stated that if two inequalities are given
i) a > b
ii) c > d

the we can combine the two inequalities to state positively that a+c > b+d.

But this does not hold true with the below problem

Example DS problem

Is p+q > r+s ?
i) p > r+s
ii) q > r+s

Now this is true as long as p, q are positive values.
If p is -4 , r is -3 , s is -3 then p > r+s
If q is -5 r is -3 , s is -3 then q > r+s
Combining the inequality
p+q > 2(r+s) so it means but p+q is greater than (r+s) as well but
with the above values p+q < r+s since p+q = -9 and r+s =-6.

Can experts please help as to when can inequalities be combined to conclude a solution and when we cannot combine.

OA is E
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by krusta80 » Wed Feb 29, 2012 2:56 pm
gmatrant wrote:I was reading the Manhattan "Equations and Inequalities" book where in it stated that if two inequalities are given
i) a > b
ii) c > d

the we can combine the two inequalities to state positively that a+c > b+d.

But this does not hold true with the below problem

Example DS problem

Is p+q > r+s ?
i) p > r+s
ii) q > r+s

Now this is true as long as p, q are positive values.
If p is -4 , r is -3 , s is -3 then p > r+s
If q is -5 r is -3 , s is -3 then q > r+s
Combining the inequality
p+q > 2(r+s) so it means but p+q is greater than (r+s) as well but
with the above values p+q < r+s since p+q = -9 and r+s =-6.

Can experts please help as to when can inequalities be combined to conclude a solution and when we cannot combine.

OA is E
You're making the assumption that 2*(r+s) HAS to be greater than (r+s) regardless of the value of r+s, but this is only true when r+s is positive.

As you state in your post, p+q > 2(r+s) as per the book, but is it NOT greater than r+s, which is not implied by the book in any way.
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by pemdas » Wed Feb 29, 2012 3:09 pm
he's not making any *assumptions* :)
he simply forgot to add r+s from the right hand side of the first inequality to the right hand side of the second inequality. Together they will make r+s + r+s and translated into his example -3-3 -3-3=-12 and this will hold true for -9>-12 always. So one can safely combine the inequalities provided the sign is the same.
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by krusta80 » Wed Feb 29, 2012 3:11 pm
pemdas wrote:he's not making any *assumptions* :)
he simply forgot to add r+s from the right hand side of the first inequality to the right hand side of the second inequality. Together they will make r+s + r+s and translated into his example -3-3 -3-3=-12 and this will hold true for -9>-12 always. So one can safely combine the inequalities provided the sign is the same.
Perhaps you're right...a big tough to follow his train of thought to be honest. :)
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by gmatrant » Wed Feb 29, 2012 10:38 pm
pemdas wrote:he's not making any *assumptions* :)
he simply forgot to add r+s from the right hand side of the first inequality to the right hand side of the second inequality. Together they will make r+s + r+s and translated into his example -3-3 -3-3=-12 and this will hold true for -9>-12 always. So one can safely combine the inequalities provided the sign is the same.
pemdas,

You state that "one can safely combine the inequalities provided the sign is the same" . But I don's see this condition specified in Manhattan's book. They just state that we can combine (add) inequalities. One needs to be guarded with inequalities only while multiplying or dividing by a variable whose sign is not known.
A kudos or thanks would do great if my answer has helped you :)
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by pemdas » Thu Mar 01, 2012 4:20 am
you need to review thoroughly inequality theory
there also reciprocals and sign flips
turn to gmat-club's math book, it's online and free i guess

-----------------
here, i've just attached the book in pdf format and check their web-site at https://gmatclub.com/forum/gmat-math-book-87417.html for improvements and GMAT examples. good luck on your cracking math section

cheers
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by Mike@Magoosh » Thu Mar 01, 2012 4:46 pm
I'll add my 2 cents to this discussion.

Just to be clear, if you know a > b and c > d, then 100% of the time, it follows that a + c > b + d, regardless of which values are positive/negative, fractions/integers, etc. If a, b, c, and d are any four real numbers, the expressions a > b and c > d lead directly and ineluctably to a + c > b + d, as well as to a - c > b - d. We can't multiply and divide without +/- sign information, but we can add and subtract all we want.

It's true, that we can't generalize any rule about u vs. 2*u ---- if we know u > 0, then 2u > u, but if we know u < 0, then 2u < u. That is, essentially, what krusta80 pointed out in the second post of this thread.

Further down in the thread, gmatrant said of the MGMAT books: "They just state that we can combine (add) inequalities. One needs to be guarded with inequalities only while multiplying or dividing by a variable whose sign is not known." This is 100% correct. Adding and subtracting inequalities proceeds problem-free, despite a dearth of knowledge about positive/negative signs of terms. Despite my friend pemdas' chastisement, I don't believe gmatrant's understanding of inequalities is lacking in any way.

I just want to make sure we are all on the same page with these concepts. Here's a somewhat germane DS question for practice:

https://gmat.magoosh.com/questions/980

When you submit your answer to that question, it will be followed by a complete video explanation.

Please let me know if anyone reading this has any question about anything I have said.

Mike :)
Magoosh GMAT Instructor
https://gmat.magoosh.com/
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by pemdas » Fri Mar 02, 2012 9:09 am
i meant sign of inequality not +/-
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by gmatrant » Fri Mar 02, 2012 1:38 pm
Mike - Thanks for the explanation . Makes it very clear now.

Pemdas - Thanks for the pdf.
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