inequalities

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marie: Newbie | Next Rank: 10 Posts; Posts: 9; Joined: Fri Sep 21, 2007 8:05 pm

by marie » Thu Oct 01, 2009 11:08 am

as part of a ds problem I had to resolve:

x^2 > 2x i.e., {x squared is greater than 2x}

is the answer: x>2 and x<0 or x>2 and x<-2??

thxs,
please show all steps!

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JeffB: Master | Next Rank: 500 Posts; Posts: 153; Joined: Tue Jun 02, 2009 2:48 pm; Location: Ohio; Thanked: 16 times

Re: inequalities

by JeffB » Thu Oct 01, 2009 2:57 pm

marie wrote:as part of a ds problem I had to resolve:

x^2 > 2x i.e., {x squared is greater than 2x}

is the answer: x>2 and x<0 or x>2 and x<-2??

thxs,
please show all steps!

x^2 > 2x

x^2 - 2x > 0
x(x - 2) > 0

0 > x > 2

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

Re: inequalities

by Stuart@KaplanGMAT » Thu Oct 01, 2009 3:37 pm

marie wrote:as part of a ds problem I had to resolve:

x^2 > 2x i.e., {x squared is greater than 2x}

is the answer: x>2 and x<0 or x>2 and x<-2??

thxs,
please show all steps!

As JeffB notes, we can simplify to:

x(x-2)>0

Let's see how we get from there to the final answer.

First, we recognize that to obtain a positive product, either both terms are positive or both are negative. Accordingly, we can set up two pairs of inequalities.

The positive case gives us:

x>0 AND x-2>0
x>0 AND x>2

When we have two inequalities pointing in the same direction, we must satisfy the more limiting inequality. In this case, that means that x>2.

The negative case gives us:

x<0 AND x-2<0
x<0 AND x<2

Again choosing the more limiting inequality, we get x<0.

So, to satisfy the original inequality, either:

x<0 OR x>2.

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

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marie: Newbie | Next Rank: 10 Posts; Posts: 9; Joined: Fri Sep 21, 2007 8:05 pm

Inequalities

by marie » Fri Oct 02, 2009 4:12 pm

Stuart,
Thank you for that. Help me understand where the following line of reasoning goes wrong.

x^2 > 2x
assume x is negative therefore
(-x)(-x) > 2(-x)
divide both sides by -x
-x < 2
multiply by -1
x > -2

Thxs

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

Re: Inequalities

by Stuart@KaplanGMAT » Fri Oct 02, 2009 7:13 pm

marie wrote:Stuart,
Thank you for that. Help me understand where the following line of reasoning goes wrong.

x^2 > 2x
assume x is negative therefore
(-x)(-x) > 2(-x)
divide both sides by -x
-x < 2
multiply by -1
x > -2

Thxs

Assuming x is negative doesn't mean we replace it with "-x" in the equation; if x is negative and you write "-x", you're actually looking at a double negative, which is positive.

If we assume x is negative, we leave it as written, we just remember to flip the inequality when we divide by x. So:

x*x>2x

dividing by x:

x<2

which is trivial - all we've shown is that if x is negative, then it's less than 2, which is true of all negative numbers.

I guess the more fundamental issue is why you're assuming that x is negative - how does that help you solve this problem?

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