Lesson: Inequalities - Part I

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Comment on Inequalities - Part I

Excellent approach

A ton of practice ques :) Are inequalities a big portion of the quant section?
gmat-admin's picture

It's a common question type. So, you should become proficient with inequalities.

Hi Brent,

If abcd ≠ 0, is abcd < 0?

1. a/b > c/d
2. b/a > d/c

the OA is C. I'm not able to understand how? Please help with a simple explanation
gmat-admin's picture

Mitch provides an awesome solution here (2nd post): http://www.beatthegmat.com/gmat-advanced-quant-t291008.html

Let me know if you need me to clarify any steps.

Cheers,
Brent

I'm a little lost here:

Given: (y - 3x)/(y - 2x) > 1
Rewrite numerator as: (y - 2x - x)/(y - 2x) > 1
Apply fraction property to get: (y - 2x)/(y - 2x) - x/(y - 2x) > 1
gmat-admin's picture

You're referring to my solution here: https://gmatclub.com/forum/is-y-3x-y-2x-233053.html

I use a well-tested concept that says (a-b)/c = a/c - b/c

For example, (7 - 5)/2 = 7/2 - 5/2

In the original question, we have the fraction (y - 3x)/(y - 2x)

Notice that we can rewrite y -3x as y - 2x - x (both of these expressions are equivalent)

So, we can take the fraction (y - 3x)/(y - 2x) and rewrite it as follows:

(y - 3x)/(y - 2x) = (y - 2x - y)/(y - 2x)
= [(y - 2x) - (y)]/(y - 2x)...I just added brackets
= (y - 2x)/(y - 2x) - y/(y - 2x)...I applied fraction property
= 1 - y/(y - 2x)...I simplified first fraction

Does that help?

Cheers,
Brent

“Target question: Is x² > 1/x ? 

We can safely divide both sides of the inequality by x² to get: 1 > 1/x” 


My question is how did you get that, meaning 1 > 1/x 
gmat-admin's picture

You're referring to my solution here: http://www.beatthegmat.com/is-x-2-1-x-t296568.html

Statement 1 tells us that x² > x

When it comes to inequalities, we must be careful to consider whether certain values are NEGATIVE or POSITIVE before we divide both sides of an inequality by that value.

For example, if we divide both sides by a NEGATIVE value, then we must REVERSE the direction of the inequality symbol. Conversely, if we divide both sides by a POSITIVE value, then the direction of the inequality symbol REMAINS as it is.

Since we know that x² is POSITIVE, we can take the inequality x² > x and divide both sides by x² to get: 1 > 1/x

ASIDE: On the left side of the inequality, we have x²/x² = 1.
On the right side of the inequality, we have x/x² = 1/x

Does that help?

Cheers,
Brent

How does x² > 1/x get re-phrased into 1 > 1/x?
1) x² > 1/x
x² x²

2) 1> 1/x (isn’t 1/x divided by x2 equal to 1/x3 ?)

gmat-admin's picture

In my solution (at http://www.beatthegmat.com/is-x-2-1-x-t296568.html), I never rephrased x² > 1/x to get 1 > 1/x

I did, however, rephrase statement 1 (x² > x) to get the equivalent inequality: 1 > 1/x

Does that help?

Thanks Brent!

For some reason I got stuck on re-phrasing the main question somehow/for some reason and couldn't work it out correctly. All is good.

Need some help on this one:

https://gmatclub.com/forum/if-x-y-z-0-is-x-135550.html
gmat-admin's picture

That's a great question.
Here's my step-by-step solution: https://gmatclub.com/forum/if-x-y-z-0-is-x-135550.html#p1937531

Cheers,
Brent

If abcd ≠ 0, is abcd < 0?

1. a/b>c/d
2. b/a>d/c
Sir this sum , I am feeling very difficult to understand.Hence Kindly help me.
gmat-admin's picture

This is a SUPER TRICKY question!
My full solution is here: https://gmatclub.com/forum/if-abcd-is-not-equal-to-zero-is-abcd-215779.h...

Cheers,
Brent

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