GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

It is currently 03 Jul 2020, 16:27

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

Inequalities Made Easy!

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Intern
Intern
avatar
B
Joined: 22 Apr 2019
Posts: 34
GMAT ToolKit User
Re: Inequalities Made Easy!  [#permalink]

Show Tags

New post 08 Jun 2019, 06:06
Bunuel wrote:

Notice that \((x+a)(x+b)(x+c)>0\) can be written as \((x-(-a))(x-(-b))(x-(-c))>0\). So, if the "roots" of (x-a)(x-b)(x-c)>0 are a, b, and c, for \((x-(-a))(x-(-b))(x-(-c))>0\) they would be -a, -b, and -c.

For example, for (x + 1)(x + 2) > 0, you put on the line -1 and -2.

Hope it helps.


Ah perfect! Thanks!
Intern
Intern
avatar
B
Joined: 15 Jul 2019
Posts: 5
Re: Inequalities Made Easy!  [#permalink]

Show Tags

New post 23 Jul 2019, 17:18
Hi Karishma & Bunuel,

First of all, let me thank you to put together such an amazing explanation of Inequalities and strategies. It helped me immensely.

I have a small doubt in Step 2 mentioned in the very first post "Inequalities with Multiple Factors" of this thread:

"Starting from the rightmost section, mark the sections with alternate positive and negative signs. The inequality will be positive in the sections where you have the positive signs and it will be negative in the sections where you have the negative signs"

My understanding is that we have to check one value from each section to understand if the equation results in a Positive or Negative value in order to assign the sign to each of the section. is this correct? or we can blindly assign Positive and Negative sign to alternate sections respectively by starting with assigning Positive sign to the rightmost section?
Intern
Intern
avatar
B
Joined: 22 Apr 2019
Posts: 34
GMAT ToolKit User
Re: Inequalities Made Easy!  [#permalink]

Show Tags

New post 23 Jul 2019, 17:22
shishirdixit wrote:
Hi Karishma & Bunuel,

First of all, let me thank you to put together such an amazing explanation of Inequalities and strategies. It helped me immensely.

I have a small doubt in Step 2 mentioned in the very first post "Inequalities with Multiple Factors" of this thread:

"Starting from the rightmost section, mark the sections with alternate positive and negative signs. The inequality will be positive in the sections where you have the positive signs and it will be negative in the sections where you have the negative signs"

My understanding is that we have to check one value from each section to understand if the equation results in a Positive or Negative value in order to assign the sign to each of the section. is this correct? or we can blindly assign Positive and Negative sign to alternate sections respectively by starting with assigning Positive sign to the rightmost section?


You can blindly assign assuming that it's set up correctly.
Intern
Intern
avatar
Joined: 19 Jun 2019
Posts: 2
Re: Inequalities Made Easy!  [#permalink]

Show Tags

New post 05 Aug 2019, 15:01
Hello Bunuel,

How did you factor this?

"Given: −2x3+17x2–30x>0−2x3+17x2–30x>0

x(−2x2+17x–30)>0x(−2x2+17x–30)>0 (taking x common)

x(2x–5)(6–x)>0x(2x–5)(6–x)>0 (factoring the quadratic)"

Thanks
Intern
Intern
User avatar
B
Joined: 26 Sep 2013
Posts: 27
GMAT ToolKit User Premium Member CAT Tests
Re: Inequalities Made Easy!  [#permalink]

Show Tags

New post 26 May 2020, 10:18
Bunuel wrote:
Or Just Use Inequalities!

BY KARISHMA, VERITAS PREP


If you are wondering about the absurd title of this post, just take a look at the above post's title. It will make much more sense thereafter. This post is a continuation of last week’s post where we discussed number plugging. Today, as per students’ request, we will look at the inequalities approach to the same official question. You will need to go through our inequalities post to understand the method we will use here.

Recall that, given \(a < b\), \((x – a)(x – b) < 0\) gives us the range \(a < x < b\) and \((x – a)(x – b) > 0\) gives us the range \(x < a\) or \(x > b\).

Question: If x is positive, which of the following could be the correct ordering of 1/x, 2x and x^2?
(I) x^2 < 2x < 1/x
(II) x^2 < 1/x < 2x
(III) 2x < x^2 < 1/x


(A) none
(B) I only
(C) III only
(D) I and II
(E) I, II and III

Solution: The question has three complex inequalities. We will take each in turn. Note that each inequality consists of two more inequalities. We will split the complex inequality into two simpler inequalities e.g. x^2 < 2x < 1/x gives us x^2 < 2x and 2x < 1/x. Next we will find the range of values of x which satisfy each of these two inequalities and we will see if the two ranges have an overlap i.e. whether there are any values of x which satisfy both these simpler inequalities. If there are, it means there are values of x which satisfy the entire complex inequality too. Things will become clearer once we start working on it so hold on.

Let’s look at each inequality in turn. We start with the first one:

(I) x^2 < 2x < 1/x

We split it into two inequalities:

(i) x^2 < 2x

We can rewrite x^2 < 2x as x^2 – 2x < 0 or x(x – 2) < 0.

We know the range of x for such inequalities can be easily found using the curve on the number line. This will give us 0 < x < 2.

(ii) 2x < 1/x

It can be rewritten as x^2 – 1/2 < 0 (Note that since x must be positive, we can easily multiply both sides of the inequality with x)

This gives us the range -1/?2 < x < 1/?2 (which is 0 < x < 1/?2 since x must be positive).

Is there a region of overlap in these two ranges i.e. can both inequalities hold simultaneously for some values of x? Yes, they can hold for 0 < x < 1/?2. Hence, x^2 < 2x < 1/x will be true for the range 0 < x < 1/?2. So this could be the correct ordering. Let’s go on to the next complex inequality.

(II) x^2 < 1/x < 2x

Again, let’s break up the inequality into two parts:

(i) x^2 < 1/x

x^1 < 1/x is rewritten as x^3 – 1 < 0 which gives us x < 1.

(ii) 1/x < 2x

1/x < 2x is rewritten as x^2 – 1/2 > 0 which gives us x < -1/?2 (not possible since x must be positive) or x > 1/?2

Can both x < 1 and x > 1/?2 hold simultaneously? Sure! For 1/?2 < x < 1, both inequalities will hold and hence x^2 < 1/x < 2x will be true. So this could be the correct ordering too.

(III) 2x < x^2 < 1/x

The inequalities here are:

(i) 2x < x^2

2x < x^2 can be rewritten as x(x – 2) > 0 which gives us x < 0 (not possible) or x > 2.

(ii) x^2 < 1/x

x^2 < 1/x gives us x^3 – 1 < 0 i.e. x < 1

Can x be less than 1 and greater than 2 simultaneously? No. Therefore, 2x < x^2 < 1/x cannot be the correct ordering.

Answer (D)

Is this method simpler?



does anyone understand why this method works? and also bunel evaluated each a<b<c as a<b and b<c why not evaluate a<c?
_________________
You are the master of your destiny
GMAT Club Bot
Re: Inequalities Made Easy!   [#permalink] 26 May 2020, 10:18

Go to page   Previous    1   2   3   [ 45 posts ] 

Inequalities Made Easy!

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne