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.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 350,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Inequalities and Roots [#permalink]
09 Aug 2011, 11:56

1

This post received KUDOS

2

This post was BOOKMARKED

Something interesting that i read while searching for material on how to solve inequalities with roots.

Though i would share it and also clarify a few doubts. I have highlighted the portions that were confusing, in blue

Kind of backbone for solving inequalities with roots, √x>y OR √x<y

• √x is undefined if x<0 • both sides can be squared when x≥0 and y≥0 • if √x>y is identically true if √x≥0 and y<0 what does identically true/false mean • But √x<y is identically false if y<0

e.g., √(2x+3) > x

when, √(2x+3) ≥ 0 and x<0

left side, 2x + 3 ≥ 0 => x ≥ -1.5

right side, x < 0

thus, -1.5 ≤ x <0 (partial solution)

2nd Condition, where both left and right side, ≥ 0

Re: Inequalities and Roots [#permalink]
09 Aug 2011, 12:22

Identically true means true for all values of the variable.

For example: sqrt(x)>y if sqrt(x)>=0 and y<0 means that this relation holds for all values of x and y that satisfy sqrt(x)>=0 and y<0.

The graph of the parabola is attached.

Please note: The graph of ax^2+bx+c: (a) Is a parabola always (b) Opens upwards (towards +y axis) if a>0 and downwards (towards -y axis) if a<0 (c) Intersects the x axis at the roots of the equation ax^2 + bx + c = 0 (d) Will not touch the x axis if the roots are not real (if b^2 - 4ac < 0) (e) Has its lowest point at x= -b/2a

Re: Inequalities and Roots [#permalink]
09 Aug 2011, 12:52

Nice.

I have forgotten how to solve these. Thanks for reminding.

You can visualize very well the solution to this by plotting the graph. Not that you can do it during the actual exam, but for reference and to remember how the typical functions look like you can use //rechneronline.de/function-graphs/ to graph any function.

Re: Inequalities and Roots [#permalink]
09 Aug 2011, 20:03

2

This post received KUDOS

Expert's post

Asher wrote:

• √x is undefined if x<0 • both sides can be squared when x≥0 and y≥0 • if √x>y is identically true if √x≥0 and y<0 what does identically true/false mean • But √x<y is identically false if y<0

Some Explanations: 1. √x implies the positive square root of x. So √x is positive. But we do not know whether the right side of the inequality is positive or negative. Hence, we cannot square the inequality. Note: Only when both sides are positive, you can square the inequality and still retain the same relation. e.g. 2 < 3 2^2 < 3^2

But -2 < 1 4 not less than 1

Similarly -4 < -2 16 not less than 4

So before you square both sides of the inequality, always ensure that both sides are positive

2. √x>y We know that √x is positive. If y is negative, then this inequality will always hold since positive > negative.

3. √x<y We know that √x is positive. If y is negative, then this inequality will never hold since positive is never less than negative. _________________

Re: Inequalities and Roots [#permalink]
09 Aug 2011, 20:05

3

This post received KUDOS

Expert's post

Let's try one of the questions you have given.

1. √(3x-2) < 2x-3 Left hand side is positive, we know. Also, the term under the root i.e. (3x - 2) should be positive or 0. 3x - 2>= 0 i.e. x >= 2/3.

What about the right hand side? Here we can say that the right hand side will definitely be positive too since left hand side (a positive number (√3x-2)) is less than the right hand side. Hence, 2x - 3 > 0 x > 3/2

Let's square both sides now: 3x - 2 < (2x - 3)^2 4x^2 -15x + 11 > 0 4(x - 1)(x - 11/4) > 0 Since the right most region is positive, we will get: positive ... 1 ... negative ... 11/4 ... positive

So, 1 > x or x > 11/4.

But we saw above that x > 3/2 So x > 11/4 _________________

Re: Inequalities and Roots [#permalink]
10 Aug 2011, 01:29

GyanOne wrote:

Identically true means true for all values of the variable.

For example: sqrt(x)>y if sqrt(x)>=0 and y<0 means that this relation holds for all values of x and y that satisfy sqrt(x)>=0 and y<0.

The graph of the parabola is attached.

Please note: The graph of ax^2+bx+c: (a) Is a parabola always (b) Opens upwards (towards +y axis) if a>0 and downwards (towards -y axis) if a<0 (c) Intersects the x axis at the roots of the equation ax^2 + bx + c = 0 (d) Will not touch the x axis if the roots are not real (if b^2 - 4ac < 0) (e) Has its lowest point at x= -b/2a

Thanks GyanOne for the explanation and the graph. I get it now. _________________

Re: Inequalities and Roots [#permalink]
10 Aug 2011, 01:38

VeritasPrepKarishma wrote:

Asher wrote:

• √x is undefined if x<0 • both sides can be squared when x≥0 and y≥0 • if √x>y is identically true if √x≥0 and y<0 what does identically true/false mean • But √x<y is identically false if y<0

Some Explanations: 1. √x implies the positive square root of x. So √x is positive. But we do not know whether the right side of the inequality is positive or negative. Hence, we cannot square the inequality. Note: Only when both sides are positive, you can square the inequality and still retain the same relation. e.g. 2 < 3 2^2 < 3^2

But -2 < 1 4 not less than 1

Similarly -4 < -2 16 not less than 4

So before you square both sides of the inequality, always ensure that both sides are positive

2. √x>y We know that √x is positive. If y is negative, then this inequality will always hold since positive > negative.

3. √x<y We know that √x is positive. If y is negative, then this inequality will never hold since positive is never less than negative.

Thanks karishma for the explanation.

i now realize that when i read the material found on one of the math websites, i kind of just tried to mug the concept without really understanding it. By now the concept it clear. _________________

If x ≥ 5/2 then -4x <= -10 ==> -4x+3 <= -7 ---------------------(2)

Now as per your question - -4x + 3 is less than 0...This is the super set of (2), so there is no other unique solution... other than x ≥ 5/2 _________________

Re: Inequalities and Roots [#permalink]
11 Aug 2011, 22:22

Expert's post

Asher wrote:

Karishma, here's me trying to solve the second problem. Let me know if i got it right.

2. √(2x - 5) > -4x + 3

Since sq.root is always positive, thus:

√(2x - 5) ≥ 0 2x - 5 ≥ 0 x ≥ 5/2

No problems till here... we know now that x must be > or equal to 5/2.

Next think, we have LHS > RHS. RHS can be negative or positive. So there are two cases possible: Case 1: -4x + 3 < 0 x > 3/4 ----- (1) Since in this case, RHS is negative and LHS will always be positive, LHS will be > than RHS. So the inequality will hold whenever x > 3/4 and x >= 5/2. So if x >= 5/2, the inequality will hold.

Case 2: -4x + 3 > 0 x < 3/4 ------(2) Now think, is it possible that x is < than 3/4 and greater than 5/2? No! So this case doesn't give any solutions.

Hence, the only solution is x >= 5/2

and yes, don't try to learn up Mathematical concepts. Try to understand them. That way, you will never forget them. _________________

Re: Inequalities and Roots [#permalink]
13 Aug 2011, 02:18

Now i get it.

Quote:

and yes, don't try to learn up Mathematical concepts. Try to understand them. That way, you will never forget them.

Thanks for this advice. This makes soo much sense. Earlier i would just learn how to solve one particular problem without really understanding the concept, but then i would always make a mistake on a different problem based on the same concept.

Anyways, my POA now is to brush my basic quant skills. _________________

Re: Inequalities and Roots [#permalink]
23 Jul 2013, 21:26

Expert's post

gmatter0913 wrote:

Could somebody help me with the below problems?

1. x+2 < sqrt (x+14)

2. x-1 < sqrt(7-x)

Use the concepts given above to try to solve these:

x+2 < \sqrt{(x+14)}

The quantity under the root must be non negative so x >= -14

Now left hand side i.e. x+2 can be positive, 0 or negative. Take two cases:

Case 1: x + 2 >= 0 x >= -2 Now both sides of the inequality are non negative so we can square it: (x + 2)^2 < (x + 14) x^2 + 3x - 10 < 0 (x + 5) (x - 2) < 0 -5 < x < 2

Since x >= -2, we get -2 <= x < 2

Case 2: x + 2 < 0 x < -2 The left hand side i.e. x + 2 is always negative in this case while right hand side is always non negative so this inequality will hold for all values in this range. -14 <= x < -2

So the overall acceptable range is -14<= x < 2 _________________

I´ve done an interview at Accepted.com quite a while ago and if any of you are interested, here is the link . I´m through my preparation of my second...

It’s here. Internship season. The key is on searching and applying for the jobs that you feel confident working on, not doing something out of pressure. Rotman has...