If |x + 5| <= |2x - 5|, which of the following specifies all the value : Problem Solving (PS)

B

Nikhil30

Current Student

Joined: 03 Jan 2019

Posts: 30

Own Kudos [?]: 29 [0]

Given Kudos: 13

Schools: Ross '23 (A)

Send PM

Re: If |x + 5| <= |2x - 5|, which of the following specifies all the value [#permalink]

04 Apr 2021, 00:38

DreamerHBS wrote:

Use options. It would hardly take a minute.

hey , i did solved the question , i am getting only x>=10 , not getting x<=0 , thats why i have posted.

please come forward with your answer , rather just asking me to solve with the help pf option , if i could have solved the question at first , i wouldn't have posted
here .

B

DreamerHBS

Intern

Joined: 09 Jun 2017

Posts: 6

Own Kudos [?]: 1 [0]

Given Kudos: 20

Send PM

Re: If |x + 5| <= |2x - 5|, which of the following specifies all the value [#permalink]

04 Apr 2021, 00:47

Nikhil30 wrote:

DreamerHBS wrote:

Use options. It would hardly take a minute.

hey , i did solved the question , i am getting only x>=10 , not getting x<=0 , thats why i have posted.

please come forward with your answer , rather just asking me to solve with the help pf option , if i could have solved the question at first , i wouldn't have posted
here .

|x+5|<=|2x-5|
case 1) both sides +
x+5 <=2x-5
x-2x+5+5<=0
-x+10<=0---> x>=10

case 2) both sides negative
-x-5<= -2x+5
-x+2x-5-5<=0
x-10<=0
x<=10

case 3) LHS positive and RHS negative
x+5<= -2x+5
x+2x+5-5<=0
3x<=0---> x<=0

B

Nikhil30

Current Student

Joined: 03 Jan 2019

Posts: 30

Own Kudos [?]: 29 [1]

Given Kudos: 13

Schools: Ross '23 (A)

Send PM

Re: If |x + 5| <= |2x - 5|, which of the following specifies all the value [#permalink]

04 Apr 2021, 01:53

1

Kudos

DreamerHBS wrote:

Nikhil30 wrote:

DreamerHBS wrote:

Use options. It would hardly take a minute.

hey , i did solved the question , i am getting only x>=10 , not getting x<=0 , thats why i have posted.

please come forward with your answer , rather just asking me to solve with the help pf option , if i could have solved the question at first , i wouldn't have posted
here .

|x+5|<=|2x-5|
case 1) both sides +
x+5 <=2x-5
x-2x+5+5<=0
-x+10<=0---> x>=10

case 2) both sides negative
-x-5<= -2x+5
-x+2x-5-5<=0
x-10<=0
x<=10

case 3) LHS positive and RHS negative
x+5<= -2x+5
x+2x+5-5<=0
3x<=0---> x<=0

hey , can we solve this question by squaring both side ,
as |a| <= |B| , can also be write as a^2<=b^2

and than proceed , by that i am getting only x>=10. can you please tell me whats wrong with that method.

L

Kinshook

GMAT Club Legend

Joined: 03 Jun 2019

Posts: 5346

Own Kudos [?]: 4000 [0]

Given Kudos: 160

Location: India

Schools: HBS '22 CBS '22 Wharton '22

GMAT 1: 690 Q50 V34 Verified score

WE:Engineering (Transportation)

Send PM

Re: If |x + 5| <= |2x - 5|, which of the following specifies all the value [#permalink]

04 Apr 2021, 03:26

Asked: If |x + 5| <= |2x - 5|, which of the following specifies all the value of x ?

|x + 5| <= |2x - 5|

Case 1: x<=-5
|x + 5| <= |2x - 5|
-x-5 < = 5-2x
x <= 10
x<=-5<=10
The inequality is valid for all x such that x<=-5

Case 2: -5<x<=2.5
|x + 5| <= |2x - 5|
x+5 <= 5-2x
3x <=0
The inequality is valid for all x such that -5<x<=0

Case 3: x>2.5
|x + 5| <= |2x - 5|
x+5 <= 2x-5
x >=10
The inequality is valid for all x such that x >=10

Combing the 3 results
x<=0 or x>=10

IMO D

B

Nikhil30

Current Student

Joined: 03 Jan 2019

Posts: 30

Own Kudos [?]: 29 [0]

Given Kudos: 13

Schools: Ross '23 (A)

Send PM

If |x + 5| <= |2x - 5|, which of the following specifies all the value [#permalink]

04 Apr 2021, 03:34

Bunuel wrote:

Nikhil30 wrote:

If |x + 5| <= |2x - 5|, which of the following specifies all the value of x ?

A. x >= 10
B. -5 <= x<= 0 or x >= 10
C. x <= -5 or x >= 10
D. x <= 0 or x >= 10
E. none.

APPROACH #1

\(|x + 5| \leq |2x - 5|\);

Square both sides: \(x^2 + 10x + 25 \leq 4x^2 - 20x + 25\)

Re-arrange and simplify: \( x^2 - 10x \geq 0\)

Factor x: \( x(x - 10) \geq 0\);

\(x \leq 0\) or \(x \geq 10\)

Answer: D.

APPROACH #2

Transition points of \(|x + 5| \leq |2x - 5|\) are -5 and 5/2. Consider three ranges:

When \(x \leq -5\), then \(x + 5 \leq 0\) and \(2x - 5 \leq 0\), so \(|x + 5| \leq |2x - 5|\) becomes \(-(x + 5) \leq -(2x -5)\). This gives \(x \leq 10\). Since we consider the case when \(x \leq -5\) then the actual solution for this range is \(x \leq -5\) (the overlap of \(x \leq -5\) and \(x \leq 10\) is \(x \leq -5\));

When \( -5 < x < \frac{5}{2}\), then \(x + 5 > 0\) and \(2x - 5 < 0\), so \(|x + 5| \leq |2x - 5|\) becomes \(x + 5 \leq -(2x -5)\). This gives \(x \leq 0\). Since we consider the case when \( -5 < x < \frac{5}{2}\) then the actual solution for this range is \( -5 < x \leq 0\) (the overlap of \(x \leq 0\) and \( -5 < x < \frac{5}{2}\) is \( -5 < x \leq 0\));

When \(x \geq \frac{5}{2}\), then \(x + 5 > 0\) and \(2x - 5 \geq 0\), so \(|x + 5| \leq |2x - 5|\) becomes \(x + 5 \leq 2x -5\). This gives \(x \leq 10\). Since we consider the case when \(x \leq -5\) then the actual solution for this range is \(x \geq 10\) (the overlap of \(x \geq 10\) and \(x \geq \frac{5}{2}\) is \(x \geq 10\)).

The three ranges \(x \leq -5\), \( -5 < x \leq 0\) and \(x \geq 10\) give the final range of x as \(x \leq 0\) or \(x \geq 10\).

Answer: D.

Hope it's clear.

hey i got your point thanks for your reply,
i am again getting confused here ,
x(x-10) >=0,
it means , x>=0 and x>=10,
would you please explain how you have written x<=0. (in approach 1)

L

Bunuel

Math Expert

Joined: 02 Sep 2009

Posts: 93260

Own Kudos [?]: 623756 [1]

Given Kudos: 81862

Send PM

Re: If |x + 5| <= |2x - 5|, which of the following specifies all the value [#permalink]

04 Apr 2021, 05:13

1

Bookmarks

Expert Reply

Nikhil30 wrote:

Bunuel wrote:

Nikhil30 wrote:

If |x + 5| <= |2x - 5|, which of the following specifies all the value of x ?

A. x >= 10
B. -5 <= x<= 0 or x >= 10
C. x <= -5 or x >= 10
D. x <= 0 or x >= 10
E. none.

APPROACH #1

\(|x + 5| \leq |2x - 5|\);

Square both sides: \(x^2 + 10x + 25 \leq 4x^2 - 20x + 25\)

Re-arrange and simplify: \( x^2 - 10x \geq 0\)

Factor x: \( x(x - 10) \geq 0\);

\(x \leq 0\) or \(x \geq 10\)

Answer: D.

APPROACH #2

Transition points of \(|x + 5| \leq |2x - 5|\) are -5 and 5/2. Consider three ranges:

When \(x \leq -5\), then \(x + 5 \leq 0\) and \(2x - 5 \leq 0\), so \(|x + 5| \leq |2x - 5|\) becomes \(-(x + 5) \leq -(2x -5)\). This gives \(x \leq 10\). Since we consider the case when \(x \leq -5\) then the actual solution for this range is \(x \leq -5\) (the overlap of \(x \leq -5\) and \(x \leq 10\) is \(x \leq -5\));

When \( -5 < x < \frac{5}{2}\), then \(x + 5 > 0\) and \(2x - 5 < 0\), so \(|x + 5| \leq |2x - 5|\) becomes \(x + 5 \leq -(2x -5)\). This gives \(x \leq 0\). Since we consider the case when \( -5 < x < \frac{5}{2}\) then the actual solution for this range is \( -5 < x \leq 0\) (the overlap of \(x \leq 0\) and \( -5 < x < \frac{5}{2}\) is \( -5 < x \leq 0\));

When \(x \geq \frac{5}{2}\), then \(x + 5 > 0\) and \(2x - 5 \geq 0\), so \(|x + 5| \leq |2x - 5|\) becomes \(x + 5 \leq 2x -5\). This gives \(x \leq 10\). Since we consider the case when \(x \leq -5\) then the actual solution for this range is \(x \geq 10\) (the overlap of \(x \geq 10\) and \(x \geq \frac{5}{2}\) is \(x \geq 10\)).

The three ranges \(x \leq -5\), \( -5 < x \leq 0\) and \(x \geq 10\) give the final range of x as \(x \leq 0\) or \(x \geq 10\).

Answer: D.

Hope it's clear.

hey i got your point thanks for your reply,
i am again getting confused here ,
x(x-10) >=0,
it means , x>=0 and x>=10,
would you please explain how you have written x<=0. (in approach 1)

The product x(x - 10) will be non-negative (\(\geq 0\)) in the following two cases:

both x and x - 10 are non-negative (\(\geq 0\)). Here notice that x - 10 is always less than x, so if x - 10 is positive, x will also be positive. Thus, the product will be non-negative if x -10 >= 0, or if x >= 10.
both x and x - 10 are non-positive (\(\leq 0\)). Here notice that x is always more than x - 10, so if x is negative, x - 10 will also be negative. Thus, the product will be non-positive if x <= 0.

Else you can use the method explained here: https://gmatclub.com/forum/solving-quad ... 70528.html

Also, you might find the following links useful:

9. Inequalities

Theory
Inequalities Made Easy!
Inequalities: Tips and hints
Arithmetic with Inequalities
Wavy Line Method Application - Complex Algebraic Inequalities
Solving Quadratic Inequalities: Graphic Approach
Inequalities - Quadratic Inequalities
Graphic approach to problems with inequalities
Inequalities trick
INEQUATIONS(Inequalities) Part 1
INEQUATIONS(Inequalities) Part 2

Questions
Inequality and absolute value questions from my collection
DS Questions
PS Questions

10. Absolute Value

Theory
Math: Absolute value (Modulus)
Absolute Value: Tips and hints
Properties of Absolute Values on the GMAT
Absolute modulus : A better understanding
GMAT Question Patterns: Abolute Value
The Reason Behind Absolute Value Questions on the GMAT

Questions
Inequality and absolute value questions from my collection
The E-GMAT Question Series on ABSOLUTE VALUE
DS Questions
PS Questions

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.

B

somyamehta777

Intern

Joined: 13 Aug 2020

Posts: 28

Own Kudos [?]: 19 [0]

Given Kudos: 23

Concentration: Marketing, International Business

Send PM

If |x + 5| <= |2x - 5|, which of the following specifies all the value [#permalink]

05 Apr 2021, 12:05

APPROACH #1

\(|x + 5| \leq |2x - 5|\);

Square both sides: \(x^2 + 10x + 25 \leq 4x^2 - 20x + 25\)

Re-arrange and simplify: \( x^2 - 10x \geq 0\)

Factor x: \( x(x - 10) \geq 0\);

\(x \leq 0\) or \(x \geq 10\)

Answer: D.

Hey Bunuel, in approach 1 when you rearrange wouldn't it be \( 3x^2 - 10x \geq 0\) and wouldn't answer choices change a bit because of it ?

L

Bunuel

Math Expert

Joined: 02 Sep 2009

Posts: 93260

Own Kudos [?]: 623756 [1]

Given Kudos: 81862

Send PM

Re: If |x + 5| <= |2x - 5|, which of the following specifies all the value [#permalink]

05 Apr 2021, 13:37

1

Kudos

Expert Reply

somyamehta777 wrote:

APPROACH #1

\(|x + 5| \leq |2x - 5|\);

Square both sides: \(x^2 + 10x + 25 \leq 4x^2 - 20x + 25\)

Re-arrange and simplify: \( x^2 - 10x \geq 0\)

Factor x: \( x(x - 10) \geq 0\);

\(x \leq 0\) or \(x \geq 10\)

Answer: D.

Hey Bunuel, in approach 1 when you rearrange wouldn't it be \( 3x^2 - 10x \geq 0\) and wouldn't answer choices change a bit because of it ?

Here is how it goes:

\(x^2 + 10x + 25 \leq 4x^2 - 20x + 25\);

\(0 \leq 3x^2 - 30x\)

\(0 \leq x^2 - 10x\)

So, you should get \(0 \leq x^2 - 10x\) (\( x^2 - 10x \geq 0\)) not \( 3x^2 - 10x \geq 0\)

B

Ujjawalm8shra

Intern

Joined: 06 Jun 2019

Posts: 6

Own Kudos [?]: 0 [0]

Given Kudos: 31

Send PM

Re: If |x + 5| <= |2x - 5|, which of the following specifies all the value [#permalink]

06 Apr 2021, 05:22

Kinshook wrote:

Asked: If |x + 5| <= |2x - 5|, which of the following specifies all the value of x ?

|x + 5| <= |2x - 5|

Case 1: x<=-5
|x + 5| <= |2x - 5|
-x-5 < = 5-2x
x <= 10
x<=-5<=10
The inequality is valid for all x such that x<=-5

Case 2: -5<x<=2.5
|x + 5| <= |2x - 5|
x+5 <= 5-2x
3x <=0
The inequality is valid for all x such that -5<x<=0

Case 3: x>2.5
|x + 5| <= |2x - 5|
x+5 <= 2x-5
x >=10
The inequality is valid for all x such that x >=10

Combing the 3 results
x<=0 or x>=10

IMO D

My query is how is the first result of x=<10 included here?

Posted from my mobile device

bumpbot

Non-Human User

Joined: 09 Sep 2013

Posts: 33010

Own Kudos [?]: 828 [0]

Given Kudos: 0

Send PM

Re: If |x + 5| <= |2x - 5|, which of the following specifies all the value [#permalink]

11 Jan 2024, 01:20

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

gmatclubot

Re: If |x + 5| <= |2x - 5|, which of the following specifies all the value [#permalink]

11 Jan 2024, 01:20

GMAT Problem Solving (PS) Questions

If |x + 5| <= |2x - 5|, which of the following specifies all the value