Last visit was: 23 Jun 2024, 22:57 It is currently 23 Jun 2024, 22:57
Toolkit
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

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.

# What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9

SORT BY:
Tags:
Show Tags
Hide Tags
Manager
Joined: 30 Dec 2015
Posts: 181
Own Kudos [?]: 728 [104]
Given Kudos: 99
Location: United States
Concentration: Strategy, Organizational Behavior
GPA: 3.88
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11420
Own Kudos [?]: 33721 [45]
Given Kudos: 320
Math Expert
Joined: 02 Sep 2009
Posts: 93964
Own Kudos [?]: 634403 [16]
Given Kudos: 82420
Math Expert
Joined: 02 Sep 2009
Posts: 93964
Own Kudos [?]: 634403 [10]
Given Kudos: 82420
What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
4
Kudos
6
Bookmarks
Complete step-by-step solution

What is the value of $$|x + 5| + |x - 3|$$ ?

The critical points (aka key points or transition points) are -5 and 3 (the values of x for which the expressions in the absolute values become 0).

Consider three ranges:

• If $$x < - 5$$, then $$x + 5 < 0$$ and $$x - 3 < 0$$, so $$|x + 5| = -(x + 5)$$ and $$|x - 3| = -(x - 3)$$. Thus in this range $$|x + 5| + |x - 3|$$ becomes $$-(x + 5) - (x - 3) = -2 - 2x$$.
• If $$- 5 \leq x \leq 3$$, then $$x + 5 \geq 0$$ and $$x - 3 \leq 0$$, so $$|x + 5| = x + 5$$ and $$|x - 3| = -(x - 3)$$. Thus in this range $$|x + 5| + |x - 3|$$ becomes $$x + 5 - (x - 3) = 8$$.
• If $$x > 3$$, then $$x + 5 > 0$$ and $$x - 3 > 0$$, so $$|x + 5| = x + 5$$ and $$|x - 3| = x - 3$$. Thus in this range $$|x + 5| + |x - 3|$$ becomes $$x + 5 + (x - 3) = 2x + 2$$.

The above mean that if x is in the first range ($$x < - 5$$) or in the third range ($$x > 3$$), then the value of $$|x + 5| + |x - 3|$$ depends on the value of x. For example:
If $$x = -10$$, then $$|x + 5| + |x - 3|= -2 - 2x=18$$;
If $$x = -7$$, then $$|x + 5| + |x - 3|= -2 - 2x=12$$;
If $$x = 4$$, then $$|x + 5| + |x - 3|= 2x + 2=10$$;
If $$x = 6$$, then $$|x + 5| + |x - 3|= 2x + 2=14$$.

But if x is in the second range ($$- 5 \leq x \leq 3$$), then the value of $$|x + 5| + |x - 3|$$ is independent of the value of x, and is ALWAYS equals to 8. For example:
If $$x = -5$$, then $$|x + 5| + |x - 3|= 8$$;
If $$x = 0$$, then $$|x + 5| + |x - 3|= 8$$;
If $$x = 3$$, then $$|x + 5| + |x - 3|= 8$$.

(1) $$x^2< 25$$:

Take the square root: $$|x| < 5$$;
Get rid of the absolute value sign: $$-5 < 0 < 5$$;
x can be in the second or third range. So, $$|x + 5| + |x - 3|$$ is either 8 or $$2x + 2$$. Not sufficient.

(2) $$x^2 > 9$$:

Take the square root: $$|x| > 3$$;
Get rid of the absolute value sign: $$x < -3$$ or $$x > 3$$;
x can be in any of the three ranges from above. So, $$|x + 5| + |x - 3|$$ is $$-2 - 2x$$, 8 or $$2x + 2$$. Not sufficient.

(1)+(2) We get $$-5 < x < -3$$ (second range) or $$3 < x < 5$$ (third range). If $$-5 < x < -3$$ (second range), then $$|x + 5| + |x - 3|=8$$ but if $$3 < x < 5$$ (third range), then $$|x + 5| + |x - 3|=2x + 2$$ (so the value will depend on the exact value of x). Not sufficient.

General Discussion
Manager
Joined: 30 Dec 2015
Posts: 181
Own Kudos [?]: 728 [0]
Given Kudos: 99
Location: United States
Concentration: Strategy, Organizational Behavior
GPA: 3.88
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
lpetroski wrote:
What is the value of |x+5| + |x-3| ?

1) $$x^2$$ < 25

2) $$x^2$$ > 9

So, at first glance I thought it was E, but when combining the inequalities I got -2 < x < 2, which all of the values -1, 0 and 1 cause the equation to equal 8 -- but the OA is E - so what did I do wrong here? Thanks!!
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 10138
Own Kudos [?]: 16886 [5]
Given Kudos: 4
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
2
Kudos
3
Bookmarks
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

What is the value of |x+5| + |x-3| ?

1) x^2 < 25

2) x^2 > 9

When you modify the original condition and the question, a case where sum of 2 absolute values is derived is that the range of in between gets a consistent answer, which is -5<=x<=3?.
There is 1 variable(x), which should match with the number of equations. so you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.
When it comes to inequality questions, if range of que includes range of con, use the fact that that con is sufficient.
For 1), in -5<x<5, the range of que doesn't include the range of con, which is not sufficient.
For 2), in x<-3 or 3<x, the range of que doesn't include the range of con, which is not sufficient.
When 1) & 2), in -5<x<-3 or 3<x<5, the range of que doesn't include the range of con, which is not sufficient.

 For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
Manager
Joined: 25 Feb 2014
Posts: 181
Own Kudos [?]: 450 [3]
Given Kudos: 147
GMAT 1: 720 Q50 V38
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
3
Kudos
we have three concerned ranges

x < -3 here equation will be -2x-2
-3 =< x < 5 here equation value is 8
x >= 5 Here equation value will be 2x + 2

so only for 2nd range we have a fixed value.

Stmt 1 implies -5 < x < 5 insufficient as it covers more than one of the three above mentioned ranges.
Stmt 2 implies x < -3 and x > 3 insufficient as it covers more than one of the three above mentioned ranges.

Combining statement 1 & 2 -5<x < -3 and 3<x<5 insufficient as it covers more than one of the three above mentioned ranges.

Ans E
Manager
Joined: 23 Jan 2016
Posts: 139
Own Kudos [?]: 81 [0]
Given Kudos: 509
Location: India
GPA: 3.2
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
Bunuel, when we simplify x^2<25, do we write |x|<5 or x<|5| ?

Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 93964
Own Kudos [?]: 634403 [6]
Given Kudos: 82420
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
2
Kudos
4
Bookmarks
OreoShake wrote:
Bunuel, when we simplify x^2<25, do we write |x|<5 or x<|5| ?

Thanks

$$x^2 < 25$$

Take the square root from both sides: $$|x| < 5$$ (recall that $$\sqrt{x^2}=|x|$$).

Hope it's clear.
Manager
Joined: 02 Nov 2015
Posts: 133
Own Kudos [?]: 58 [0]
Given Kudos: 121
GMAT 1: 640 Q49 V29
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
Bunuel wrote:
lpetroski wrote:
lpetroski wrote:
What is the value of |x+5| + |x-3| ?

1) $$x^2$$ < 25

2) $$x^2$$ > 9

So, at first glance I thought it was E, but when combining the inequalities I got -2 < x < 2, which all of the values -1, 0 and 1 cause the equation to equal 8 -- but the OA is E - so what did I do wrong here? Thanks!!

First of all the solution is -5<x<-3 or 3<x<5. Next, you assume with no ground that x is an integer.

Hope it helps.

An eye opener...

Sent from my Lenovo TAB S8-50LC using GMAT Club Forum mobile app
GMAT Club Legend
Joined: 03 Jun 2019
Posts: 5226
Own Kudos [?]: 4087 [0]
Given Kudos: 160
Location: India
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
lpetroski wrote:
What is the value of |x+5| + |x-3| ?

1) $$x^2$$ < 25

2) $$x^2$$ > 9

Asked: What is the value of |x+5| + |x-3| ?

1) $$x^2$$ < 25
|x| < 5
-5<x<5
For the region -5<x<=3
|x+5| + |x-3| = 8
But for the region 3<x<5
|x+5| + |x-3| = 8 + 2(x-3) = 2+2x
NOT SUFFICIENT

2) $$x^2$$ > 9
|x| > 3
x<-3 or x>3
|x+5| + |x-3| varies with the value of x
NOT SUFFICIENT

(1) + (2)
1) $$x^2$$ < 25
2) $$x^2$$ > 9
3<|x|<5
-5<x<-3 or 3<x<5
For the region -5<x<-3
|x+5| + |x-3| = 8
But for the region 3<x<5
|x+5| + |x-3| = 2 + 2x
NOT SUFFICIENT

IMO E
Intern
Joined: 18 May 2021
Posts: 43
Own Kudos [?]: 3 [0]
Given Kudos: 51
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
chetan2u wrote:
lpetroski wrote:
lpetroski wrote:
What is the value of |x+5| + |x-3| ?

1) $$x^2$$ < 25

2) $$x^2$$ > 9

So, at first glance I thought it was E, but when combining the inequalitiesI got -2 < x < 2, which all of the values -1, 0 and 1 cause the equation to equal 8 -- but the OA is E - so what did I do wrong here? Thanks!!

Hi
the highlighted portion is wrong..

1) $$x^2$$ < 25

this gives -5<x<5
|x+5| + |x-3|
if 4 then |4+5|+|4-3|=10
if -4 then |-4+5|+|-4-3|=8
Insuff

2) $$x^2$$ > 9
this gives x<-3 or x>3
same this is also not suff

combined
either -5<x<-3 OR 3<x<5..
substitute 4 ans is 10
substitute -4 ans is 8..
Insuff
E

I tested a few numbers and got the answer right.
Although when I combined the two equations I interpreted it this way:
stat 1 says -5<x<5 and stat 2 says x>3 or x<-3

Thus, combining we can see from stat 1 that x<5 and from stat 2 that x<-3, thus combining these we see x<-3
Similarly combining the cases of what x should be bigger than, we get x>3
Thus, the final combined range becomes x>3 or x<-3 (essentially the same as stat 2), amd since stat 2 was anyway insuff, I marked E

Where did I go wrong in combining the two stats ? Experts pls help chetan2u VeritasKarishma Bunuel MathRevolution
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11420
Own Kudos [?]: 33721 [1]
Given Kudos: 320
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
1
Kudos
Chitra657, that will not be the correct interpretation.

-5<x<5
…….-5|xxxxxxxxxxxxxx|5…..
x<-3 or x>3
xxxxxxxxx-3|………|3xxxxxxxxxx

x is the marked range of x
What is common in two that fits in both the ranges
…..-5|xxxxx|-3……..3|xxxxxxx|5…..
Tutor
Joined: 16 Oct 2010
Posts: 15015
Own Kudos [?]: 66161 [2]
Given Kudos: 435
Location: Pune, India
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
2
Kudos
Chitra657 wrote:
Where did I go wrong in combining the two stats ? Experts pls help chetan2u VeritasKarishma Bunuel MathRevolution

Do one thing - mark the inequalities you have obtained on the number line.
The blue line shows -5 < x < 5
The two black arrows show x > 3 or x < -3.
Attachment:

Screenshot 2021-11-02 at 14.17.32.png [ 21.09 KiB | Viewed 16015 times ]

Which areas fall under both inequalities? (since both inequalities need to be satisfied)?
The 3 < x < 5 and -5 < x < -3
Intern
Joined: 02 Mar 2022
Posts: 1
Own Kudos [?]: 2 [2]
Given Kudos: 6
Location: United Kingdom
GMAT 1: 650 Q43 V35
What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
2
Kudos
Hi there,
Out of interest, why would one assume that x is an integer when no where in the question does is state that x is an integer?
It seems like it is necessary to get to the answer in the right way, however I was under the impression that unless a question states that 'x is an integer', you should not assume it?
Thanks!
Senior Manager
Joined: 22 Nov 2019
Posts: 255
Own Kudos [?]: 125 [0]
Given Kudos: 212
GPA: 4
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
Bunuel wrote:
If $$-5 < x < -3$$ (second range), then $$|x + 5| + |x - 3|=8$$

Bunuel

Could I please ask, how we can be sure that x = -4 here, as we are not given that x is an integer. Cant it be any value say -4.2, -3.5 etc?
Math Expert
Joined: 02 Sep 2009
Posts: 93964
Own Kudos [?]: 634403 [1]
Given Kudos: 82420
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
1
Kudos
TargetMBA007 wrote:
Bunuel wrote:
If $$-5 < x < -3$$ (second range), then $$|x + 5| + |x - 3|=8$$

Bunuel

Could I please ask, how we can be sure that x = -4 here, as we are not given that x is an integer. Cant it be any value say -4.2, -3.5 etc?

For the range -5 ≤ x ≤ -3, the value of |x + 5| + |x - 3| is 8 regardless of the exact value of x. You can verify this by testing any x within -5 < x < -3. This happens because when -5 ≤ x ≤ 3, then x + 5 ≥ 0 and x - 3 ≤ 0, leading to |x + 5| = x + 5 and |x - 3| = -(x - 3). Therefore, in this range, |x + 5| + |x - 3| simplifies to x + 5 - (x - 3) = 8, which resolves to 8 = 8. Hence, |x + 5| + |x - 3|= 8 is valid for any value within this range.

Hope it's clear.
Manager
Joined: 23 May 2023
Posts: 62
Own Kudos [?]: 32 [1]
Given Kudos: 305
Location: India
Concentration: Real Estate, Sustainability
GPA: 3.7
WE:Other (Other)
What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
1
Kudos
Bunuel wrote:
Complete step-by-step solution

What is the value of $$|x + 5| + |x - 3|$$ ?

The critical points (aka key points or transition points) are -5 and 3 (the values of x for which the expressions in the absolute values become 0).

Consider three ranges:

• If $$x < - 5$$, then $$x + 5 < 0$$ and $$x - 3 < 9$$, so $$|x + 5| = -(x + 5)$$ and $$|x - 3| = -(x - 3)$$. Thus in this range $$|x + 5| + |x - 3|$$ becomes $$-(x + 5) - (x - 3) = -2 - 2x$$.
• If $$- 5 \leq x \leq 3$$, then $$x + 5 \geq 0$$ and $$x - 3 \leq 9$$, so $$|x + 5| = x + 5$$ and $$|x - 3| = -(x - 3)$$. Thus in this range $$|x + 5| + |x - 3|$$ becomes $$x + 5 - (x - 3) = 8$$.
• If $$x > 3$$, then $$x + 5 > 0$$ and $$x - 3 > 9$$, so $$|x + 5| = x + 5$$ and $$|x - 3| = x - 3$$. Thus in this range $$|x + 5| + |x - 3|$$ becomes $$x + 5 + (x - 3) = 2x + 2$$.

The above mean that if x is in the first range ($$x < - 5$$) or in the third range ($$x > 3$$), then the value of $$|x + 5| + |x - 3|$$ depends on the value of x. For example:
If $$x = -10$$, then $$|x + 5| + |x - 3|= -2 - 2x=18$$;
If $$x = -7$$, then $$|x + 5| + |x - 3|= -2 - 2x=12$$;
If $$x = 4$$, then $$|x + 5| + |x - 3|= 2x + 2=10$$;
If $$x = 6$$, then $$|x + 5| + |x - 3|= 2x + 2=14$$.

But if x is in the second range ($$- 5 \leq x \leq 3$$), then the value of $$|x + 5| + |x - 3|$$ is independent of the value of x, and is ALWAYS equals to 8. For example:
If $$x = -5$$, then $$|x + 5| + |x - 3|= 8$$;
If $$x = 0$$, then $$|x + 5| + |x - 3|= 8$$;
If $$x = 3$$, then $$|x + 5| + |x - 3|= 8$$.

(1) $$x^2< 25$$:

Take the square root: $$|x| < 5$$;
Get rid of the absolute value sign: $$-5 < 0 < 5$$;
x can be in the second or third range. So, $$|x + 5| + |x - 3|$$ is either 8 or $$2x + 2$$. Not sufficient.

(2) $$x^2 > 9$$:

Take the square root: $$|x| > 3$$;
Get rid of the absolute value sign: $$x < -3$$ or $$x > 3$$;
x can be in any of the three ranges from above. So, $$|x + 5| + |x - 3|$$ is $$-2 - 2x$$, 8 or $$2x + 2$$. Not sufficient.

(1)+(2) We get $$-5 < x < -3$$ (second range) or $$3 < x < 5$$ (third range). If $$-5 < x < -3$$ (second range), then $$|x + 5| + |x - 3|=8$$ but if $$3 < x < 5$$ (third range), then $$|x + 5| + |x - 3|=2x + 2$$ (so the value will depend on the exact value of x). Not sufficient.

Bunuel

Can you please explain why you are checking if (x-3) is lesser than or greater than 9 and not 0 in each of the three ranges?
Math Expert
Joined: 02 Sep 2009
Posts: 93964
Own Kudos [?]: 634403 [1]
Given Kudos: 82420
Re: What is the value of |x + 5| + |x - 3| ? (1) x^2 < 25 (2) x^2 > 9 [#permalink]
1
Kudos
Dumsy_1711 wrote:
Bunuel wrote:
Complete step-by-step solution

What is the value of $$|x + 5| + |x - 3|$$ ?

The critical points (aka key points or transition points) are -5 and 3 (the values of x for which the expressions in the absolute values become 0).

Consider three ranges:

• If $$x < - 5$$, then $$x + 5 < 0$$ and $$x - 3 < 9$$, so $$|x + 5| = -(x + 5)$$ and $$|x - 3| = -(x - 3)$$. Thus in this range $$|x + 5| + |x - 3|$$ becomes $$-(x + 5) - (x - 3) = -2 - 2x$$.
• If $$- 5 \leq x \leq 3$$, then $$x + 5 \geq 0$$ and $$x - 3 \leq 9$$, so $$|x + 5| = x + 5$$ and $$|x - 3| = -(x - 3)$$. Thus in this range $$|x + 5| + |x - 3|$$ becomes $$x + 5 - (x - 3) = 8$$.
• If $$x > 3$$, then $$x + 5 > 0$$ and $$x - 3 > 9$$, so $$|x + 5| = x + 5$$ and $$|x - 3| = x - 3$$. Thus in this range $$|x + 5| + |x - 3|$$ becomes $$x + 5 + (x - 3) = 2x + 2$$.

The above mean that if x is in the first range ($$x < - 5$$) or in the third range ($$x > 3$$), then the value of $$|x + 5| + |x - 3|$$ depends on the value of x. For example:
If $$x = -10$$, then $$|x + 5| + |x - 3|= -2 - 2x=18$$;
If $$x = -7$$, then $$|x + 5| + |x - 3|= -2 - 2x=12$$;
If $$x = 4$$, then $$|x + 5| + |x - 3|= 2x + 2=10$$;
If $$x = 6$$, then $$|x + 5| + |x - 3|= 2x + 2=14$$.

But if x is in the second range ($$- 5 \leq x \leq 3$$), then the value of $$|x + 5| + |x - 3|$$ is independent of the value of x, and is ALWAYS equals to 8. For example:
If $$x = -5$$, then $$|x + 5| + |x - 3|= 8$$;
If $$x = 0$$, then $$|x + 5| + |x - 3|= 8$$;
If $$x = 3$$, then $$|x + 5| + |x - 3|= 8$$.

(1) $$x^2< 25$$:

Take the square root: $$|x| < 5$$;
Get rid of the absolute value sign: $$-5 < 0 < 5$$;
x can be in the second or third range. So, $$|x + 5| + |x - 3|$$ is either 8 or $$2x + 2$$. Not sufficient.

(2) $$x^2 > 9$$:

Take the square root: $$|x| > 3$$;
Get rid of the absolute value sign: $$x < -3$$ or $$x > 3$$;
x can be in any of the three ranges from above. So, $$|x + 5| + |x - 3|$$ is $$-2 - 2x$$, 8 or $$2x + 2$$. Not sufficient.

(1)+(2) We get $$-5 < x < -3$$ (second range) or $$3 < x < 5$$ (third range). If $$-5 < x < -3$$ (second range), then $$|x + 5| + |x - 3|=8$$ but if $$3 < x < 5$$ (third range), then $$|x + 5| + |x - 3|=2x + 2$$ (so the value will depend on the exact value of x). Not sufficient.