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# In the equation |6x+9| = |3x+25|, what is the sum of all possible valu

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Math Expert
Joined: 02 Sep 2009
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In the equation |6x+9| = |3x+25|, what is the sum of all possible valu  [#permalink]

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04 May 2016, 23:57
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35% (medium)

Question Stats:

76% (02:30) correct 24% (02:30) wrong based on 211 sessions

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In the equation |6x+9| = |3x+25|, what is the sum of all possible values of x?

A. 4/9
B. 4/5
C. 14/9
D. 15/7
E. 17/3

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Posts: 8609
Re: In the equation |6x+9| = |3x+25|, what is the sum of all possible valu  [#permalink]

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05 May 2016, 02:43
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Bunuel wrote:
In the equation |6x+9| = |3x+25|, what is the sum of all possible values of x?

A. 4/9
B. 4/5
C. 14/9
D. 15/7
E. 17/3

since it is MOD on both sides we can square both sides..
thereafter we get the two terms on ONE side and we will have a QUADRATIC equation..
In a Quadratic equation, the value of sum of roots /x is$$-\frac{(coeff of x}{coeff of x^2)}$$..
so lets concentrate ONLY on x^2 and x values..

$$|6x+9| = |3x+25|$$..
$$(6x+9)^2 = (3x+25)^2$$..
$$36x^2 +108 x+ 9^2 = 9x^2+150x +25^2$$..
$$(36-9)x^2 + (108-150)x +9^2+25^2$$..
sum of the roots =$$\frac{-b}{a} = -(\frac{-42}{27}) = \frac{14}{9}$$
C
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Re: In the equation |6x+9| = |3x+25|, what is the sum of all possible valu  [#permalink]

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05 May 2016, 05:32
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Hello...
I followed the Absolute value equation Strategies... got answer c - 14/9

Following is the explanation:

We need to test four cases overall: positive/positive, positive/negative, negative/positive, and negative/negative.
(1) The positive/positive case: (6x+9) = (3x+25)
(2) The positive/negative case: (6x+9) = -(3x+25)
(3) The negative/positive case: —(6x+9) = (3x+25)
(4) The negative/negative case: —(6x+9) = -(3x+25)

case (1) and case (4) yield the same equation. Likewise, case (2) and case (3) yield the same equation. Thus only need to consider two real cases: one in which neither expression changes sign, and another in which one expression changes sign.

CASE A: Same sign
6x+9 = 3x+25
6x-3x = 25-9
3x = 16
x=16/3

CASE B: Different Sign
-(6x+9) = (3x+25)
-6x-9 = 3x+25
-6x-3x=25+9
-9x=34
x=-34/9

Sum of all possible values of x = 16/3+(-34/9)

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Re: In the equation |6x+9| = |3x+25|, what is the sum of all possible valu  [#permalink]

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05 May 2016, 02:38
x<-25/3
- 6x - 9 = -3x - 25
=> 3x = 16
=> x = 16/3

-25/3 < x < -9/6
6x + 9 = -(3x+25)
=> 6x+9 = -3x - 25
=> 9x = -34
=> x = -34/9

x> -9/6
6x+9 = 3x+25
=> 3x = 16
=> x = 16/3

Sum of all possible values of a = 16/3 + (-34/9)
=48/9 - 34/9
=14/9

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Joined: 10 Dec 2019
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Re: In the equation |6x+9| = |3x+25|, what is the sum of all possible valu  [#permalink]

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06 Feb 2020, 05:11
Skywalker18 wrote:
x<-25/3
- 6x - 9 = -3x - 25
=> 3x = 16
=> x = 16/3

-25/3 < x < -9/6
6x + 9 = -(3x+25)
=> 6x+9 = -3x - 25
=> 9x = -34
=> x = -34/9

x> -9/6
6x+9 = 3x+25
=> 3x = 16
=> x = 16/3

Sum of all possible values of a = 16/3 + (-34/9)
=48/9 - 34/9
=14/9

Hi

I added all 3 values and ended up getting 16/3 + 16/3 - 16/9
which resulted in to 80/9.

Probably it could be silly mistake from my end but can you please help me out here?
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Joined: 22 Jun 2019
Posts: 7
Re: In the equation |6x+9| = |3x+25|, what is the sum of all possible valu  [#permalink]

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21 Mar 2020, 02:42
1
Anurag06 wrote:
Skywalker18 wrote:
x<-25/3
- 6x - 9 = -3x - 25
=> 3x = 16
=> x = 16/3

-25/3 < x < -9/6
6x + 9 = -(3x+25)
=> 6x+9 = -3x - 25
=> 9x = -34
=> x = -34/9

x> -9/6
6x+9 = 3x+25
=> 3x = 16
=> x = 16/3

Sum of all possible values of a = 16/3 + (-34/9)
=48/9 - 34/9
=14/9

Hi

I added all 3 values and ended up getting 16/3 + 16/3 - 16/9
which resulted in to 80/9.

Probably it could be silly mistake from my end but can you please help me out here?

Anurag06
When x<-25/3, x = 16/3 is not possible.
So, you should add 2 values(16/3 and -34/9) only
Re: In the equation |6x+9| = |3x+25|, what is the sum of all possible valu   [#permalink] 21 Mar 2020, 02:42