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# |x+3| - |4-x| = |8+x|. How many solutions does the equation

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Joined: 12 Dec 2016
Posts: 1107

Kudos [?]: 20 [1], given: 1056

Location: United States
GMAT 1: 700 Q49 V33
GPA: 3.64
Re: |x+3| - |4-x| = |8+x|. How many solutions does the equation [#permalink]

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16 Apr 2017, 18:46
1
KUDOS
this problem has a very simple solution, give me your kudos after reading it, thanks

so, first move the -[4-x] to the right,
2nd step, square both sides of the equation
3rd, take all variables to one side, simplify the result, (right now, one side should have 0)
4th, you can see one side always is greater than 0, then the equation cannot hold, then there is no value of x

pls, give me kudos, thanks

Kudos [?]: 20 [1], given: 1056

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Posts: 194

Kudos [?]: 17 [0], given: 447

|x+3| - |4-x| = |8+x|. How many solutions does the equation [#permalink]

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26 Jul 2017, 09:58
VeritasPrepKarishma wrote:
guerrero25 wrote:
I am trying to understand the Modules questions - I took this from GMAT club's quant book .

Q. |x+3| - |4-x| = |8+x|. How many solutions does the equation have?

I could not follow why the equal signs are considered ? e.g -8 <= x < -3 , -3 <= x < 4, x >=4 ..Appreciate if someone can explain the logic?

Solution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:

a) x < -8. -(x+3) - (4-x) = -(8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not less than -8)

b) -8 <= x < -3. -(x+3) - (4-x) = (8+x) --> x = -15. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)

c) -3 <= x < 4 (x+3) - (4-x) = (8+x) --> x = 9. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)

d) x >=4. (x+3) + (4-x) = (8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

thanks !

|x| = x when x >= 0 (x is either positive or 0)
|x| = -x when x < 0 (note here that you can put the equal to sign here as well x <= 0 because if x = 0,
|0| = 0 = -0 (all are the same)
So the '=' sign can be put with x > 0 or with x < 0. We usually put it with 'x > 0' for consistency.

When we are considering ranges, say,
x < -8 ------ x is less than -8
-8 <= x < -3 ------- x is greater than or equal to -8 but less than -3
-3 <= x < 4 ------- x is greater than or equal to -3 but less than 4
x >=4 -------- x is greater than or equal to 4

We need to include the transition points (-8, -3, 4) somewhere so we include them with greater than sign.

Mind you, we could have taken the ranges as
x <= -8
-8 < x <= -3
-3 < x <= 4
x > 4

The only point is that we don't include the transition points twice.

Hope the role of '=' sign is clear.

hi mam

in "a" when assumed that x is less than -8, the RHS modulus (8+x) is correctly negated. I am okay with this operation. Side by side, however, the LHS modulus (x + 3) is also negated...

what is the main logic underlying this negation ..? is this such that to make the LHS negative, some number should be subtracted from some bigger negatives..? if so, how can we conclude that (x+3) is bigger in value than is (4-x) ..?

Kudos [?]: 17 [0], given: 447

Manager
Status: love the club...
Joined: 24 Mar 2015
Posts: 194

Kudos [?]: 17 [0], given: 447

Re: |x+3| - |4-x| = |8+x|. How many solutions does the equation [#permalink]

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26 Jul 2017, 10:33
ssislam wrote:
VeritasPrepKarishma wrote:
guerrero25 wrote:
I am trying to understand the Modules questions - I took this from GMAT club's quant book .

Q. |x+3| - |4-x| = |8+x|. How many solutions does the equation have?

I could not follow why the equal signs are considered ? e.g -8 <= x < -3 , -3 <= x < 4, x >=4 ..Appreciate if someone can explain the logic?

Solution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:

a) x < -8. -(x+3) - (4-x) = -(8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not less than -8)

b) -8 <= x < -3. -(x+3) - (4-x) = (8+x) --> x = -15. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)

c) -3 <= x < 4 (x+3) - (4-x) = (8+x) --> x = 9. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)

d) x >=4. (x+3) + (4-x) = (8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

thanks !

|x| = x when x >= 0 (x is either positive or 0)
|x| = -x when x < 0 (note here that you can put the equal to sign here as well x <= 0 because if x = 0,
|0| = 0 = -0 (all are the same)
So the '=' sign can be put with x > 0 or with x < 0. We usually put it with 'x > 0' for consistency.

When we are considering ranges, say,
x < -8 ------ x is less than -8
-8 <= x < -3 ------- x is greater than or equal to -8 but less than -3
-3 <= x < 4 ------- x is greater than or equal to -3 but less than 4
x >=4 -------- x is greater than or equal to 4

We need to include the transition points (-8, -3, 4) somewhere so we include them with greater than sign.

Mind you, we could have taken the ranges as
x <= -8
-8 < x <= -3
-3 < x <= 4
x > 4

The only point is that we don't include the transition points twice.

Hope the role of '=' sign is clear.

hi mam

in "a" when assumed that x is less than -8, the RHS modulus (8+x) is correctly negated. I am okay with this operation. Side by side, however, the LHS modulus (x + 3) is also negated...

what is the main logic underlying this negation ..? is this such that to make the LHS negative, some number should be subtracted from some bigger negatives..? if so, how can we conclude that (x+3) is bigger in value than is (4-x) ..?

wow..i think i have got it...

when x is less than -8, (x+3) got negative and hence this is negative ... hooray..

Kudos [?]: 17 [0], given: 447

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Joined: 16 Oct 2010
Posts: 7738

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Re: |x+3| - |4-x| = |8+x|. How many solutions does the equation [#permalink]

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26 Jul 2017, 20:54
2
KUDOS
Expert's post
ssislam wrote:
VeritasPrepKarishma wrote:
guerrero25 wrote:
I am trying to understand the Modules questions - I took this from GMAT club's quant book .

Q. |x+3| - |4-x| = |8+x|. How many solutions does the equation have?

I could not follow why the equal signs are considered ? e.g -8 <= x < -3 , -3 <= x < 4, x >=4 ..Appreciate if someone can explain the logic?

Solution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:

a) x < -8. -(x+3) - (4-x) = -(8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not less than -8)

b) -8 <= x < -3. -(x+3) - (4-x) = (8+x) --> x = -15. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)

c) -3 <= x < 4 (x+3) - (4-x) = (8+x) --> x = 9. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)

d) x >=4. (x+3) + (4-x) = (8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

thanks !

|x| = x when x >= 0 (x is either positive or 0)
|x| = -x when x < 0 (note here that you can put the equal to sign here as well x <= 0 because if x = 0,
|0| = 0 = -0 (all are the same)
So the '=' sign can be put with x > 0 or with x < 0. We usually put it with 'x > 0' for consistency.

When we are considering ranges, say,
x < -8 ------ x is less than -8
-8 <= x < -3 ------- x is greater than or equal to -8 but less than -3
-3 <= x < 4 ------- x is greater than or equal to -3 but less than 4
x >=4 -------- x is greater than or equal to 4

We need to include the transition points (-8, -3, 4) somewhere so we include them with greater than sign.

Mind you, we could have taken the ranges as
x <= -8
-8 < x <= -3
-3 < x <= 4
x > 4

The only point is that we don't include the transition points twice.

Hope the role of '=' sign is clear.

hi mam

in "a" when assumed that x is less than -8, the RHS modulus (8+x) is correctly negated. I am okay with this operation. Side by side, however, the LHS modulus (x + 3) is also negated...

what is the main logic underlying this negation ..? is this such that to make the LHS negative, some number should be subtracted from some bigger negatives..? if so, how can we conclude that (x+3) is bigger in value than is (4-x) ..?

Consider this:
If x < -8 (i.e. x can be -9, -10, -11, -11.4, -20 and so on...),
what can you say about the sign of (x + 3) ?
Can we say that when x is less than -3, (x + 3) is negative? Sure. So if we know that x is less than -8, then obviously, it is less than -3 too. So in this case, (x+3) will certainly be negative.
Hence |x + 3| will also translate into -(x+3).

To see it easily, drawing a number line helps.

......................... (-8) ...................... (-3) ..................................................... (4) ...................................

For (x + 3), all values of x to the left of -3 will give |x + 3| = -(x + 3). So if x < -8, it is to the left of -3 and hence will lead to |x+3| = -(x + 3).
To the right of -3, all values of x will give |x + 3| = (x + 3).

For more on this, check: https://www.veritasprep.com/blog/2014/0 ... -the-gmat/
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|x+3| - |4-x| = |8+x|. How many solutions does the equation [#permalink]

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29 Jul 2017, 02:34
@VeritasPrepKarishma/ @Bunnel

..While I understand this approach perfectly, i take around 3.5 minutes to solve it this way every time. This has me worried. Am I overlooking some concept that is causing me to take so much time?

Bunuel wrote:
guerrero25 wrote:
|x+3| – |4-x| = |8+x| How many solutions will this equation have?

A. 0
B. 1
C. 2
D. 3
E. 4

[Reveal] Spoiler:
I am trying to understand the Modules questions - I took this from GMAT club's quant book .

Q. |x+3| - |4-x| = |8+x|. How many solutions does the equation have?

I could not follow why the equal signs are considered ? e.g -8 <= x < -3 , -3 <= x < 4, x >=4 ..Appreciate if someone can explain the logic?

Solution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:

a) x < -8. -(x+3) - (4-x) = -(8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not less than -8)

b) -8 <= x < -3. -(x+3) - (4-x) = (8+x) --> x = -15. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)

c) -3 <= x < 4 (x+3) - (4-x) = (8+x) --> x = 9. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)

d) x >=4. (x+3) + (4-x) = (8+x) --> x = -1. We reject the solution because our condition is not satisfied (-1 is not more than 4)

thanks !

Responding to pm.

Absolute value properties:

When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$.

STEP BY STEP SOLUTION:

We have three transition points for $$|x+3| - |4-x| = |8+x|$$: -8, -3, and 4 (transition point is the value of x for which an expression in the modulus equals to zero). Thus we have four ranges to check:

1. $$x<-8$$;
2. $$-8\leq{x}\leq{-3}$$;
3. $$-3<x<4$$
4. $$x\geq{4}$$

Note that it does not matter in which range(s) you include the transition points with "=" sign as long you include them.

1. When $$x<-8$$, then $$x+3$$ is negative, $$4-x$$ is positive and $$8+x$$ is negative. Thus $$|x+3|=-(x+3)$$, $$|4-x|=4-x$$ and $$|8+x|=-(8+x)$$.

Therefore for this range $$|x+3| - |4-x| = |8+x|$$: transforms to $$-(x+3) - (4-x) =-(8+x)$$: --> $$x=-1$$. This solution is NOT OK, since $$x=-1$$ is NOT in the range we consider ($$x<-8$$).

2. When $$-8\leq{x}\leq{-3}$$, then $$x+3$$ is negative, $$4-x$$ is positive and $$8+x$$ is positive. Thus $$|x+3|=-(x+3)$$, $$|4-x|=4-x$$ and $$|8+x|=8+x$$.

Therefore for this range $$|x+3| - |4-x| = |8+x|$$: transforms to $$-(x+3) - (4-x) =8+x$$: --> $$x=-15$$. This solution is NOT OK, since $$x=-15$$ is NOT in the range we consider ($$-8\leq{x}\leq{-3}$$).

3. When $$-3<x<4$$, then $$x+3$$ is positive, $$4-x$$ is positive and $$8+x$$ is positive. Thus $$|x+3|=x+3$$, $$|4-x|=4-x$$ and $$|8+x|=8+x$$.

Therefore for this range $$|x+3| - |4-x| = |8+x|$$: transforms to $$x+3 - (4-x) =8+x$$: --> $$x=9$$. This solution is NOT OK, since $$x=9$$ is NOT in the range we consider ($$-3<x<4$$).

4. When $$x\geq{4}$$, then $$x+3$$ is positive, $$4-x$$ is negative and $$8+x$$ is positive. Thus $$|x+3|=x+3$$, $$|4-x|=-(4-x)=x-4$$ and $$|8+x|=8+x$$.

Therefore for this range $$|x+3| - |4-x| = |8+x|$$: transforms to $$x+3 - (x-4) =8+x$$: --> $$x=-1$$. This solution is NOT OK, since $$x=-1$$ is NOT in the range we consider ($$x\geq{4}$$).

Thus no value of x satisfies $$|x+3| - |4-x| = |8+x|$$.

Hope it's clear.

Kudos [?]: [0], given: 1

|x+3| - |4-x| = |8+x|. How many solutions does the equation   [#permalink] 29 Jul 2017, 02:34

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