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If y = ||x – 3| – 2|, for how many values of x is y = 4?

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Director
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If y = ||x – 3| – 2|, for how many values of x is y = 4?  [#permalink]

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New post Updated on: 24 Mar 2015, 02:19
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Difficulty:

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Question Stats:

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If y = ||x – 3| – 2|, for how many values of x is y = 4?

A) 1
B) 2
C) 3
D) 4
E) 5

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Thanks,
Lucky

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Originally posted by Lucky2783 on 23 Mar 2015, 22:35.
Last edited by Bunuel on 24 Mar 2015, 02:19, edited 1 time in total.
RENAMED THE TOPIC.
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Re: If y = ||x – 3| – 2|, for how many values of x is y = 4?  [#permalink]

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New post 24 Mar 2015, 01:03
1
\(y=||x-3|-2|\)

We can distinguish the two cases:
a) \(x-3\ge 0 \Rightarrow x\ge 3\)
b) \(x-3<0 \Rightarrow x<3\)

case a): \(x\ge 3\)

\(y=||x-3|-2|\) becomes \(y=|x-3-2|=|x-5|\)
subcase i) \(x \ge 5\)
\(y=x-5 \Rightarrow 4=x-5 \Rightarrow x=9\)
subcase ii) \(3\le x<5\)
\(y=5-x \Rightarrow 4=5-x \Rightarrow x=1\) ELIMINATE

case b): \(x<3\)

\(y=||x-3|-2|\) becomes \(y=|3-x-2|=|-x+1|\)
subcase iii) \(x<1\)
\(y=-x+1 \Rightarrow 4=-x+1 \Rightarrow x=-3\)
subcase iv) \(1 \le x<3\)
\(y=x-1 \Rightarrow 4=x-1 \Rightarrow x=5\) ELIMINATE.

There are only two solutions. B
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Re: If y = ||x – 3| – 2|, for how many values of x is y = 4?  [#permalink]

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New post 09 Apr 2015, 06:56
Lucky2783 wrote:
If y = ||x – 3| – 2|, for how many values of x is y = 4?

A) 1
B) 2
C) 3
D) 4
E) 5



easy solution


y = ||x – 3| – 2| can be 4 only and only when X-3= +/-6. so there are 2 values of X . answer B.
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Re: If y = ||x – 3| – 2|, for how many values of x is y = 4?  [#permalink]

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New post 21 Apr 2015, 20:56
Lucky2783 wrote:
If y = ||x – 3| – 2|, for how many values of x is y = 4?

A) 1
B) 2
C) 3
D) 4
E) 5



\(||x – 3| – 2|=Y\)..........Eqn (1)

Substitute Y=4 and solve

\(||x – 3| – 2|=4\)

Case I :\(|x – 3| – 2 = 4\)........................................OR.......................................Case II :\(|x – 3| – 2 = -4\)
\((x – 3)=6\)....OR....\((x – 3)=-6\).........................OR....................................... \((x – 3)=-2\)....OR....\((x – 3)=2\)

Substituting value of (x-3) in eqn (1),

only (x-3)=6, -6 gives the value of Y=4


Answer is B
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Re: If y = ||x – 3| – 2|, for how many values of x is y = 4?  [#permalink]

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New post 21 Mar 2016, 23:00
The answer is B : 2 values that is 9 and -3
I got there by solving the equation, by putting the value of y = 4.

The problem is I took around 2:36 min. anyway I could have done that in less time (assuming i am a bit quick in arithmetic maths) or the time taken is sufficient for a 700+ level question.

TIA
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Re: If y = ||x – 3| – 2|, for how many values of x is y = 4?  [#permalink]

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New post 02 May 2018, 19:56
||x-3|-2|=4

|x-3|-2=4

has two solutions: x=9, x=-3

|x-3|-2=-4 means |x-3|=-2, but absolute value cannot be negative, NO SOLUTONS

B fits
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Re: If y = ||x – 3| – 2|, for how many values of x is y = 4?  [#permalink]

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New post 09 Aug 2018, 03:13
Lucky2783 wrote:
If y = ||x – 3| – 2|, for how many values of x is y = 4?

A) 1
B) 2
C) 3
D) 4
E) 5


Let |x – 3| = a

The original equation becomes
|a - 2| = 4
So, a = 6, -2

But a can't be negative, since a is |x-3| and modulus of a number can't be negative.

Substituting, |x-3| = 6, we get 2 values of x.

(B)
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Re: If y = ||x – 3| – 2|, for how many values of x is y = 4?  [#permalink]

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New post 11 Aug 2018, 20:12
1
Take |x-3|=k
So,
|k-2|=4.
So, k=6,-2
Now, |x-3|= -2 which is not possible as modulus is always positive.
|x-3| = 6. So, x=9 and 3.

So, there are 2 solutions.
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Re: If y = ||x – 3| – 2|, for how many values of x is y = 4? &nbs [#permalink] 11 Aug 2018, 20:12
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