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

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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 15 Nov 2019, 01:54
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A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

60% (01:35) correct 40% (02:01) wrong based on 113 sessions

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Math Expert
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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 15 Nov 2019, 06:39
1
Bunuel 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


Are You Up For the Challenge: 700 Level Questions


Don't get into finding values of x through Mods, just think logically..
\(4 = ||x – 3| – 2|\)......For this |x-3|=6 or -2, but |x-3| cannot be negative so |x-3|=6..

|x-3|=6
x-3=6...x=9
3-x=6....x=-3
So, TWO values -( 9 and -3)

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 15 Nov 2019, 10:16
2
1

Solution



Given
    • y=||x–3|–2|

To find
    • The number of values x fo4 y=4

Approach and Working out
    • y=||x–3|–2| and y=4
    • 4=||x–3|–2|

Case 1-) |x–3|–2 = 4
    • |x–3| = 6
    • x-3 = +6 and -6
    • x = 9 and -3

Case 2-) |x–3|–2 = -4
    • |x-3|= -2 which is not possible.

So, only 2 values of x are possible.

Thus, option B is the correct answer.
Correct Answer: Option 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 16 Nov 2019, 09:17
1
Bunuel 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


y = ||x – 3| – 2|
—> 4 = ||x – 3| – 2|
—> lx – 3| – 2 = ± 4
—> lx - 3l - 2 = -4 or lx - 3l - 2 = 4
—> lx - 3l = -2 or lx - 3l = 6
—> lx - 3l = -2 is NOT possible as mod can never be negative
—> lx - 3l = 6
—> x - 3 = -6 or x - 3 = 6
—> x = -3 or 9
—> 2 different values

IMO Option 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 19 Nov 2019, 04:11
Bunuel 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|=4
|x-3|≥0…x≥3…(pos):|x-3-2|=4…|x-5|=4
|x-5|≥0…x≥5…(pos):(x-5)=4…x=9…(x≥5)…valid
|x-5|<0…x<5…(neg):(x-5)=-4…x=1…(x≥3)…invalid
|x-3|<0…x<3…(neg):|-(x-3)-2|=4…|-x+1|=4
|-x+1|≥0…-x≥-1…x≤1…(pos):(-x+1)=4…-x=3…x=-3…(x≤1)…valid
|-x+1|<0…-x<-1…x>1…(neg):(-x+1)=-4…-x=-5…x=5…(1<x<3)…invalid
x={9,-3}=2

Ans (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 22 Nov 2019, 07:10
Bunuel 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


Are You Up For the Challenge: 700 Level Questions

Let lx-3l=X
y=lX-2l
if y=4
lX-2l=4
X-2=4 or X-2=-4
X=6 or X=-2( Not possible since X=lx-3l
X=6
lx-3l=6
x-3=6 or x-3=-6
x=9 or x=-3
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 22 Nov 2019, 07:16
If \(y = ||x – 3| – 2|\), for how many values of x is \(y = 4\)?

A. 1
B. 2--> correct
C. 3
D. 4
E. 5

Solution:
y = ||x – 3| – 2| =4
=> |x – 3| – 2 = +4 or -4
=> |x-3| = 6 or -2
|x-3| >=0, so |x-3| != -2 and |x-3| = 6
=> x-2 = +6 or -6 --> x has two values--> answer: B
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Re: If y = ||x – 3| – 2|, for how many values of x is y = 4?   [#permalink] 22 Nov 2019, 07:16
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