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# Is |x| < 1? (1) |x + 2| = 3 |x − 1| (2) |2x − 5| ≠ 0

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Is |x| < 1? (1) |x + 2| = 3 |x − 1| (2) |2x − 5| ≠ 0  [#permalink]

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09 Oct 2018, 03:15
00:00

Difficulty:

75% (hard)

Question Stats:

56% (02:31) correct 44% (02:17) wrong based on 76 sessions

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Is |x| < 1?

(1) |x + 2| = 3|x − 1|
(2) |2x − 5| ≠ 0

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Re: Is |x| < 1? (1) |x + 2| = 3 |x − 1| (2) |2x − 5| ≠ 0  [#permalink]

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09 Oct 2018, 03:50
Bunuel wrote:
Is |x| < 1?

(1) |x + 2| = 3|x − 1|
(2) |2x − 5| ≠ 0

Question: Is |x| < 1?

Question REPHRASED: Is -1 < x < 1?

Statement 1: |x + 2| = 3|x − 1|

Case 1: x+2 = 3(x-1)
i.e. x = 5/2 = 2.5 i.e. Outside the asked range

Case 2: -(x+2) = 3(x-1)
i.e. x = 1/4 i.e. In the asked range

P.S. There may be two more cases but they will be replica of the two cases mentioned here

NOT SUFFICIENT

Statement 2: |2x − 5| ≠ 0

i.e. 2x - 5 ≠ 0
i.e. x ≠ 5/2
i.e. x ≠ 2.5

NOT SUFFICIENT

Combining the two statements

Now we can eliminate x = 2.5 due to second statement so the only possible value of x that we are left with is x = 1/4 hence
SUFFICIENT

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Re: Is |x| < 1? (1) |x + 2| = 3 |x − 1| (2) |2x − 5| ≠ 0  [#permalink]

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09 Oct 2018, 04:00
Bunuel wrote:
Is |x| < 1?

(1) |x + 2| = 3|x − 1|
(2) |2x − 5| ≠ 0

(1) one way to solve this equation is if both the expressions in absolute values are positive - that is, as x+2=3(x-1). Simplifying, we get x+2=3x-3 ==> 5=2x ==> x=2.5. Therefore, |x| = 2.5 >1 - the answer to the question is no!
Now let's try solving if one of the expression in absolute values is negative: -x-2=3(x-1)==> -x-2=3x-3 ==> 1=4x ==> x = 0.25. Therefore,Therefore, |x| = 0.25 <1 - the answer to the question is yes! two contradicting answer > insufficient data! eliminate A & D

(2) Without even simplifying, it is clear that just eliminating a single value for x won't tell us if it larger or smaller than 1 - insufficient! eliminate B

Combined: now let's look at the equation in (2): whether we treat the expression as positive (2x-5) or negative (5-2x), we get the same thing: 2x ≠ 5 ==> x ≠ 2.5. Combined with the data from (1) - which told us x equals either 2.5 or 0.25 - this eliminates one option (2.5), and leaves only the other (0.25). x has a definite value > the expression will have one solution only! sufficient! the answer is C.
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Re: Is |x| < 1? (1) |x + 2| = 3 |x − 1| (2) |2x − 5| ≠ 0  [#permalink]

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09 Oct 2018, 04:24
Bunuel wrote:
Is |x| < 1?

(1) |x + 2| = 3|x − 1|
(2) |2x − 5| ≠ 0

Rewriting question is -1<x<1

Statement 1

Squaring both side

X^2+4x+4=9x^2-18x+9
Solving gives two value of x
(2x-5)(4x-1)=0
5/2 and 1/4
Not sufficient

Statement 2
2x-5 not equal to zero.
Not sufficient

Combine

From st1 we know (2x-5)(4x-1)=0
From 2 2x-5 not= to zero
So we know 4x-1=0
Which gives one value of x hence sufficient

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Re: Is |x| < 1? (1) |x + 2| = 3 |x − 1| (2) |2x − 5| ≠ 0  [#permalink]

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09 Oct 2018, 06:26
Bunuel wrote:
Is |x| < 1?

(1) |x + 2| = 3|x − 1|
(2) |2x − 5| ≠ 0

1)

$$x + 2 = 3x - 3$$

$$-2x = -5$$

$$x = \frac{5}{2}$$

or $$-x -2 = 3x - 3$$

$$-4x = -1$$

$$x = \frac{1}{4}$$

$$\frac{5}{2}$$ < 1 no

$$\frac{1}{4}$$ < 1 yes

Insufficient.

2)

if x = 0 then the absolute value would be 5 ≠ 0

0 < 1 yes

if x = 3 then the absolute value would be 1 ≠ 0

3 < 1 no

insufficient.

Combine both

we know $$\frac{5}{2}$$ is not a possible value from statement 2)

hence the value is $$\frac{1}{4}$$

$$\frac{1}{4}$$ < 1 sufficient.

Re: Is |x| < 1? (1) |x + 2| = 3 |x − 1| (2) |2x − 5| ≠ 0   [#permalink] 09 Oct 2018, 06:26
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