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# Which of the following inequalities is equal to |x-3|x||<8?

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Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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25 Aug 2017, 01:17
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Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0
[Reveal] Spoiler: OA

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Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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25 Aug 2017, 01:44
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0

I believe there is a TYPO in Q and Q means |x-3|*|x|<8

best way is working on the choices
substiute x as 0..
$$|0-3||0|<8....0<8$$.. True
so choice A and B, which exclude 0 can be eliminated
Now substitute x as 3
$$|3-3||3|=0<8$$... true
eliminate B and D as they exclude 3

ans C
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Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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25 Aug 2017, 01:52
chetan2u wrote:
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0

I believe there is a TYPO in Q and Q means |x-3|*|x|<8

best way is working on the choices
substiute x as 0..
$$|0-3||0|<8....0<8$$.. True
so choice A and B, which exclude 0 can be eliminated
Now substitute x as 3
$$|3-3||3|=0<8$$... true
eliminate B and D as they exclude 3

ans C

No, it's correct. $$|x-3|x||<8$$ is equivalent to $$-2<x<4$$.
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Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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25 Aug 2017, 03:11
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Bunuel can show the process that how can we get the range of x in an absolute inequality ?

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Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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25 Aug 2017, 03:32
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0

Case 1: if $$x >0$$ then $$|x| = x$$, hence the equation can be written as
$$|x-3x|<8$$, or $$|-2x|<8$$ or $$2x<8$$
therefore $$x<4$$-----(1)

Case 2: if $$x<0$$, then $$|x| = -x$$, hence the equation can be written as
$$|x -3(-x)|<8$$, or $$|x+3x|<8$$, or $$|4x|<8$$ or $$|x|<2$$
therefore $$-x<2$$ or $$x>-2$$----(2)

Case 3: if $$x = 0$$, then it will always satisfy the inequality

combining cases 1, 2 & 3 we get
$$-2<x<4$$

Option $$C$$

Last edited by niks18 on 25 Aug 2017, 03:38, edited 1 time in total.

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Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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25 Aug 2017, 03:34
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MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0

$$|x-3|x||<8 \iff -8 < x-3|x| < 8$$

Case 1: $$x \geq 0$$

We have $$-8 < x - 3x < 8 \implies -8 < -2x < 8 \implies 4 > x > -4$$
Thus $$0 \leq x < 4$$

Case 2: $$x < 0$$

We have $$-8 < x + 3x < 8 \implies -8 < 4x < 8 \implies -2 < x < 2$$
Thus $$-2 < x < 0$$

Combine both 2 cases, we have $$-2 < x < 4$$. Answer C
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Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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25 Aug 2017, 10:34
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0

Answered this question using plugin method. Plug -1 and 3, if both of the numbers satisfy the equation, then we can choose C.

Absolute in absolute makes me confused
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Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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25 Aug 2017, 10:39
niks18 wrote:
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0

Case 1: if $$x >0$$ then $$|x| = x$$, hence the equation can be written as
$$|x-3x|<8$$, or $$|-2x|<8$$ or $$2x<8$$
therefore $$x<4$$-----(1)

Case 2: if $$x<0$$, then $$|x| = -x$$, hence the equation can be written as
$$|x -3(-x)|<8$$, or $$|x+3x|<8$$, or $$|4x|<8$$ or $$|x|<2$$
therefore $$-x<2$$ or $$x>-2$$----(2)

Case 3: if $$x = 0$$, then it will always satisfy the inequality

combining cases 1, 2 & 3 we get
$$-2<x<4$$

Option $$C$$

Dear niks18 , can you please explain more the one that I highlighted?
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Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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25 Aug 2017, 10:49
2
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septwibowo wrote:
niks18 wrote:
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0

Case 1: if $$x >0$$ then $$|x| = x$$, hence the equation can be written as
$$|x-3x|<8$$, or $$|-2x|<8$$ or $$2x<8$$
therefore $$x<4$$-----(1)

Case 2: if $$x<0$$, then $$|x| = -x$$, hence the equation can be written as
$$|x -3(-x)|<8$$, or $$|x+3x|<8$$, or $$|4x|<8$$ or $$|x|<2$$
therefore $$-x<2$$ or $$x>-2$$----(2)

Case 3: if $$x = 0$$, then it will always satisfy the inequality

combining cases 1, 2 & 3 we get
$$-2<x<4$$

Option $$C$$

Dear niks18 , can you please explain more the one that I highlighted?

Hiseptwibowo
its a property of mod function which can be explained as follows -

let $$x = -3$$, then
$$|-3| = 3$$ this is same as $$-(-3)$$

hence if $$x<0$$ (i.e negative), then $$|x| = -x$$

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Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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25 Aug 2017, 10:57
niks18 wrote:
Hiseptwibowo
its a property of mod function which can be explained as follows -

let $$x = -3$$, then
$$|-3| = 3$$ this is same as $$-(-3)$$

hence if $$x<0$$ (i.e negative), then $$|x| = -x$$

Ah!! thanks niks18 , at first I thought that how can an absolute results a negative number. Now I got it, (-X) is not negative because X itself is negative.

Many thanks!
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Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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27 Aug 2017, 19:11
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Expert's post
=> -8 < x-3|x| <8

1) x >= 0
-8 < x-3x < 8
-8 < -2x < 8
-4 < x < 4
0 <= x < 4

2) x < 0
-8 < x-3|x| < 8
-8 < x+3x < 8
-8 < 4x < 8
-2 < x < 2
-2 < x < 0

Thus, -2 < x < 4
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Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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12 Sep 2017, 11:48
niks18 wrote:
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0

Case 1: if $$x >0$$ then $$|x| = x$$, hence the equation can be written as
$$|x-3x|<8$$, or $$|-2x|<8$$ or $$2x<8$$
therefore $$x<4$$-----(1)

Case 2: if $$x<0$$, then $$|x| = -x$$, hence the equation can be written as
$$|x -3(-x)|<8$$, or $$|x+3x|<8$$, or $$|4x|<8$$ or $$|x|<2$$
therefore $$-x<2$$ or $$x>-2$$----(2)

Case 3: if $$x = 0$$, then it will always satisfy the inequality

combining cases 1, 2 & 3 we get
$$-2<x<4$$

Option $$C$$

Hey
Have I interpreted it correctly?

1) If x>=0
|x|= x

2) If x<0

|x|=-x because -(-(-x))

3) If x<0 and the sign is also given, then

|-x|=x

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Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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12 Sep 2017, 12:24
Shiv2016 wrote:
niks18 wrote:
MathRevolution wrote:
Which of the following inequalities is equal to |x-3|x||<8?

A. 0<x<4
B. 0<x<2
C. -2<x<4
D. -2<x<2
E. -2<x<0

Case 1: if $$x >0$$ then $$|x| = x$$, hence the equation can be written as
$$|x-3x|<8$$, or $$|-2x|<8$$ or $$2x<8$$
therefore $$x<4$$-----(1)

Case 2: if $$x<0$$, then $$|x| = -x$$, hence the equation can be written as
$$|x -3(-x)|<8$$, or $$|x+3x|<8$$, or $$|4x|<8$$ or $$|x|<2$$
therefore $$-x<2$$ or $$x>-2$$----(2)

Case 3: if $$x = 0$$, then it will always satisfy the inequality

combining cases 1, 2 & 3 we get
$$-2<x<4$$

Option $$C$$

Hey
Have I interpreted it correctly?

1) If x>=0
|x|= x

2) If x<0

|x|=-x because -(-(-x))

3) If x<0 and the sign is also given, then

|-x|=x

Hi Shiv2016

The highlighted part is not correct.
As you have already mentioned that $$x<0$$ i.e $$x$$ is negative and we know that mod function is always positive
so as per your equation $$LHS=RHS$$ implies that $$positive = negative$$ which is incorrect.
assume that x = -2
so per statement 3: |-(-2)| = -2, or |2|=-2 Not possible

Also you have added one extra "-" in statement 2

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Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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12 Sep 2017, 12:38
So does that mean we have to always take |-x|= x when x<0 for the simple reason that || always gives positive value?

I am actually confused because it took me some time to understand absolute values and I started solving questions this way only.
I always took || as positive but in some questions, || gives negative value e.g. if x<0, then |x|= -x which is said to be positive in some solutions.
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Re: Which of the following inequalities is equal to |x-3|x||<8? [#permalink]

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12 Sep 2017, 12:49
Shiv2016 wrote:
So does that mean we have to always take |-x|= x when x<0 for the simple reason that || always gives positive value?

I am actually confused because it took me some time to understand absolute values and I started solving questions this way only.
I always took || as positive but in some questions, || gives negative value e.g. if x<0, then |x|= -x which is said to be positive in some solutions.

Hi Shiv2016

1. Mod is always positive
2. Negative sign inside mod function can be converted to positive i.e. $$|-x|$$ is same as $$|x|$$
3. In an Equality $$LHS = RHS$$

when we are saying that $$x<0$$, then $$|-x|$$ cannot be equal to $$x$$ but we can say that $$|-x|=-x$$
assume $$x=-2; |-(-2)|=-(-2)$$, or $$|2|=2$$, which is perfectly fine
Hence whenever $$x<0$$, we can write mod function as $$|x|=-x$$
for $$x>0; |x|=x$$

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Re: Which of the following inequalities is equal to |x-3|x||<8?   [#permalink] 12 Sep 2017, 12:49
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