Which of the following inequalities is equivalent to |2x - |x||

Question

Which of the following inequalities is equivalent to  |2x-|x||<3 ?
 
A.  0<x<2

B.  0<x<3

C.  -1<x<3

D.  0<x<1

E.  -3<x<1

singh.00 · Answer

|2x−|x||<3

It can be written as 
1. 2x−|x|<3 
2. -(2x-|x|)<3 or -2x+|x|<3

In 1, if x is positive: x<3 and if x is negative: -3x<3 or x>-1. Hence, −1<x<3.
In 2, if x is positive: x<3 and if x is negative: 3x<3 or x<3. Hence, x<3.

In both cases, −1<x<3 is common.

IMO, C

scartedduf01 · Answer

Could you please explain further?

Sroy1 · Answer

Consideering value of 'X' +ve or -ve two solution set will be there. First lets consider X>0, Then |2X-|X||<3 |2X-X|<3 |X|<3 X<3 (as X is+ve) Thus first solution 0-1 Thus second solution becomes -1

Ophelia__ · Answer

Asked: Which of the following inequalities is equal to |2x-|x||<3?

|2x-|x||<3
-3 <2x - |x| < 3

Case 1: x>=0
-3 < 2x - x < 3
-3 < x < 3
Since x>0
0 <= x < 3

Case 2: x<0
-3 < 2x + x < 3
-3 < 3x < 3
-1 < x < 1
Since x<0
-1 < x < 0

Combining above 2:
-1 < x < 3

Oppenheimer1945 · Answer

Making graph of  y1=-|x|  and  y2=2x  , then adding both gives  f(x)=y1+y2

Check interval when  f(x)<3  gives  -1<x<3

SHSAHOO · Answer

chetan2u  , I have gone through you post on Absolute modulus. However, I am finding difficulty to how to approach this question. Could you please explain in your way or which would be the best way to solve this problem?

chetan2u · Answer

SHSAHOO

Here, I will not use any method but make use of options. 
I’ll choose 2 first, so |2*2-|2||=2<3…true. Discard A, D and E
Next choose 0, so 0<3…true. Discard B
Answer: C

Cuttlefish · Answer

Can someone please explain why we get to merge the two solutions?

Posted from my mobile device

Bunuel · Answer

When solving, we get two ranges for which the given inequality holds: -1 < x < 0 and 0 ≤ x < 3. So, -1 < x < 0  or  0 ≤ x < 3. However, notice that these two solution sets represent a  single continuous  set -1 < x < 3. So, it's more convenient to write it this way. Also, this is how the correct answer is represented.

Hope it's clear.

Cuttlefish · Answer

Apologies, I thought I understood but the reality is I still have some lingering questions. How come 0 gets to be eliminated? If we combine the two solution ranges, then it's - 1 < x < 0>= x <3
I just don't get how x can be both non-negative and negative at the same time.

Bunuel · Answer

-1 < x < 0 implies:

--------(-1) ---- (0)------------(3)

0 ≤ x < 3 implies:

--------(-1)---- (0)------------ (3)

Now, combine the two ranges to get -1 < x < 3:

--------(-1) ----(0)------------ (3)

Hope it's clear.

napolean92728 · Answer

We need to solve the inequality:
 ∣2x−∣x∣∣<3

Step 1: Consider Absolute Value Definition 
We define cases based on xxx:
 
  Case 1:  x≥0  
 Since  ∣x∣=x , the expression simplifies to:  ∣2x−x∣=∣x∣<3 
which results in:  −3<x<3  
  Case 2:  x<0  
 Since  ∣x∣=−x , the expression simplifies to:
 ∣2x−(−x)∣=∣3x∣<3 
which results in:  −1<x<1

Step 2: Combine the Two Cases 
From the first case:  −3<x<3 
From the second case:  −1<x<1 
The valid range satisfying both cases is:  −1<x<3

Step 3: Identify the Correct Answer 
Looking at the answer choices, the correct answer is: C. −1<x<3

Which of the following inequalities is equivalent to |2x - |x|| < 3?

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