Is |x-2|+|x+2|

Question

Is |x-2|+|x+2|<4?
1). x<0
2). x is within (-2, 2)

Please explain solve and explain ur answers.

I started of writing 4 different eqns considering each modulus value + ve or - ve and ended up getting 4<4, -4<4 ,2x<4 and -2x<4.4 <4 is not true.Therfore E?

Please explain

Peer

vishy007 · Answer

I am getting B for this.

I tried with values between -2 & 2 always leading to 4 on LHS.

By the way how are you getting -4 on LHS?

FN · Answer

what does witihin mean?

if x=-2..then the stem=4..

if x=-1.7 then the stem<4

I think E is best

walker · Answer

|-1.7-2|+|-1.7+2|=|-3.7|+|0.3|=3.7+0.3=4

rpmodi · Answer

I am voting for D .

1.. x<0

x=-1 : |x-2|+|x+2| = 4
 x=-2 : |x-2|+|x+2| =4
 x=-3 : |x-2|+|x+2| > 4 
.....
.......
............

for any -ve value it will be >=4 .

Other posts have given the reason why statement 2 is sufficient .............

FN · Answer

grrr...yes you are correct..how can i get rid of these silly mistakes??? looks like i will never score over 50 on Q...

Yeah i get B for this..

greenoak · Answer

Hi, Peer!
Other posters have already answered you. However, I think that the answer should be D.

You needn’t to consider all four possibilities for modulus in this case. In fact, each of the statements gives you data needed to decide which possibilities you should consider.

1) x<0 => (x-2)<0, so this left us only with two possibilities:
 -2<=x<0 : -x+2+x+2 = 4 so we can answer the question
 x<-2: -x+2 – x – 2 =-2*x ...

Here I almost fell into the trap: I overlooked that -2*x for x<-2 gives us value always greater than 4! So, we in fact can answer the question despite the fact we don't know the value of the expression on the left side. We know enough to say that it _won't be_ less than 4.

So 1) is sufficient.

2) that x is within (-2,2) gives us the single possibility for modulus: first one is (-x+2) and the second one (x+2). So, we have:
 
-x+2+x+2 = 4 and we can answer the question.

So 2) is sufficient and it’s D.

****
Edited: Now I began to doubt it myself... Maybe I missed something here?

walker · Answer

My approach to this problem.

|x-2|+|x+2|<4

1. Find key points, points where one of modules equals zero: x=-2 and x=2

1.1 x e (-∞,-2) --> |x-2|+|x+2| = -x+2-x+2 = -2x >4
1.2 x e   --> |x-2|+|x+2| = -x+2+x-2 = 4
1.3 x e (2,+∞) --> |x-2|+|x+2| = x-2+x+2 = 2x >4

So, the expression equals 4 only if x e  . the second condition is enough and therefore B

rpmodi · Answer

Walker , I did not quite get this explanation , but can you tell me what is the flaw in my explanation if condition A is not sufficient ?

copying my explanation posted here :

"I am voting for D .

1.. x<0

x=-1 : |x-2|+|x+2| = 4
x=-2 : |x-2|+|x+2| =4
x=-3 : |x-2|+|x+2| > 4 
.....
.......
............

for any -ve value it will be >=4 .

Other posts have given the reason why statement 2 is sufficient ............."

walker · Answer

Rpmodi, You are right, it is D. The expression always >=4. It is my silly mistake as I forgot my rule: reread the problem a few times.

vishy007 · Answer

I also made a silly mistake. I agree with D.

FN · Answer

damn..i agree with D..

totally overlooked 1

Is |x-2|+|x+2|<4? 1). x<0 2). x is within (-2, 2)

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