If |a + b| = |a - b|, then ab must be equal to:

Question

A. 1
B. -1
C. 0
D. 2
E. -2

Bunuel · Accepted Answer

If |a + b| = |a - b|, then ab must be equal to:

A. 1
B. -1
C. 0
D. 2
E. -2

Square both sides:

(a+b)^2=(a-b)^2 ;

a^2+2ab+b^2=a^2-2ab+b^2 ;

4ab=0 ;

ab=0 .

Answer: C.

Hope it's clear.

Bunuel · Answer

Check this post: elimination-of-radials-confused-138409.html As for inequalities: A. We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality). For example: 2<4 --> we can square both sides and write: 2^2<4^2 ; 0\leq{x}<{y} --> we can square both sides and write: x^2-2 if we square we'll get 1>4 which is not right. So if given that x>y then we can not square both sides and write x^2>y^2 if we are not certain that both x and y are non-negative. B. We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality). For example: -2<-1 --> we can raise both sides to third power and write: -2^3=-8<-1=-1^3 or -5<1 --> -5^2=-125<1=1^3 ; x we can raise both sides to third power and write: x^3

tusharacc · Answer

This I solved by trial and error...with a logic that what RHS has (a - b) and LHS has addition of the same terms. So for this to be true one has to be be zero hence answer will be zero.

Regards,
Tushar

pike · Answer

Clearly if  a  or  b  equal zero,  ab = 0

so, let  b
eq{0} 
Distance of  a  from  b , equals the distance of  a  from - b 
Draw this on a number line,  a  must equal zero

same logic holds for  a
eq{0}

So either  a  or  b  = 0,  ab = 0

or  just solve using our normal absolute value method, two cases:

(a+b) = (a-b) 
 b = 0

-(a+b) = (a-b) 
 -a-b = a-b 
 a = 0

so  ab = 0

honggil · Answer

Bunuel,

Can you always just square absolute values like you did on this problem?
I am wondering if squaring is a valid operation on absolute values.

EvaJager · Answer

If  b=0 , the equality obviously holds.
 |a+b|=|a-b|  means the distance between  a  and - b  is the same as the distance between  a  and  b .
For  b
eq0,  it means that  a  is the average of - b  and  b  (or the midpoint between - b  and  b ), so necessarily  a=0. 
Altogether, the product  ab  must be  0.

Answer C

CA88 · Answer

I did this problem a little differently. Since both sides have absolute values, I used positives and negatives to get the following four possibilities:

a+b = a-b
a+b = -(a-b)
-(a+b) = a-b
-(a+b) = -(a-b)

Simplifying each equation gets:

b = -b
a = -a
-a = a
-b = b

From there, I concluded that a*b must be zero since the only way a = -a is if a = 0 and the only way b = -b is if b = 0. Did I do this problem incorrectly?

EvaJager · Answer

It is absolutely correct.
Good job!

pacifist85 · Answer

I am not sure if the way I thought of the absolute value was correct, but here it is.

QUESTION: If |a+b|=|a-b|, then a*b must be equal to:

I thought that when there is an absolute value, we think of the "value" within it with no regards to the sign. So, I thought it could be:
a+b= a-b. If we move a to the other side, both a's get cancelled, which leads to b=-b. Moving -b to the other side it becomes 2b=0. So, b=0, which means that a*b must be equal to 0.

Bunuel · Answer

You are missing the second case there.

If a+b and a-b have the same sign, then we'd have a+b = a-b --> b=0 --> ab=0;
If a+b and a-b have the opposite signs, then we'd have a+b = -(a-b) --> a=0 --> ab=0.

RahulJain293 · Answer

Bunuel

I did it as below.. Is it correct?

l a+b l = l a-b l

Since l x l = root (x2)

root (a+b)square = root (a-b) square

Reduces to a+b = a-b

Hence a-a+b+b = 0

We get 2b = 0

b = 0

Hence, a*b --> a*0 = 0 (opt c)

Bunuel · Answer

|a + b| = |a - b| does not mean that a + b = a - b, so no, your method is not correct. The point is that  \sqrt{(a + b)^2} = \sqrt{(a - b)^2}  simplifies back to |a + b| = |a - b|, not to a + b = a - b.

|a + b| = |a - b| means that either a + b = a - b, which occurs when a + b and a - b have the same sign, OR a + b = -(a - b), which occurs when a + b and a - b have opposite signs.

Hope it's clear.

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If |a + b| = |a - b|, then ab must be equal to:

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If |a + b| = |a - b|, then ab must be equal to:

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