Which of the following expressions CANNOT have a negative va

Question

Which of the following expressions CANNOT have a negative value?

A) |a + b| – |a – b|
B) |a + b| – |a|
C) |2a + b| – |a + b|
D) a^2 + b^2 – 2|ab|
E) |a^3 + b^3| – a – b

Bunuel · Accepted Answer

Notice that  a^2 + b^2 - 2|ab|=(|a|-|b|)^2  --> the square of any number is greater than or equal to 0.

Answer: D.

KarishmaB · Answer

You can eliminate by considering situations in which the expressions will be negative. We need to make the first term smaller than the second term in each case.

A) |a + b| – |a – b|
If a is negative and b positive, absolute value of (a - b) will be more than that of (a + b). So this expression will be negative.

B) |a + b| – |a|
If b is negative and a positive, absolute value of (a + b) will be less than that of a. So this expression will be negative.

C) |2a + b| – |a + b|
If a is negative and b positive (greater than 2a), (a + b) would be greater than (2a + b). So this expression will be negative.

E) |a^3 + b^3| – a – b
= |a^3 + b^3| – (a + b)
If a and b are positive numbers less than 1, (a + b) will be greater than (a^3 + b^3). So this expression will be negative.

TirthankarP · Answer

Reason to choose D is fine.
But is there any way to eliminate each option one by one (not preferably by plugging in values)?

Bunuel · Answer

I wouldn't use algebra to discard other options. Picking numbers probably would be easier.

TeamGMATIFY · Answer

The best way to find out that an expression can have negative value or not is by converting the expression into a perfect square
Because perfect squares can never be negative

On checking the options, we see that option D can be written in the form of a perfect square

a^2 + b^2 – 2|ab| = (|a| - |b|)^2
Hence the correct answer is Option D

OptimusPrepJanielle · Answer

Whenever you are required to find if the equation is a perfect square or not, try to bring everything in the form of a perfect square as they are always positive.

Of the given options, only option D can be written in from of a perfect square.

a^2+b^2-2|ab| = |a|^2 + |b|^2 - 2|a||b| = (|a| - |b|)^2. 
This will always be positive.

Correct Option: D

mbaapp1234 · Answer

Picking numbers here is a fine approach to use.

A) For the first one, pick a negative A so that the second absolute value expression becomes smaller and makes the entire expression negative.

B) Here, you can also make A negative.

C) For this one, it's best to use a negative A again (-1) and B = 2.

D) Correct.

E) Here you can use 1/2 and 1/2 for A and B because fractions cubed will become smaller and approach zero. Meanwhile, the original fractions are then subtracted from these smaller terms.

vaibhav1221 · Answer

Because the overall value will depend on the values of A and B as well, which are unknown, the expression should either equal zero or have an even power.
Option D is the only one which would render a positive outcome every time given any real value of A and B.

ScottTargetTestPrep · Answer

Let's test each answer choice:

A) |a + b| - |a - b|

If we let a = 2 and b = -2, then |a + b| = 0, but |a - b| = 4. Thus, |a + b| - |a - b| = -4. Eliminate A.

B) |a + b| - |a|

Let a = 1 and b = -1. Then, |a + b| = 0 and |a| = 1. Thus, |a + b| - |a| = -1. Eliminate B.

C) |2a + b| - |a + b|

Let a = 1 and b = -2. Then, |2a + b| = 0, and |a + b| = 1. Thus, |2a + b| - |a + b| = -1. Eliminate C.

D) a^2 + b^2 - 2|ab|

If ab  \geq  0, then |ab| = ab. In this case, the given expression becomes a^2 + b^2 - 2ab = (a - b)^2.

If ab < 0, then |ab| = -ab. In this case, the given expression becomes a^2 + b^2 - 2(-ab) = a^2 + b^2 + 2ab = (a + b)^2.

We see that the given expression is equivalent to either (a - b)^2, or (a + b)^2. Since the square of some number can never be negative, the given equation cannot be negative. This is the correct answer.

While we found our answer, let's verify that answer choice E can be negative for the sake of completeness.

E) |a^3 + b^3| - a - b

Let a = 1/2 and b = 0. Then, |a^3 + b^3| = 1/8. Substituting in the given expression, we get 1/8 - 1/2 - 0 = -3/8. We see that this expression can be negative.

Shubhradeep · Answer

If we can show that for any case an expression is negative we can conclude that it does not have any positive value.
1. |a+b|-|a-b| -> for some case it will have a value of (a+b)-(a-b)=2b [when a+b>0 and a-b>0]. Now if b<0, we have our number 1 candidate for elimination.
2.|a+b|-|a| -> Again very easy to eliminate. This will have a value of b when a+b>0 and a>0. But if b<0, the expression is negative
3. |2a+b|-|a+b| -> this can have a value of a which again can be negative
5. |a^3+b^3|-a-b -> This one needs a bit of work but is easy again. This expression boils down to |(a+b)*(a^2+b^2-ab)|-(a+b).
Very easily we say that a+b can have a value greater than (a^3+b^3) [ a=1, b=-2 or a=0.1, b=0.2]. So this expression is also liable to be a negative one.

For a^2+b^2-2|ab| we can either have a value of a^2+b^2-2ab or a^2+b^2+2ab which are perfects squares [ (a-b)^2 and (a+b)^2 respectively].
Alternatively, a^2+b^2-2|ab| can be written as (|a|-|b|)^2 which can never have a negative value.

Hence, correct answer option is D.

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Which of the following expressions CANNOT have a negative va

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Which of the following expressions CANNOT have a negative va

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