Problem Solving

If a - b > a + b, where a and b are integers, which of the following must be true?

I. a < 0
II. b < 0
III. ab < 0

The answer is - only (II)

But I could derive III as well by multiplying both the sides by a resulting in :
-ab > ab or 2ab <0 ; ab<0

Could anyone tell me where i went wrong?

ShruAj wrote:If a - b > a + b, where a and b are integers, which of the following must be true?

I. a < 0
II. b < 0
III. ab < 0

The answer is - [spoiler]only (II)[/spoiler]

But I could derive III as well by multiplying both the sides by a resulting in :
-ab > ab or 2ab <0 ; ab<0

Could anyone tell me where i went wrong?

III says that ab < 0.

This means that neither a nor b can equal 0.

It also means that either a > 0 and b < 0 or a < 0 and b > 0.

a > 0 and b < 0 is obviously possible, as we can see by plugging in a = 2 and b = -2

a - b = 4 > 0 = a + b

However, III does not have to be the case. a and b could both be less than 0, and a - b > a + b, would still work.

Let's use a = -2 and b = -4

a - b = 2 > -6 = a + b

Where you went wrong is in multiplying an inequality by a variable without knowing whether the variable represents a positive number or a negative number.

If a is negative, then when you multiply both sides by a, the inequality becomes reversed, as inequalities always do when multiplied by negative numbers.

So in multiplying by a you get two possibilities -ab > ab or -ab < ab.

Thank you, Marty!

ShruAj wrote:If a - b > a + b, where a and b are integers, which of the following must be true?

I. a < 0
II. b < 0
III. ab < 0

a - b > a + b
-b > b
0 > 2b
0 > b
b < 0.

Since a can be any integer value, only II must be true.

Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.


If a - b > a + b, where a and b are integers, which of the following must be true?

I. a < 0
II. b < 0
III. ab < 0

a-b>a+b, erasing a from both sides gives us -b>b, 0>b+b, 0>2b, 0>b therefore only (II) is the answer..

But I could derive III as well by multiplying both sides by a resulting in :
-ab > ab or 2ab <0 ; ab<0

Could anyone tell me where i went wrong?

==> since we don't know whether a is positive or negative, we can't assure that the inequality sign will stay the same if you multiply a to both sides.



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