If a – b > a + b, where a and b are integers, which of the following

You don't need number plugging at all.

Given: \(a-b>a+b\) --> \(a\) cancels out (which means that from given info we can say nothing about it: it can be positive, negative or zero, so it can be ANY number) --> \(2b<0\) --> \(b<0\).

As we know nothing about \(a\) and know that \(b<0\), so only II must be true.

Answer: B.
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Unless you know whether a is positive or not you cannot determine how the inequality will look like after you multiply it by a. You cannot still prove III to be true.

Lets consider the two cases here:
Suppose a is positive (3, say)
3-b>3+b
Multiply the inequality by 3
9-3b>9+3b
6b<0
b<0

hence, ab<0


Now, suppose a=-3
plug in the values:

-3-b>-3+b
Multiply by a(-3)
9+3b<9-3b
6b<0
b<0

hence, ab > 0

Thus the sign of ab will vary depending on the sign of a. Hope this clears up the confusion

Simplify the inequality:
a – b > a + b
b<0

We know for certain that II is true.

We don't have info about the value of a. Hence, I cannot be derived.

For III ab can be positive or negative depending on the value of a. Since we don't know about a, III cannot be inferred.

Hence, B is the answer

can we not sqaure both sides of a – b > a + b ?

I definitely found this confusing, but let me try and explain how I went through this.

a - b > a + b --> subtract a from both sides.
- b > b --> b is clearly negative.
0 > 2b
0 > b --> yup, negative; (II) is correct.

What else?

Well can 'a' be positive? Yes, a positive number (-) a negative number is always greater than the sum of two negative numbers. Can a be negative? Ok, well what if a = -1...

(-1 - (-2)) > (-1 + (-2))
(-1 + 2) > (-1 - 2)
-1 > -3 --> ok so yes.

In this case, the answer to both (I) and (III) is 'NO' as a could be positive or negative.

More conceptually however, it might be more edifying to convince oneself that a negative number minus another negative number (so the negative cancel out and it becomes addition) is greater (less negative) and a negative number plus another negative number (which just goes more negative).

There's a problem above (in red)
While it's true that a-b > a+b, we cannot divide both sides of by a+b to conclude that (a-b)/(a+b) > 1, because it COULD be the case that (a+b) is negative, in which case the inequality becomes (a-b)/(a+b) < 1

More in this video:

If b < 0, then ab < 0 only when a > 0, but as I said in the solution we know nothing about a, it can be positive, negative or zero and since we are asked which of the following must be true, then we cannot be sure that ab < 0.
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well agreed.....however, if we multiply both side of the equation by "a", we get to know that III is also true.....which follows that I is also true?

Q: 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

Through inequality; I just derived that b is -ve
a – b > a + b
-b > b --> subtracting 'a' from both sides
-2b > 0 --> subtracting 'b' from both sides
-b > 0 --> dividing both sides by 2
b < 0 --> multiplying -ve sign on both sides

The rest two; I ruled out using numbers

Let b, as we know is -ve, to be equal to -1

Case I
a=+ve, say 100
a - b = 100 – (-1) = 101
a + b = 100 + (-1) = 99

a-b > a+b

Case II
a=-ve, say -100
a - b = -100 – (-1) = -100 + 1 = -99
a + b = -100 + (-1) = -100 - 1 = -101

So, a - b > a + b

Thus, a - b > a + b is true for both +ve and -ve 'a'

We just proved that a-b>a+b is true for both +ve and -ve values of a.

We can't conclusively say that a < 0. Statement I is ruled out.

for a=+ve; ab = +ve * -ve = -ve
and
for a=+ve; ab = -ve * -ve = +ve

So, we can't conclusively say ab < 0. Statement III is ruled out.

Ans: B

1. No. 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). Check for more here: inequalities-tips-and-hints-175001.html

2. Why do you want to square?
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You are only complicating the simple expression given by squaring it.

Also, as a general rule of inequalities, squaring variables in the inequalities is not a good idea until you know for sure the sign of a-b as multiplication by a negative number reverses the inequality.
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Sorry I got my mistake
-3>-5
but 9< 25

actually after squaring we get 0>4ab , and thought III is also true

Hi Bunuel,

I cannot understand, how could we cancel 'a' here when we do not know about its sign? Please explain this.


Thanks.

There is a difference between reducing (dividing/multiplying) an inequality by a variable and subtracting/adding a variable to both parts of an inequality. The first one we cannot do unless we know the sign of the variable but subtracting/adding is always applicable.

Check below links for more:
Inequalities Made Easy!

Solving Quadratic Inequalities - Graphic Approach
Inequality tips

DS Inequalities Problems
PS Inequalities Problems

700+ Inequalities problems
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Hi, can someone please advise on the below approach. I'm able to quantify a as positive and can't seem o negate my method,

for Option III - ab<0

We know that b<0 by reducing the equation a-b > a+b from the question stem. I took this a step further as below,

a-b > a+b
(a-b)/(a+b) > 1
(a-b)/(a+b) - 1 > 0
(a-b -a -b)/(a+b) > 0
= -2b/(a+b) > 0

Since b is negative I know that the numerator is a positive value and thus the denominator must be positive such that the above equation hold true.

thus, (a+b) > 0
a > - b

as b is negative, say -4 -->

a > - (-4), a > 4 and thus positive.

I know its a stretch, but I really need someone to tell me where I went wrong!

TIA,
Sahil

a – b > a + b
=> 0>2b
=>b<0

Now, we dont know about a.
If a> 0 , ab<0; but if a<0, ab>0;
So, only II must be true.
Ans B.
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Hi All,

In Roman Numeral questions, the answers choices are often designed to offer you a 'shortcut' (and a way to avoid some of the 'work' involved). Beyond the algebra that you can do in this question, you can also solve it by TESTing VALUES:

We're told that A-B > A+B and that A and B are INTEGERS. We're asked which of the following MUST be true.

IF...
A = 0
B = -1
0 - (-1) > 0 + (-1)
We can eliminate Roman Numerals I and III.

There's only one answer remaining...

Final Answer:

GMAT assassins aren't born, they're made,
Rich

a-b>a+b
So -b>b,
0>2b
b<0
So Answer is B

If B is less than 0 then wouldn't that make AB also less than 0? Where did I go wrong?