# Lesson: Operations with Signed Numbers - Part II

## Comment on Operations with Signed Numbers - Part II

### Hi Brent

Hi Brent
I am commenting on GMATPrepNow - SVP answer on this question https://gmatclub.com/forum/is-13n-a-positive-number-215241.html
Statement 1: How we can be certain that the Negative sign in -21N is for the number 21 not for N. for example 21 x (-N) = -21N, as it is written as a result and not as an equation, I mean if it was written like this -21 x N, I would say for sure N is positive. Can you clarify please.

Let's examine both possible cases:

case a: -21N is the same as (-21)(N)
So, (-21)(N) = a negative value
In other words, (negative number)(N) = a negative value
In this case, we can conclude that N must be POSITIVE

case b: -21N is the same as (21)(-N)
So, (21)(-N) = a negative value
In other words, (positive number)(N) = a negative value
So, we can conclude that -N must be negative
If -N is negative, then it must be the case that N is POSITIVE

As you can see, we reach the same conclusion (N is POSITIVE) in either case.

Does that help?

Cheers,
Brent

It does. Thanks

### Hello Brent,

Hello Brent,

I am having a difficult understanding the solution for the question

https://gmatclub.com/forum/if-p-is-a-negative-integer-which-of-the-following-must-be-true-of-1-p-210890.html

### Hi kunmibode,

Hi kunmibode,

I have a solution here: https://gmatclub.com/forum/if-p-is-a-negative-integer-which-of-the-follo...

We're told that p is a negative integer

So, p COULD be -1, -2, -3, -4, etc

The keyword here is MUST, as in "which of the following MUST be true of 1/p?"

Well, p COULD equal -2, which means 1/p = 1/(-2) = -1/2

So, "Which of the following MUST be true of 1/p?"

A. 1/p > −1
MUST this be true?
Replace p with -2 to get the inequality: -1/2 > 1
This inequality is NOT true
So, we can eliminate A.

We can now apply the same strategy to eliminate answer choice B....

We can keep testing values of p until we eliminate all answer choices except one.

Does that help?

Cheers,
Brent

### amazing videos. Loved your

amazing videos. Loved your work Brent!!

### Hello brent,

Hello Brent,

you have given a list of practice questions at the bottom of this video. What if i only study from your videos and not from anywhere else? Will it still suffice?

### The answer to that question

The answer to that question really depends on your target score and your understanding of the various complexities of the concepts covered in that particular video lesson. For example, if you have a 700 target score, then you should probably make sure that you can answer related practice questions in the 650-800 range.

Cheers,
Brent

### The universities I’m

The universities I’m targeting is asking want 600 & some want 650. Accordingly how should i see my questions?

### That really depends on you

That really depends on you strengths and weaknesses with regard to the Verbal and Quantitative sections.
If you're stronger in Quant than you are in Verbal, you might want to answer 650-800 level Quant questions and perhaps 500-650 level Verbal questions.
Conversely, if you're stronger in Verbal than you are in Quant, you might want to answer 650-800 level Verbal questions and perhaps 500-650 level Quant questions.

I hope that helps.

Cheers,
Brent

### Is |a - b| < |a| + |b| ?

Is |a - b| < |a| + |b| ?

(1) ab< 0

(2) a^b < 0

### Nice question!!

Nice question!!
Key concept: |x - y| = the DISTANCE between x and y on the number line.

So, we can say that |a| = |a - 0| = the DISTANCE between a and 0 on the number line.
Likewise, |b| = |b - 0| = the DISTANCE between b and 0 on the number line.

TARGET QUESTION: Is |a - b| < |a| + |b| ?

REPHRASED TARGET QUESTION: Is the distance between a and b equal to the SUM of the distance between a and 0 AND the distance between b and 0?

STATEMENT 1: ab < 0
This means that one value is POSITIVE and the other is NEGATIVE
So, our number line looks like this: ___a____0____b___ or this: ___b____0____a___
Is the distance between a and b equal to the SUM of the distance between a and 0 AND the distance between b and 0?
In both cases, the answer is YES
Statement 1 is sufficient.

STATEMENT 2: a^b < 0
This tells us that a is negative, and b is an ODD integer that can be either positive or negative.
Consider these two conflicting cases:

CASE a: a = -1 and b = -1. Statement 2 is satisfied, because (-1)^(-1) = -1.
In this case, |a - b| = |(-1) - (-1)| = |0| = 0, and |a| + |b| = |-1| + |-1| = 1 + 1 = 2
So, the answer to the original target question is "YES, |a - b| < |a| + |b|

CASE b: a = -1 and b = 1. Statement 2 is satisfied, because (-1)^1 = -1.
In this case, |a - b| = |(-1) - (1)| = |-2| = 2, and |a| + |b| = |-1| + |1| = 1 + 1 = 2
So, the answer to the original target question is "NO, it is not the case that |a - b| < |a| + |b|

Statement 2 is NOT sufficient.

Cheers,
Brent

Hi Brent! Can I please ask, why did you pick a=0 for the second statement? how did you come to that conclusion?

Thank you!

### Wow! I have no idea how I

Wow! I have no idea how I reached that conclusion!
In fact, it's totally incorrect.
I have edited my response above to be much more logical :-)

Cheers,
Brent

### Hi Brent,

Hi Brent,

a^b <0, I could see clearly that a is negative and b is odd but I didnt see that b can be negative. If b would be negative would not be like this a^1/b? What am I missing here?

### Rule for negative numbers: x^

Rule for negative numbers: x^(-n) = 1/(x^n)

So, for example, 5^(-3) = 1/(5^3) = 1/125
Likewise, (-2)^(-3) = 1/(-2)^3 = = 1/-8 = -1/8

Does that help?

Cheers,
Brent

### Thank you Brent,

Thank you Brent,
I remember the exponent now. However, in this question we assume that "b" may be odd and negative, because we are dealing with absolute values? If the question wouldn't be an absolute value related then the statement will be such as a^(-b)<0? In that case we can see that a is negative and be is negative as well.

### If a^b < 0, then a is

If a^b < 0, then a is negative, and b is an ODD integer that is either positive or negative.
So, for statement 2, we aren't assuming that b is odd and negative. It COULD be the case that b is odd and negative, but it could also be the case that b is odd and positive.

I'm wondering if the notation is confusing you up a little bit.
For example, some students feel that -k always represents a negative number.
However, if k = -1, then -k is actually a positive number, since -k = -(-1) = 1

Does that help?

### Hi Brent! Hope you are well!

Hi Brent! Hope you are well! I am facing difficulty in the absolute number concent. Please refer to the question link below -
https://gmatclub.com/forum/is-xy-213592.html

|X| signifies absolute number, which means a positive number, right? Then how can use x = -1.1, -1.2...etc (negative numbers) in statement 1) 1<|x|<4?

Thanks in advance for the help!

Hi Harleen,
You're right to say that "|X| signifies absolute number, which means a positive number"

This means that:
|-1.1| = 1.1
|-1.2| = 1.2
.
.
.
|-3.8| = 3.8
|-3.9| = 3.9

So, it is true that 1 < |-1.1| < 4, because this becomes 1 < 1.1 < 4 (which is true)
And 1 < |-1.2| < 4, because this becomes 1 < 1.2 < 4 (which is true)
.
.
.
And 1 < |-3.9| < 4, because this becomes 1 < 3.9 < 4 (which is true)

So, as you can see, it's possible for x to have a negative value.

Does that help?

Cheers,
Brent

### Yes, it does. Thanks so much!

Yes, it does. Thanks so much!

Harleen Kaur

### Target question: Is xy > 16?

Target question: Is xy > 16?

Statement 1: 1 < |x| < 4
Since there's no information about y, we cannot determine whether xy is greater than 16
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y > 16
Since there's no information about x, we cannot determine whether xy is greater than 16
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
There are several values of x and y that satisfy BOTH statements. Here are two:
Case a: x = 2 and y = 20, in which case xy = 40. In this case, the answer to the target question is YES, xy IS greater than 16
Case b: x = -2 and y = 20, in which case xy = -40. In this case, the answer to the target question is NO, xy is NOT greater than 16
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Please tell me in case B why we took value of x as -2 when it is given x as absolute value

Thanks

### We're told that 1 < |x| < 4

We're told that 1 < |x| < 4
This doesn't mean that x must be positive. It just means that |x| must be positive.

For example |-3| = 3
So, we can write: 1 < |-3| < 4
So, x = -3 is a solution to the inequality 1 < |x| < 4

Likewise, |-2| = 2
So, x = -2 is a solution to the inequality 1 < |x| < 4

Does that help?

Cheers,
Brent

### |-4| (|-20| - |5|) =

|-4| (|-20| - |5|) =

A. –100
B. –60
C. 60
D. 75
E. 100
Why C?? OA given... it can be E too.??

### |-4|(|-20| - |5|) = |-4| (20

|-4|(|-20| - |5|) = |-4| (20 - 5) [deal with the part in brackets]
= |-4| (15) [simplify the part in brackets]
= (4)(15) [simplify |-4| to get 4]
= 60
= C

Does that help?

Cheers, Brent

Yes it did

### If |x| = |y| and xy < 0,

If |x| = |y| and xy < 0, which of the following must be true?

A. xy^2 > 0

B. yx^2 > 0

C. x + y = 0

D. x/y +1=2

E. 1/x + 1/y = 1/2]

Couldn't x and y be the same positive fraction, and then the answer choice C would not hold?

### If x and y were the same

If x and y were the same positive fraction, then xy would be positive (since positive times positive equals positive).

|x| = |y| tells us that x and y have the same MAGNITUDE.
That is, x and y are the same distance from 0 on the number line.

xy < 0 tells us that the product xy is NEGATIVE.
This means one value (x or y) is POSITIVE, and the other value is NEGATIVE.

So, the two values have the same MAGNITUDE, and one value is POSITIVE, and the other value is NEGATIVE.

For example, it could be the case that x = 5 and y = -5

If we test x = 5 and y = -5, we see that answer choices B, D and E are not true. So, we can eliminate those answer choices.
-------------------

Let's test another pair of values....

It could also be the case that x = -3 and y = 3

If we test x = -3 and y = 3 with the REMAINING answer choices, we see that A is not true. So, we can eliminate A.

This leaves us with C, the correct answer.

Cheers,
Brent

### Hi Brent,

Hi Brent,
Given that per PEMDAS and BEDMAS, order of division and multiplication is interchangeable. In this question how can we assume that we will divide first and then multiply?

Be careful. Multiplication and division are considered equal. So when an expression contains both multiplication and addition, we perform those operations from left to right. The same applies to addition and subtraction.

This is discussed at 5:55 in the above video.

Cheers,
Brent

### Dear Brent,

Dear Brent,
Question 1: https://gmatclub.com/forum/if-p-and-q-are-integers-such-that-p-0-q-and-s-is-a-128798.html#p1055833

I solved in correctly, however I don’t understand why expert consider s=0 if in the question it is stated that S is no-negative integer. So why do they chose a value zero for S if zero is either negative or positive?

### Great question!

Great question!

Numbers can be negative, positive, or zero (which is neither positive nor negative)

So, if we say a number is non-negative, then that number can be either positive or zero.

By the way, here's my full solution: https://gmatclub.com/forum/if-p-and-q-are-integers-such-that-p-0-q-and-s...

Cheers,
Brent

### Hi Brent,

Hi Brent,
Question: https://gmatclub.com/forum/is-x-y-188469.html
Is x>yx>y?
(1) 6x>5y
(2) xy<0

Could you please tell where I am wrong in statement 1?
6x>5y
x>(5/6)y
in that case I though that x anyway will be greater than y. What did I miss?

Saying that x > (5/6)y, is the same as saying x > 0.83y (approximately)
So, it could be the case that x = 1 and y = 1 (since 1 > 0.83)
It could also be the case that x = 1 and y = 1.01 (since 1 > 0.8383)
It could also be the case that x = 100 and y = 1 (since 1 > 0.83)

In the first case, x = y
In the second case, x < y
In the third case, x > y

Does that help?

### https://gmatclub.com/forum/is

https://gmatclub.com/forum/is-x-y-188469.html

Hi Brent,

I did not understand why not the answer option E.

Even though we combine both statements, in case-a it satisfies the target question but in case-b, it does not satisfy the target question.

In one case it is positive and in another case it is negative. So we are getting two unique answers, then it should be E.

Be careful. That's not quite what my solution says.

From Statement 2, we know there are two possible cases:
Case a: x is POSITIVE and y is NEGATIVE .
Case b: x is NEGATIVE and y is POSITIVE.

Case a also satisfies Statement 1. So, we know that it's possible for x to be POSITIVE and y to be NEGATIVE .

HOWEVER, case b does NOT satisfy Statement 1.
So, it CAN'T be the case that x is NEGATIVE and y is POSITIVE.

In other words, ONLY CASE A IS TRUE, which means the answer to the target question must be YES, x IS DEFINITELY greater than y

Does that help?

### Gotcha, Thank you.

Gotcha, Thank you.

### Hi Brent, for this question

Hi Brent, for this question.https://gmatclub.com/forum/is-the-product-abcd-negative-229409.html

Is this deduction okay. Statement 1 gives no information about the equation because of the fact that it is not less than an actual figure. therefore the variables could be any number on the number line so we cannot determine if it is negative or positive. Therefore the statement is insufficient

Statement 2: ad>0. This is interpreted as a and d being positive seperately.No information is provided about b and c so it is insufficient

combining both statement
Statemnt 2 is interpreted as a and d being positive seperately. And since in statement 1 a<b and b<c it means b and c greater than A but less than d. This implies that definelity b and c are positive numbers and then we can answer the target question.

This was really how i got to choosing C as my answer.

Is this okay?

Your reasoning is perfect except for one point:
Statement 2 tells us that ad > 0. In other words the product of a and d is POSITIVE.
This means EITHER a and d are both positive, OR a and d are both negative.

If a and d are both positive, then we we can be certain of that b and c are also positive, since statement 1 tells us that a < b < c < d.
Likewise, if a and d are both negative, then we we can be certain of that b and c are also negative, since statement 1 tells us that a < b < c < d.

In both cases, the product abcd is positive.

Cheers, Brent

### consecutive integers -10 to

consecutive integers -10 to 10 inclusive, shouldn't the number be 10-(-10)+1, which is 21 according to the formula?

### I believe you're referring to

I believe you're referring to this question: https://gmatclub.com/forum/from-the-consecutive-integers-10-to-10-inclus...

Yes, there are 21 integers from -10 to 10 inclusive.
The question asks us to choose 20 of those 21 numbers (repetitions allowed). That is, the question isn't telling us there are 20 integers from -10 to 10 inclusive.

### https://gmatclub.com/forum/is

https://gmatclub.com/forum/is-13n-a-positive-number-215241.html
Hi Brent,
Can we take 0^2 in 2nd statement?
Also if possible could you please let me know in which scenarios we can take 0 as a plugging in value.
Thanks

Target question: Is 13N a positive number?

(1) -21N is a negative number
(2) N² < 1

Yes, N = 0 definitely satisfies statement two, since 0² = 0, and 0 < 1
In this case, the answer to the target question is NO.

When it comes to testing particular values, the main restriction is that the denominator of a fraction cannot equal 0.
For example, if we're told 5/x < 1, x cannot equal 0, since 5/0 is not a defined number.

Does that help?

### yes it does, Thanks for the

yes it does, Thanks for the clarification!