If ab - 2b = (4 - a)b, what is the value of b? (1) |a^2 - 9| ≤ 0 (2)

Solution

Step 1: Analyse Question Stem

• We have, \( ab-2b = (4-a)b\)

• We need to find the value of b.

• Now, \( ab-2b = (4-a)b\)

\( ⟹ b(a-2) – b(4-a) = 0 \)
\( ⟹ b( a- 2 – 4 + a)= 0 \)
\( ⟹ b(2a – 6)= 0 \)
\( ⟹ 2b(a-3)= 0 …….Eq.(i) \)

 So, if \(a ≠ 3\), in that case b must be 0.
 And, if \(a = 3\), in that case b can take any real number value.
• So, if we can determine that \(a ≠ 3\), we can find the unique value of b.

Now, let’s analyse the statements.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE

Statement 1: \(|a^2 – 9| ≤ 0\)

• We know that absolute values cannot be less than 0

o This, means, \(|a^2 – 9| = 0\) \(⟹ a = -3,\) or \(3\)

 If a = 3, in that case b = 0, unique solution
 However, if a = -3, in that cases b can take any real number value.

Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.

Statement 2: \(a< 0\)

• From this statement, we can be sure that \(a ≠ 3\).

o Therefore, \(b = 0\)

Hence, statement 2 is sufficient.
Thus, the correct answer is Option B.

from question:
ab-2b=4b-ab
2ab=6b

statement 1: tells us either a=<3, -3 or -3, 3<=a, nothing about B, hence Insuff
statement 2: tells us nothing about b, hence insuff
together: nothing about b.. hence IMO ans: E

If ab - 2b = (4 - a)b, what is the value of b?

(1) |a^2 - 9| ≤ 0
(2) a < 0

ab - 2b = 4b - ab
2ab = 6b
2ab - 6b = 0
b(a - 3) = 0

So, either b = 0 and a - 3 ≠ 0
OR
b ≠ 0 and a - 3 = 0
OR
b = 0 and a - 3 = 0

(1) |a^2 - 9| ≤ 0
Since |mod| ≥ 0
|a^2 - 9| = 0 only
a^2 - 9 = 0
a = 3 => b can take any value
OR -3 => b = 0
However, b can take any value.

INSUFFICIENT.

(2) a < 0
Here, since a < 3 so b can take only one value i.e.
b = 0, so that b(a - 3) = 0

SUFFICIENT.

Answer B.

Given: ab - 2b = (4 - a)b which upon simplification becomes b(3-a)=0. Thus, either b = 0 or a = 3 or both.

Statement 1: |a^2 - 9| ≤ 0, as absolute (modulus) values are either 0 or Positive Numbers. Therefore a^2 = 9.
We get a = 3 or a = -3. Thus we cannot be sure about the value of 'b'. Insufficient

Statement 2: a < 0, thus a≠3. Therefore, b = 0. Sufficient.

Choice B is the correct answer.

If ab - 2b = (4 - a)b, what is the value of b?

(1) |a^2 - 9| ≤ 0
(2) a < 0

Given, b (a-2) = b (4 -a), so, when b =0, a can assume any value. If b is not 0, a will be 3.

1) a^2 - 9 =0, as the absolute value cannot be less than 0. so, a is either -3 or +3. Both of the values will satisfy the equation given b is 0 or not. not sufficient.

2) no information about b. not sufficient.

Together, a can assume only the value -3. so b is 0. Sufficient.

C is the answer.

(a - 2)b = (4 - a)b
Either a=3 or b=0

Q. what is the value of b?

(1) |a^2 - 9| ≤ 0
a=-3 --> b=0
a=3 --> b can be any
NOT SUFFICIENT

(2) a < 0
a=-3 --> b=0
Any other a --> b=0
SUFFICIENT

Ans (B)

Posted from my mobile device

E - Neither sufficient

ab - 2b = (4 - a)b ==> ab - 2b = 4b - ab ==> 2ab = 6b. ==> a = 3

then 3b - 2b = (4-3)b ===> b=b

So whatever value we chose for B, will give us B and since neither statement 1 or 2 talk about option B we can conclude the answer is E

If ab - 2b = (4 - a)b, what is the value of b?

(1) |a^2 - 9| ≤ 0
(2) a < 0



This is a question in which the question stem gives more information than we usually give it credit for!

ab - 2b = 4b - ab
=> 2ab - 6b = 0
=> (a-3)b = 0

So either a = 3 or b = 0


st1) From the inequality equation, it is clear that \(a\neq{3} \), so b has to be equal to 0 (SUFFICIENT)

st2) clearly tells us that \(a\neq{3}\) , so b = 0 (SUFFICIENT)

So, the answer should be D

IMO B.

ab-2b = 4b-ab

2ab = 4b + 2b

2ab =6b


ab =3b

ab-3b = 0
b(a-3) = 0

either a=3 or b= 0 or both

Stmt 1
suggests that a^2 = 9
since it is absolute
a = 3

(a = -3 does not hold true after we simplify the question stem.)

3b = 3b

we get b = b
We dont get value so insuff.

Stmt 2 says a is negative.
Since this condition is false after we simplify the question stem.
so b has to be 0.
Suff.

B it is

Posted from my mobile device

From statement 1
a =3, so b can take up 0 or any other value
Not sufficient
From statement 2
a is less than 0 that means b can only be 0.
Sufficient
Answer is B option.

Posted from my mobile device

We have \(ab-2b=(4-a)b\)

\((a-2)b=(4-a)b\)
\(ab-2b=4b-ab\)
\(2ab-6b=0\)
\(ab-3b=0\)
\(b(a-3)=0\)

From the above we can infer the following two cases -
(i) if \(a=3\), \(b\) can be any real number
(ii) if \(a\neq{3}\), \(b\) = 0

From Statement (1) we have,
\(|a^2 - 9|\leq{0}\)
We know that a modulus function always gives a positive number as it's output. Hence \(|a^2 - 9|\) cannot be negative and can only be equal to 0 at the best.

\(a^2-9=0\)
\(a^2=9\)
\(a=3\) or \(a=-3\)

Unique value of b cannot be arrived at by either value of a. Hence Options (A) and (D) may be eliminated.

From Statement (2) we have,
\(a<0\)

If \(a<0\) then \(a\neq{3}\). Then b should be equal to 0 as we have found in (ii) above. Since we have a unique value of b in this case, (B) is the answer.

And C

With statement A
You get two values of a= 3, -3
This is not sufficient to find b

With statement 2
A<0
We are not clear what A is and what can be possible value for b. So this is insufficient

1+2
it is clear that A= -3, as A<0, with this find B from given equation
-3B -2B = -5B = 7B
So B=0

Posted from my mobile device

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.