What is the value of m? (1) |m| = −36/m (2) 2m+2|m| = 0

Target question: What is the value of m?

Statement 1: |m| = −36/m
Key property: If |something| = k (where k is positive), then something = k or something = -k
When we apply the property we get: m = -36/m or m = 36/m
Let's solve each equation separately...

Take: m = -36/m
Multiply both sides of the equation by m to get: m² = -36
Since the square of a number (e.g., m²) is always greater than or equal to 0, there are no real solutions to the equation m² = -36

Take: m = 36/m
Multiply both sides of the equation by m to get: m² = 36
So, either m = 6 or m = -6

IMPORTANT: Before we conclude that statement 1 is not sufficient, we must test each possible solution by plugging it into the original equation

Let's first test m = 6.
We get |6| = -36/6, which simplifies to be |6| = -6
NOT TRUE.
So, m = 6 is an extraneous root that we must ignore.

Now we will test m = -6.
We get |-6| = -36/-6, which simplifies to be |-6| = 6
WORKS
So, m = -6 is the only solution

Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 2m+2|m| = 0
Subtract 2m from both sides of the equation to get: 2|m| = -2m
Divide both sides of the equation by 2 to get: |m| = -m

Applying the above property we get two equations: m = -m and m = m

Once again, we'll solve each equation separately.
Take: m = -m
Add m to both sides to get: 2m = 0
Solve: m = 0
Let's make sure this is an actual solution by plugging m = 0 into the original equation to get: 2(0)+2|0| = 0...WORKS!

Take: m = m
We can see right away that all values of m will satisfy the equation m = m
So, one possible value is m = -1
Let's make sure this is an actual solution by plugging m = -1 into the original equation to get: 2(-1)+2|-1| = 0...WORKS!

Since we've already identified the two possible values of m (m = 0 and m = -1), we know that statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent
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(1) |m| = −36/m
Because |m| > 0 => m < 0 so that -36/m > 0
=> m|m| = -36
=> m = -6
--> sufficient.

(2) 2m+2|m| = 0
<=> m + |m| = 0
<=> |m| = -m
==> m: inumerous
--> insufficient.

=> IMO: A

What is the value of m?

(1) |m| = −36/m
clearly |m| has to +ve always.
So m = -6
Sufficient.

(2) 2m+2|m| = 0
2|m| = -2m
|m| = -m
Clearly m has to be negative but it can take infinite values.
Not sufficient.

IMO A

What is the value of m?

(1) |m| = −36/m

\(|m|\geq{0} \)
\(=> -36/m > 0\)
While \(m\neq{0} => m>0\)
\(=>m*|m| = -36 => m = -6\)
=> sufficient.

(2) 2m+2|m| = 0

\(=> m + |m| = 0\\
=> |m| = -m\\
=> m < 0\)
=> insufficient.

=> A is Correct
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