Data Sufficiency

greenwich wrote:Why in Statement 1: |x+3|=4x-3
here LHS is Absolute value, which is non-negative?

I think I miss something by solving the problem as follows:

x+3=4x-3
3x=6
x=2

x+3=-4x+3
5x=0
x=0

What did I do wrong in the above problem?

Your approach is solid. However, whenever you deal with absolute value equations, you must check for EXTRANEOUS ROOTS (solutions)
When we plug x = 0 into the original equation, we get:
|0+3| = 4(0)-3
3 = -3 (no good)
So, the solution x = 0 is an extraneous root, which means we must disregard it.
This means that x = 2 is the ONLY solution to the original equation.

I hope that helps.

Cheers,
Brent

greenwich wrote:Is x>0?

(1) |x + 3| = 4x - 3
(2) |x + 1| = 2x - 1

When solving questions involving ABSOLUTE VALUE, there are 3 steps:
1. Apply the rule that says: If |x| = k, then x = k and/or x = -k
2. Solve the resulting equations
3. Plug in the solutions to check for extraneous roots

Now here's one approach:

Target question: Is x > 0?

Statement 1: |x + 3| = 4x - 3
When we apply the rule above, we get two equations
case a: x + 3 = 4x - 3
Rearrange: 6 = 3x
Solve: x = 2
Check for extraneous roots by plugging x = 2 into solution
|2 + 3| = 4(2) - 3
5 = 5
Great, we'll KEEP the solution x = 2

case b: x + 3 = -(4x - 3)
Expand: x + 3 = -4x + 3
Rearrange: 5x = 0
Solve: x = 0
Check for extraneous roots by plugging x = 0 into solution
|0 + 3| = 4(0) - 3
3 = -3
No good. We'll DISREGARD the solution x = 0

So, x MUST equal 2, which means x is definitely greater than 0
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: |x + 1| = 2x - 1
When we apply the rule above, we get two equations
case a: x + 1 = 2x - 1
Rearrange: 2 = x
Solve: x = 2
Check for extraneous roots by plugging x = 2 into solution
|2 + 1| = 2(2) - 1
3 = 3
Great, we'll KEEP the solution x = 2

case b: x + 1 = -(2x - 1)
Expand: x + 1 = -2x + 1
Rearrange: 3x = 0
Solve: x = 0
Check for extraneous roots by plugging x = 0 into solution
|0 + 1| = 2(0) - 1
1 = -1
No good. We'll DISREGARD the solution x = 0

So, x MUST equal 2, which means x is definitely greater than 0
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent

Hi All,

This DS question has a great "concept" shortcut built into it; if you can spot it, then you can avoid doing ALL of the Algebra.

We're asked if X > 0. This is a YES/NO question.

Fact 1: |X+3| = 4X-3

There's a great Number Property rule about absolute values: the result of an absolute value calculation can NEVER be negative.

Here, we're given an equation with an absolute value on "the left" and normal algebra on "the right." Focus on the "right side" of the equation.....

If X = ANY NEGATIVE, then (4X - 3) will be NEGATIVE. Since |X+3| CAN'T be negative, there cannot be a negative answer to this equation.

In that same way, if X = 0, then (4X-3) will also be NEGATIVE. As before, |X+3| CAN'T be negative, so X cannot be 0 either. Once you eliminate the negatives and 0 as possible answers, all that is left are POSITIVE ANSWERS. Since the question asks if X > 0, we don't even need to find the answer(s) to this equation; we KNOW that it/they MUST be positive, so the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT.

Fact 2: |X+1| = 2X - 1

Here, we have the exact same situation as in Fact 2.

If X = ANY NEGATIVE, then (2X-1) will be NEGATIVE
If X = 0, then (2X-1) will be NEGATIVE

Neither of these options is possible when dealing with an absolute value, so X MUST be POSITIVE and the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT.

Final Answer: D

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