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Source: — Data Sufficiency |
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by GMATGuruNY » Tue Mar 06, 2018 3:20 am
Gmat_mission wrote:What is x?

(1) |x| < 2

(2) |x| = 3x - 2
Statement 1:
Here, x can be any value between -2 and 2.
INSUFFICIENT.

Statement 2:
Case 1: signs unchanged
x = 3x - 2
2 = 2x
1 = x
x = 1.

Case 2: signs changed in the absolute value
-x = 3x - 2
2 = 4x
2/4 = x
x = 1/2.

When an equation has absolute value only ON ONE SIDE, plug the two solutions back into the original equation to ensure that both are valid.

If we plug x=1 into |x| = 3x-2, we get:
|1| = 3*1 - 2
1 = 1.
This works.
x=1 is a valid solution for |x| = 3x-2.

If we plug x=1/2 into |x| = 3x-2, we get:
|1/2| = 3(1/2) - 2
1/2 = -1/2.
Doesn't work.
x=1/2 is NOT a valid solution for |x| = 3x-2.

Thus, Statement 2 has only one valid solution:
x=1.
SUFFICIENT.

The correct answer is B.
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by Vincen » Tue Mar 06, 2018 4:56 am
Hello.

(1) |x|<2.

this implies that x can be 0, 1, 1.5, -10, . . . . . NOT Sufficient.

(2) |x|=3x-2.

Using the definition of absolute value we get two cases: $$(a)\ \ x=3x-2\ \Leftrightarrow\ 2=2x\ \Leftrightarrow\ x=1.$$ $$(b)\ \ -x=3x-2\ \Leftrightarrow\ 2=4x\ \Leftrightarrow\ x=\frac{1}{2}.$$ Now, let's plug the solutions in the original equation: $$\left|1\right|=3\left(1\right)-2\ \Leftrightarrow\ \ 1=3-2\ \Leftrightarrow\ 1=1.$$ $$\left|\frac{1}{2}\right|=3\left(\frac{1}{2}\right)-2\ \Leftrightarrow\ \ \frac{1}{2}=\frac{3}{2}-2\ \Leftrightarrow\ \frac{1}{2}\ne-\frac{1}{2}.$$ Hence, x=1/2 is not a solution.

Therefore, we get one unique solution x=1. Thus, this statement is SUFFICIENT.

The correct answer is the option B.
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by Jeff@TargetTestPrep » Thu Mar 08, 2018 5:05 pm
Gmat_mission wrote:What is x?

(1) |x| < 2

(2) |x| = 3x - 2
Statement One Alone:

|x| < 2

We see that x < 2 or:

-x < 2

x > -2

So, -2 < x < 2.

Statement one alone is not sufficient to answer the question.

Statement Two Alone:

|x| = 3x - 2

When x is positive we have:

x = 3x - 2

-2x = -2

x = 1

When x is negative we have:

-x = 3x - 2

2 = 4x

1/2 = x

We see that x = 1 matches our assumption that x is positive whereas x = ½ does not match our assumption that x is negative (because x = ½, which is a positive value). Thus, x can be only 1.

Answer: B

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