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Which of the following inequalities is equivalent to

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by Max@Math Revolution » Wed Mar 28, 2018 12:53 am
[GMAT math practice question]

Which of the following inequalities is equivalent to |2x-|x|| < 3?

A. 0 < x < 2
B. 0 < x < 3
C. -1 < x < 3
D. 0 < x < 1
E. -3 < x < 1
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by GMATGuruNY » Wed Mar 28, 2018 2:55 am
Max@Math Revolution wrote:[GMAT math practice question]

Which of the following inequalities is equivalent to |2x-|x|| < 3?

A. 0 < x < 2
B. 0 < x < 3
C. -1 < x < 3
D. 0 < x < 1
E. -3 < x < 1
Since x=0 satisfies the given inequality, the correct answer must include 0 within its range.
Eliminate A, B and D.
Since x=2 satisfies the given inequality, the correct answer must include 2 within its range.
Eliminate E.

The correct answer is C.
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by Max@Math Revolution » Fri Mar 30, 2018 1:00 am
=>

|2x-|x||<3
=> -3 < 2x – |x| < 3

Case 1: If x ≥ 0, then |x| = x, and so
-3 < 2x – |x| < 3
=> -3 < x < 3
=> 0 ≤ x < 3, since x ≥ 0.

Case 2: If x < 0, then |x| = - x, and so
-3 < 2x – |x| < 3
=> -3 < 3x < 3
=> - 1< x < 1
=> - 1< x < 0, since x < 0.

Thus, -1 < x < 3.

Therefore, the answer is C.


Answer: C
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by Scott@TargetTestPrep » Wed Apr 04, 2018 4:52 pm
Max@Math Revolution wrote:[GMAT math practice question]

Which of the following inequalities is equivalent to |2x-|x|| < 3?

A. 0 < x < 2
B. 0 < x < 3
C. -1 < x < 3
D. 0 < x < 1
E. -3 < x < 1
The given inequality is one with an absolute value expression within another absolute value expression. In general, this type of inequality is difficult to solve. Therefore, instead of solving it algebraically, we will analyze each answer choice to determine the correct answer. Notice that the correct answer should have the two endpoints of the interval (for example, the endpoints of choice A are 0 and 2) that makes |2x-|x|| equal to 3, and thus every value in between the endpoints will make |2x-|x|| less than 3.

A. 0 < x < 2

x = 0: |2(0) - |0|| = 0 .....This is not 3.

Choice A can’t be the right answer.

B. 0 < x < 3

Choice B can’t be the right answer either since when x = 0, |2x-|x|| is not equal to 3.

C. -1 < x < 3

x = -1: |2(-1) - |-1|| = |-2 - 1| = |-3| = 3

x = 3: |2(3) - |3|| = |6 - 3| = |3| = 3

We see that both endpoints make |2x-|x|| equal to 3. Thus, every value between -1 and 3, i.e., -1 < x < 3, will make 2x-|x|| less than 3.

Answer: C

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