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# Which of the following is true if -3<x<9?

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Math Revolution GMAT Instructor
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Which of the following is true if -3<x<9? [#permalink]

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15 Nov 2017, 01:54
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Difficulty:

5% (low)

Question Stats:

86% (01:18) correct 14% (00:51) wrong based on 57 sessions

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[GMAT math practice question]

Which of the following is true $$if -3<x<9$$?

A. $$|x-3|<4$$
B. $$|x-3|<5$$
C. $$|x-3|<6$$
D. $$|x-3|>4$$
E. $$|x-3|>5$$
[Reveal] Spoiler: OA

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Kudos [?]: 3137 [0], given: 0

Director
Joined: 25 Feb 2013
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Which of the following is true if -3<x<9? [#permalink]

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15 Nov 2017, 01:59
MathRevolution wrote:
[GMAT math practice question]

Which of the following is true $$if -3<x<9$$?

A. $$|x-3|<4$$
B. $$|x-3|<5$$
C. $$|x-3|<6$$
D. $$|x-3|>4$$
E. $$|x-3|>5$$

$$-3<x<9$$, add both sides of the inequality $$-3$$

$$=> -6<x-3<6$$

or $$|x-3|<6$$

Option C

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VP
Joined: 22 May 2016
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Which of the following is true if -3<x<9? [#permalink]

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15 Nov 2017, 10:09
MathRevolution wrote:
[GMAT math practice question]

Which of the following is true $$if -3<x<9$$?

A. $$|x-3|<4$$
B. $$|x-3|<5$$
C. $$|x-3|<6$$
D. $$|x-3|>4$$
E. $$|x-3|>5$$

1) Find the distance between, then the midpoint of, the endpoints:
$$9 -(-3) = 12$$
$$\frac{12}{2}$$= 6 = midpoint

2) Change the inequality so it relates to 6 and -6. Subtract 3 from each side:
(-3 - 3) < x < (9 - 3)
-6 < x < 6

3) The -3 becomes part of the absolute value expression:
-6 < |x - 3| < 6, or
|x - 3| < 6

The last step just takes the two cases created by the absolute value inequality |x - 3| < 6, and combines them.

That is, -6 < |x - 3| < 6 says "the absolute value of something is less than 6 (and absolute value is nonnegative)," so change it to
|x - 3| < 6

To check: The two cases for |x - 3| < 6 are

LHS: x - 3 > - 6
RHS: x - 3 < 6
Combined:
-6 < x - 3 < 6
-3 < x < 9 = original. Correct

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Math Revolution GMAT Instructor
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Re: Which of the following is true if -3<x<9? [#permalink]

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17 Nov 2017, 01:22
=>

|x-a|<b
⟺ -b < x – a < b
⟺ a - b < x < a + b
So, a is the midpoint of the end points a – b and a + b, and
b is half the distance between the end points a – b and a + b.

The midpoint of -3 and 9 is $$\frac{[(-3)+9]}{2} = \frac{6}{2} = 3$$. So, $$a = 3$$.
Since the distance between -3 and 9 is 12, $$b = \frac{12}{2} = 6$$.
Thus, |x-3| < 6.

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Re: Which of the following is true if -3<x<9?   [#permalink] 17 Nov 2017, 01:22
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