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

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

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New post 15 Nov 2017, 01:54
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C
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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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Which of the following is true if -3<x<9? [#permalink]

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New post 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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Which of the following is true if -3<x<9? [#permalink]

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New post 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

Answer C

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

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New post 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.

Therefore, the answer is C.

Answer: C
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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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