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D
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12 Days of Christmas 🎅 GMAT Competition with Lots of Questions & Fun
What is the median of all values of x which satisfy \(|x − 2| = |x − 4| + |x − 6|\)?
A. 2
B. 4
C. 5
D. 6
E. 8
What is the median of all values of x which satisfy \(|x − 2| = |x − 4| + |x − 6|\)?
A. 2
B. 4
C. 5
D. 6
E. 8
|
21 Dec 2023, 08:46
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Bunuel
Use the concept of distance for absolute values discussed here: https://youtu.be/oqVfKQBcnrs
\(|x − 2| = |x − 4| + |x − 6|\)
Draw the number line. We are given that 'Distance of x from 2 = Distance of x from 4 + Distance of x from 6'
-----------------2 ---------------4 -------------6----------------
x cannot be to the left of 4 because it is closer to 2 in that case. x can be at 4 because then
Distance of x from 2 = Distance of x from 4 + Distance of x from 6
2 = 0 + 2
To the right of 4 the sum of distances from 4 and 6 is lower compared with distance from 2 but as we go more to the right, sum of distances from 4 and 6 in increasing more as compared to distance from 2. At x = 8, again
Distance of x from 2 = Distance of x from 4 + Distance of x from 6
6 = 4 + 2
When we go further to the right of 8, for each step we take, sum of distance from 4 and 6 will increase by 2 but distance from 2 will increase by 1 only. So the distance will never become equal again.
Hence x can take only two values 4 and 8 and their median is 6.
Answer (D)
21 Dec 2023, 05:42
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GMAT Club Official Explanation:
What is the median of all values of x which satisfy \(|x − 2| = |x − 4| + |x − 6|\)?
A. 2
B. 4
C. 5
D. 6
E. 8
The critical points (aka key points or transition points) are 2, 4, and 6 (the values of \(x\) for which the expressions in the absolute values become 0).
So, we should consider the following four ranges:
If \(x < 2\), then:
- \(x - 2 < 0\), and thus \(|x - 2| = -(x - 2)\);
\(x - 4 < 0\), and thus \(|x - 4| = -(x - 4)\);
\(x - 6 < 0\), and thus \(|x - 6| = -(x - 6)\).
Hence, in this range \(|x − 2| = |x − 4| + |x − 6|\) becomes \(-(x − 2) = -(x − 4) - (x − 6)\). This gives \(x = 8\). Since this value is out of the range we consider, then for this range the given equation does not have a solution.
If \(2 \leq x < 4\), then:
- \(x - 2 \geq 0\), and thus \(|x - 2| = x - 2\);
\(x - 4 < 0\), and thus \(|x - 4| = -(x - 4)\);
\(x - 6 < 0\), and thus \(|x - 6| = -(x - 6)\).
Hence, in this range \(|x − 2| = |x − 4| + |x − 6|\) becomes \(x − 2 = -(x − 4) - (x − 6)\). This gives \(x = 4\). Since this value is out of the range we consider, then for this range the given equation does not have a solution.
If \(4 \leq x < 6\), then:
- \(x - 2 \geq 0\), and thus \(|x - 2| = x - 2\);
\(x - 4 \geq 0\), and thus \(|x - 4| = x - 4\);
\(x - 6 < 0\), and thus \(|x - 6| = -(x - 6)\).
Hence, in this range \(|x − 2| = |x − 4| + |x − 6|\) becomes \(x − 2 = x − 4 - (x − 6)\). This gives \(x = 4\). Since this value is within the range we consider, then \(x = 4\) is a valid solution.
If \(x \geq 6\), then:
- \(x - 2 \geq 0\), and thus \(|x - 2| = x - 2\);
\(x - 4 \geq 0\), and thus \(|x - 4| = x - 4\);
\(x - 6 \geq 0\), and thus \(|x - 6| = x - 6\).
Hence, in this range \(|x − 2| = |x − 4| + |x − 6|\) becomes \(x − 2 = x − 4 + x − 6\). This gives \(x = 8\). Since this value is within the range we consider, then \(x = 8\) is a valid solution.
Therefore, two values satisfy \(|x − 2| = |x − 4| + |x − 6|\), 4 and 8. The median of these values is (4 + 8)/2 = 6.
Answer: D.
