Events & Promotions
|
|
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
May 0312:30 AM EDT
-01:30 AM EDT
Learn how Kamakshi achieved a GMAT 675 with an impressive 96th %ile in Data Insights. Discover the unique methods and exam strategies that helped her excel in DI along with other sections for a balanced and high score.
May 0301:30 AM EDT
-02:30 AM EDT
Struggling to find the right strategies to score a 99 %ile on GMAT Focus? Riya (GMAT 715) boosted her score by 100-points in just 15 days! Discover how the right mentorship, tailored strategies, and an unwavering mindset can transform your GMAT prep.
May 0306:30 AM PDT
-08:30 AM PDT
Verbal trouble on GMAT? Fix it NOW! Join Sunita Singhvi for a focused webinar on actionable strategies to boost your Verbal score and take your performance to the next level.
May 0510:00 AM PDT
-11:00 AM PDT
GMAT Inequalities is a high-frequency topic in GMAT Quant, but many students struggle because the concepts behave differently from standard algebra. Understanding the right rules, patterns, and edge cases can significantly improve both speed and accuracy.
May 0612:30 AM EDT
-01:30 AM EDT
GMAT 615 to 715 in Just 15 Days! Riya’s 100-pt Improvement Strategy- May 06
08:30 AM PDT
-09:30 AM PDT
In Episode 3 of our GMAT Ninja Critical Reasoning series, we tackle Discrepancy, Paradox, and Explain an Oddity questions. You know the feeling: the passage gives you two facts that seem completely contradictory.... - May 08
08:00 AM PDT
-08:30 AM PDT
Join the special YouTube live-stream for selecting the winners of GMAT Club MBA Scholarships sponsored by Juno live. Watch who gets these coveted MBA scholarships offered by GMAT Club and Juno.
31 Dec 2023, 08:46
Kudos
Bookmarks
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
A. 2
B. 4
C. 5
D. 6
E. 8
31 Dec 2023, 08:46
Kudos
Bookmarks
Official Solution:
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 \( \geq 6\), 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 \(\frac{4 + 8}{2} = 6\).
Answer: D
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 \( \geq 6\), 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 \(\frac{4 + 8}{2} = 6\).
Answer: D
