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08 Jan 2025, 01:28
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If \(p\) and \(q\) are, respectively, the least and greatest prime number solutions to \(\sqrt{x^2 - 4x + 4} < \sqrt{(x-20)^2}\), what is the value of \(p - q\)?
A. -9
B. -7
C. -5
D. -3
E. -1
A. -9
B. -7
C. -5
D. -3
E. -1
08 Jan 2025, 01:28
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Official Solution:
If \(p\) and \(q\) are, respectively, the least and greatest prime number solutions to \(\sqrt{x^2 - 4x + 4} < \sqrt{(x-20)^2}\), what is the value of \(p - q\)?
A. -9
B. -7
C. -5
D. -3
E. -1
Simplify:
\(\sqrt{x^2 - 4x + 4} < \sqrt{(x-20)^2}\)
\(\sqrt{(x - 2)^2} < \sqrt{(x-20)^2}\)
Recall that \(a^2 = |a|\):
\(|x - 2| < |x - 20|\)
This implies that \(x\) is closer to 2 than it is to 20. The halfway point between 2 and 20 is \(\frac{2 + 20}{2} = 11\). Thus, \(x\) must be somewhere in the green interval:
----------------(2)----------------(11)----------------(20)----------------
Thus, \(x < 11\).
Since \(p\) and \(q\) are, respectively, the least and greatest prime numbers in this range, \(p = 2\) and \(q = 7\). Therefore, \(p - q = 2 - 7 = -5\).
Answer: C
If \(p\) and \(q\) are, respectively, the least and greatest prime number solutions to \(\sqrt{x^2 - 4x + 4} < \sqrt{(x-20)^2}\), what is the value of \(p - q\)?
A. -9
B. -7
C. -5
D. -3
E. -1
Simplify:
\(\sqrt{x^2 - 4x + 4} < \sqrt{(x-20)^2}\)
\(\sqrt{(x - 2)^2} < \sqrt{(x-20)^2}\)
Recall that \(a^2 = |a|\):
\(|x - 2| < |x - 20|\)
This implies that \(x\) is closer to 2 than it is to 20. The halfway point between 2 and 20 is \(\frac{2 + 20}{2} = 11\). Thus, \(x\) must be somewhere in the green interval:
----------------(2)----------------(11)----------------(20)----------------
Thus, \(x < 11\).
Since \(p\) and \(q\) are, respectively, the least and greatest prime numbers in this range, \(p = 2\) and \(q = 7\). Therefore, \(p - q = 2 - 7 = -5\).
Answer: C
