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If 8 < x < 9, and x^2 = (10 – y)(10 + y), which of the following is a

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Bunuel
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If 8 < x < 9, and x^2 = (10 – y)(10 + y), which of the following is a [#permalink] New post  21 Oct 2021, 07:30
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If \(8 < x < 9\), and \(x^2 = (10 – y)(10 + y)\), which of the following is a possible value for y?

(A) –7
(B) –6
(C) 3
(D) 4
(E) 5
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BrentGMATPrepNow
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Re: If 8 < x < 9, and x^2 = (10 – y)(10 + y), which of the following is a [#permalink] New post  21 Oct 2021, 08:12
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Bunuel wrote:
If \(8 < x < 9\), and \(x^2 = (10 – y)(10 + y)\), which of the following is a possible value for y?

(A) –7
(B) –6
(C) 3
(D) 4
(E) 5


If \(8 < x < 9\), then \(8^2 < x^2 < 9^2\).
Simplify to get: \(64 < x^2 < 81\)

Since \(x^2 = (10 – y)(10 + y)\), we can substitute this value into the above inequality to get: \(64 < (10 – y)(10 + y) < 81\)
Expand to get: \(64 < 100 – y^2 < 81\)
Subtract \(100\) from all sides to get: \(-36 < –y^2 < -19\)
Multiply all sides by \(-1\) to get: \(36 > y^2 > 19\) [since we multiplied all sides of the inequality by a NEGATIVE value, we REVERSED the direction of the inequality symbols]

At this point, we can easily see that \(y = 5\) satisfies the inequality \(36 > y^2 > 19\)

Answer: E

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Re: If 8 < x < 9, and x^2 = (10 – y)(10 + y), which of the following is a [#permalink] New post  21 Oct 2021, 07:47
8 < x < 9

\(X^2\) = (10 - y)(10 + y)

\(X^2\) = \(10^2\) - \(y^2\)

X = √(\(10^2\) - \(y^2\))

Substitute value given in answer option, for correct option it has to be in between 8 and 9.

Random substitute (E)

X = √(\(10^2\) - \(5^2\))

X = √(100 - 25) = √75

(8^2 = 64 and 9^2 = 81)
75 Value lies between \(8^2\) and \(9^2\)

Correct Option E

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Re: If 8 < x < 9, and x^2 = (10 y)(10 + y), which of the following is a [#permalink] New post  10 Mar 2023, 04:36
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Re: If 8 < x < 9, and x^2 = (10 y)(10 + y), which of the following is a [#permalink]
10 Mar 2023, 04:36
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