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If |x - 5| = 2|x - 8|, then what is the value of x? (1) |x2 – 100| >

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If |x - 5| = 2|x - 8|, then what is the value of x? (1) |x2 – 100| > [#permalink] New post   31 Jan 2020, 11:42
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If |x - 5| = 2|x - 8|, then what is the value of x?

(1) \(|x^2 – 100| > 50\)

(2) \(|x^2 – 49| =0\)
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D
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If |x - 5| = 2|x - 8|, then what is the value of x? (1) |x2 – 100| > [#permalink] New post   31 Jan 2020, 12:24
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If |x - 5| = 2|x - 8|, then what is the value of x?
--> square the both sides:
\((x-5)^{2} = (2(x-8))^{2}\)

--> \(x^{2}-10x+25= 4(x^{2}-16x+64)= 4x^{2}-64x+ 256\)

\(3x^{2}-54x+231=0\)

\(x^{2}-18x+77=0\)
--> (x-7)(x-11)=0
x could be 7 or 11.

(Statement1): \(|x^{2}–100|>50\)
--> if x=7, then |49-100|=51 >50 (√)
--> if x=11, then |121-100|= 21 >50 (ø)

x is equal to 7
Sufficient

(Statement2): \(|x^{2}–49|=0\)
--> if x=7, then |49-49| =0 (√)
--> if x=11, then |121-49| ≠0
x is equal to 7
Sufficient

The answer is D.
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If |x - 5| = 2|x - 8|, then what is the value of x? (1) |x2 – 100| > [#permalink] New post   31 Jan 2020, 12:40
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lnm87 wrote:
If |x - 5| = 2|x - 8|, then what is the value of x?

(1) \(|x^2 – 100| > 50\)

(2) \(|x^2 – 49| =0\)



Let us solve the given equation in the problem stem:

\(|x - 5| = 2|x - 8|\)

Removing the absolute value sign from both sides and introducing \(±\):

\(=> (x - 5) = ± 2(x - 8)\)

\(=> (x - 5) = +2(x - 8)\) OR \((x - 5) = -2(x - 8)\)

\(=> x - 5 = 2x - 16\) OR \(x - 5 = -2x + 16\)

\(=> x = 11\) OR \(x = 7\) ... (i)


Let us now check the statements:

Statement 1: \(|x^2 – 100| > 50\)

Verifying with the values of \(x\) obtained in (i):

\(x = 11: |121 - 100| = 21 > 50\) --- false

\(x = 7: |49 - 100| = |-51| = 51 > 50\) --- true

Thus, we have \(x = 7\) (unique) - Sufficient


Statement 2: \(|x^2 – 49| =0\)

Verifying with the values of \(x\) obtained in (i):

\(x = 11: |121 - 49| = 0\) --- false

\(x = 7: |49 - 49| = 0\) --- true

Thus, we have \(x = 7\) (unique) - Sufficient


Answer D
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Re: If |x - 5| = 2|x - 8|, then what is the value of x? (1) |x2 – 100| > [#permalink] New post   31 Jan 2020, 19:00
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lnm87 wrote:
If |x - 5| = 2|x - 8|, then what is the value of x?

(1) \(|x^2 – 100| > 50\)

(2) \(|x^2 – 49| =0\)


I've got some math in the question stem, so I'll start by writing it down and then simplifying it.

|x - 5| = 2|x - 8|

I like to think of absolute values containing subtraction as being "distances". For example, |x - 5| is the distance between x and 5 on a number line, and |x - 8| is the distance between x and 8 on a number line.

So, this question stem says "the distance between x and 5 on a number line is twice the distance between x and 8 on a number line."

Or in other words, "x is twice as far from 5 as it is from 8."

Sketch a number line to see where this will happen:

-----------5---------x----8-------

-----------5--------------8--------------x--------

A little guessing and testing confirms that x = 7 and x = 11 are the two cases where this is true.

So, the question stem says "either x = 7, or x = 11." That's a much easier translation. It also makes the statements easier to work with: we only ever need to test the values 7 and 11! Now on to the statements.

Statement 1: Given that |x^2 - 100| is greater than 50, can I exclude either 7 or 11?

Yes. x can't equal 11, because |11^2 - 100| is less than 50. And it can't be any other number, aside from 7, since the question stem says that x is definitely either 7 or 11. Therefore, x = 7, and the statement is sufficient.

Statement 2: This lets me exclude x = 11 as well. Therefore, x = 7, and the statement is sufficient.

The correct answer is D.
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Re: If |x - 5| = 2|x - 8|, then what is the value of x? (1) |x2 100| > [#permalink] New post   30 Jul 2023, 01:59
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Re: If |x - 5| = 2|x - 8|, then what is the value of x? (1) |x2 100| > [#permalink]
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