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

Question

If |x - 5| = 2|x - 8|, then what is the value of x?

(1)  |x^2 – 100| > 50

(2)  |x^2 – 49| =0

lacktutor · Accepted Answer

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.

sujoykrdatta · Answer

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

ccooley · Answer

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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