If x is a non-negative integer and |8 - |x + 3|| = 7, then find the nu

Question

If x is a non-negative integer and |8 - |x + 3|| = 7, then find the number of values of x that satisfy the given absolute value inequality?

A. 0 
 B. 1 
 C. 2 
 D. 3 
 E. 4

Bunuel · Answer

|8 - |x + 3|| = 7

8 - |x + 3| = 7 or 8 - |x + 3| = -7

The first one gives  |x + 3| = 1, which implies x = -2 or x = -4. Since given that x is a non-negative integer, we discard both values.

The second one give |x + 3| = 15, which implies x = 12 or x = -18. Again, we discard -18.

So, only one value: x = 12.

Answer: B.

GMATinsight · Answer

We have been given
 |8 - |x + 3|| = 7

- Now, the absolute values of Left hand side must be 7
- But the value of  |x + 3| is Non-negative due to modulus sign
- Therefore, |x + 3| has to be 1 or 15 for the absolute value of left side part to be 7
- for |x + 3| to be 1, x can be -4 or -2 (rejected due to being negative) and for x+3 to be -15, x can be -18 (Rejected) and for x+3 to be 15, x can be 12 hence Only one non-negative possible value of x
 
 Answer: Option B

Related Video:

https://www.youtube.com/watch?v=dnWC37yPs7s

Bunuel · Answer

That's not correct.

|x + 3| can also be 15, for |8 - |x + 3|| to be 7.

haianh · Answer

x>=0,  so: |x+3|=x+3, then:
|8-|x+3|| = |8-(x+3)| = |5-x|=7
solve it:
5-x=7 or 5-x=-7
  	o   x=-2 (discard) or x=12 (valid). 
Answer: B

SkyBulb · Answer

can we do it simply as in, we know x is non-negative so surely x+3 is non negative too and open |x+3| to (x+3)
so we get 8-(x+3) = 7 and 8-(x+3) = -7 which gives -2 and 12 as answers. -2 is not possible as its non negative, so only 12 left.

Bunuel · Answer

Yes, this is a valid shortcut.

Since x is non-negative, x + 3 is definitely positive, so |x + 3| = x + 3.

Thus:

|8 - |x + 3|| = 7

becomes:

|8 - (x + 3)| = 7

So:

8 - (x + 3) = 7 or 8 - (x + 3) = -7

This gives x = -2 or x = 12. Since x must be non-negative, only x = 12 works.

MehakKaushal · Answer

Hi  Bunuel 
Looks like the options are incorrect. I dont see a 12 as option B.

Bunuel · Answer

Please check the question carefully:

...find  the number of values of x  that satisfy the given equation

So the question does not ask for the value of x. It asks how many values of x are possible.

Since the only possible value is x = 12, the number of values is 1. Therefore, option B is correct.

If x is a non-negative integer and |8 - |x + 3|| = 7, then find the nu

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GMAT Problem Solving (PS) Questions

If x is a non-negative integer and |8 - |x + 3|| = 7, then find the nu

Prep Toolkit

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