If |p 6| + p = 6, which of the following must be true?

Question

If |p − 6| + p = 6, which of the following must be true?

A. p = 0
B. p = -6
C. p = 6
D. p > -6
E. p <= 6

Bunuel-Can you please explain! and Can we write |p−6| as (p-6) and -(p-6)( I mean when modulus is withdrawn) ?

Bunuel · Accepted Answer

|p-6|+p=6

|p-6|=6-p

|p-6|=-(p-6)

|x|=-x  means, that  x\leq{0} . So:

p-6\leq{0}

p\leq{6} .

Answer: E.

bagdbmba · Answer

Bunuel-Can we write |p−6| as (p-6) and -(p-6)( I mean when modulus is withdrawn) ?

Bunuel · Answer

When  p\leq{6} , then  |p-6|=-(p-6)=6-p .
When  p\geq{6} , then  |p-6|=p-6 .

bagdbmba · Answer

So, only 6 satisfies both |p−6|= (p-6) and |p−6|=-(p-6)

Then why the answer is not ONLY 6 for the above qs?

Bunuel · Answer

These are not simultaneous equations.

If...
If...

Since given that  |p-6|=-(p-6)=6-p , then  p\leq{6} .

bagdbmba · Answer

Ah! got it...

so the question itself already limits the scope of modulus. Is it below 700 qs? BTW,don't get the "If....If...." part in your above reply !

Bunuel · Answer

I think it's a sub-600 question, because it's about basic property (definition) of absolute value:

When  x \leq {0}  then  |x|=-x , or more generally when  some \ expression\leq{0}  then  |some \ expression|={-(some \ expression)} .

When  x \geq {0}  then  |x|=x , or more generally when  some \ expression\geq{0}  then  |some \ expression|={some \ expression} .

P.S. What highlighted part?

WholeLottaLove · Answer

If |p−6|+p=6, which of the following must be true?

There are two ways to solve, one of which is to just pick numbers.

The other way:
|p-6|+p=6
|p-6| = 6-p
|p-6| = -(p-6)
|x| = -x (i.e. the value inside the absolute value bars is negative)
So, (p-6) is negative when p≤6

(E)

A. p=0
B. p=-6
C. p=6
D. p>-6
E. p<=6

nancerella · Answer

without plugging in numbers, how would you know that this |p-6| = -(p-6) would be p<=6 and not just p<6? 
the way i'm solving it, this |x| = -x condition would be met when (p-6)<0 and therefore -(p-6)<0 yields p<6. but why also p=6?

thanks!

Bunuel · Answer

Because when  x \leq {0}  then  |x|=-x , or more generally when  some \ expression\leq{0}  then  |some \ expression|={-(some \ expression)} .

We have  |p-6|=-(p-6) , thus  p-6\leq{0}  -->  p\leq{6} .

WholeLottaLove · Answer

Why can't we (or how would we know not to) solve in the following manor:

If |p−6|+p=6, which of the following must be true?

|p−6|+p=6
p≥6
p-6+p=6
p=6

p<6
-(p-6)+p=6
-p+6+p=6
6=6

Thanks!

Bunuel · Answer

Actually we can. You got the correct result.

When p<6, you got that the equation holds for any value of p.
When p≥6, you got that the equation holds for p=6.

So, the equation holds for p<6 and p=6 --> p<=6.

Hope it's clear.

WholeLottaLove · Answer

because for p<6 and p≥6 the intersection is p ≤6, correct?

Bunuel · Answer

No. Because p<6 and p=6 means p<=6.

clipea12 · Answer

or simply |p−6|+p=6 
|p−6|=6-p
means 6-p >=0 therefore 6>=p or p<= 6
hope it helps

arshu27 · Answer

Dear  Bunuel ,

if you substitute p=0 in the given equation then also you get LHS=RHS.

am i missing out anything here?

Bunuel · Answer

The question is  "which of the following  must  be true" , NOT  "which of the following  could  be true" .

Check other Must or Could be True Questions  HERE .

arshu27 · Answer

Hi  Bunuel ,

if we substitute p=0 then the answer must be 6. There can be no other answer right?

Bunuel · Answer

Not sure I understand what you mean...

p = 0, as well as any other value of p less than or equal 6 satisfies |p − 6| + p = 6.

If |p 6| + p = 6, which of the following must be true?

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If |p 6| + p = 6, which of the following must be true?

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