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If |p-5| =3 and |q-3| = 5, which of the following statements must be

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Bunuel
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If |p-5| =3 and |q-3| = 5, which of the following statements must be [#permalink] New post   09 Jun 2020, 05:15
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E

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If \(|p-5| =3\) and \(|q-3| = 5\), which of the following statements must be true?


A. \(p + q > 0\)

B. \(pq ≥ 0\)

C. \(|p| = |q|\)

D. \(|p| ≥ |q|\)

E. \(-6 ≤ p-q ≤ 10\)
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E
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If |p-5| =3 and |q-3| = 5, which of the following statements must be [#permalink] New post   09 Jun 2020, 05:46
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Bunuel wrote:
If \(|p-5| =3\) and \(|q-3| = 5\), which of the following statements must be true?


A. \(p + q > 0\)

B. \(pq ≥ 0\)

C. \(|p| = |q|\)

D. \(|p| ≥ |q|\)

E. \(-6 ≤ p-q ≤ 10\)


\(|p-5| =3\)

i.e. p = 8 or 2



\(|q-3| = 5\)

i.e. q = 8 or -2



i.e. 0 ≤ p+q ≤ 16 Option A OUT

i.e. -16 ≤ p*q ≤ 64 Option B OUT

Option C and D are Out because lpl may be 2 and lql may be 8

i.e. -6 ≤ p-q ≤ 10

Answer: Option E
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Re: If |p-5| =3 and |q-3| = 5, which of the following statements must be [#permalink] New post   08 Jul 2023, 07:04
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Given that \(|p-5| =3\) and \(|q-3| = 5\) and we need to find which of the following statements must be true?

|p-5| = 3
=> p - 5 = 3 or p - 5 = -3
=> p = 8 or p = 2

|q-3| = 5
=> q - 3 = 5 or q - 3 = -5
=> q = 8 or q = -2

A. p + q > 0
=> This is true for p = 8, q = 8 => p + q = 16
And false for p = 2, q = -2 => p + q = 0
=> FALSE

B. pq > 0
=> This is true for p = 8, q = 8 => pq = 64
And false for p = 8, q = -2 => pq = -16
=> FALSE

C. |p| = |q|
This is true for p = q = 8
And false for p = 2 and q = 8
=> FALSE

D. |p| ≥ |q|
This is true for p = q = 8
And false for p = 2 and q = 8
=> FALSE

E. -6 ≤ p-q ≤ 10
This is true for all 4 possible cases
p = 8, q = 8 => p - q = 0
p = 8, q = -2 => p - q = 10
p = 2, q = 8 => p - q = -6
p = 2, q = -2 => p - q = 4
=> TRUE

So, Answer will be E
Hope it helps!

Watch the following video to learn the Basics of Absolute Values

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Re: If |p-5| =3 and |q-3| = 5, which of the following statements must be [#permalink] New post   09 Jun 2020, 07:13
=> |P-5| = 3
=> Case 1: +P-5 = 3 => P = 8
=> Case 2: -P+5 = 3 => P = 2

=> |Q-3| = 5
=> Case 1: +Q-3 = 5 => Q = +8
=> Case 2: -Q+3 = 5 => Q = -2

Putting values in options one by one
A. P+Q > O
=> Case 1: P+Q = 8+8 > 0 (correct)
=> Case 2: p+Q = -2+2 => 0 = 0 (incorrect)

B. PQ≥0
=> Case 1: PQ = 8*8 > 0 (correct)
=> Case 2: PQ = -2*2 < 0 (incorrect)

C. |P| = |Q|
=> Case 1: |2| = |-2| = 2=2 (correct)
=> Case 2: |8| = |-2| = 8≠2 (incorrect)

D. |P| ≥ |Q|
=> Case 1: |8| ≥ |8| = (correct)
=> Case 2: |2| < |8| = (incorrect)

E. -6≤P-Q≤10
=> Case 1: P-Q = 8-(-2) = 10 (maximum value) (correct)
=> Case 2: P-Q = 2-8 = -6 (minimum value) (correct)

Answer is E

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If |p-5| =3 and |q-3| = 5, which of the following statements must be [#permalink] New post   03 Oct 2020, 00:07
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Solving for the 2 Absolute Value Inequalities:

I.
-(3) = p - 5 = +3


p = +8 ; OR ; p = +2


II.
-(5) = q - 3 = +5


q = +8 ; OR ; q = -(2)



WOTF MUST be True - if we find a Case in which the Statement can be FALSE, we Eliminate



A. p + q > 0


IF: p = +2 and q =-(2)

then this Statement is FALSE ---ELIMINATE


B. pq >/= 0

Same case as used in Answer A proves this Statement FALSE ---- ELIMINATE



C. [p] = [q]

IF: p = +2 and q = +8

Then this Statement is FALSE ---- ELIMINATE



D. [p] >/= [q]

Same Case as used in Answer C proves this Statement can be FALSE ---- ELIMINATE




By process of Elimination it must be E, but just to check:


E. -(6) </= p - q </= +10



p can equal = +8 OR +2

q can equal = +8 OR -(2)


(1st) the MAXIMUM Value of the Expression (p - q) is obtained when the Value of p is at its MAXIMUM Value and the Value of q is at its MINIMUM Value --- this happens where:

p = +8
q = (-)2

p - q = 8 - -(2) = 8 + 2 = +10

+10 is the MAXIMUM Value that (p - q) can take

therefore; (p - q) </= +10


(2nd)the MINIMUM Value of the Expression (p - q) occurs where the Value of p is at its MINIMUM and the Value of q is at its MAXIMUM --- this occurs where:

p = +2
q = +8

p - q = 2 - 8 = -(6)

-(6) is the MINIMUM Value that (p - q) can take


therefore: -(6) </= p - q



TOGETHER, -E- is proven that it must be TRUE:


-(6) </= p - q < /= + 10

-E-
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If |p-5| =3 and |q-3| = 5, which of the following statements must be [#permalink]
03 Oct 2020, 00:07
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