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

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If |p-5| =3 and |q-3| = 5, which of the following statements must be [#permalink]
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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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Re: If |p-5| =3 and |q-3| = 5, which of the following statements must be [#permalink]
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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