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If |a – 5| = 11 and |2b – 6| = 8, what is the minimum possible value

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If |a – 5| = 11 and |2b – 6| = 8, what is the minimum possible value  [#permalink]

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New post 28 Jul 2017, 17:49
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If \(|a – 5| = 11\) and \(|2b – 6| = 8\), what is the minimum possible value of \(\frac{a}{b}\)?

(A) \(\frac{-6}{7}\)
(B) 6
(C) \(\frac{16}{7}\)
(D) -6
(E) -16

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If |a – 5| = 11 and |2b – 6| = 8, what is the minimum possible value  [#permalink]

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New post 28 Jul 2017, 19:07
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The expression |c - d| always means "the distance between c and d on the number line".

So the equation |a - 5| = 11 means "the distance between a and 5 is equal to 11". So a can be in either of these two places, 11 away from 5:

-a----------5------------a-----

and either a = -6 or a = 16.

Similarly, if we have (factoring out a 2 from the absolute value and canceling it) |b - 3| = 4, we know that the distance between b and 3 is equal to 4:

-----b-----3-----b------

and b = -1 or b = 7.

You could also use the 'cases method' here to find the possible values of a and b.

We can make a/b negative by using one negative and one positive value, and since we want the minimum value of a/b, we'll want a/b to be negative. So we can consider the two pairs of values that will make a/b negative (one positive value, one negative value) :

- if a = -6 and b = 7, then a/b = -6/7

- if a = 16 and b = -1, then a/b = 16/(-1) = -16

The smaller of these is -16, so that's the minimum value of a/b.
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Re: If |a – 5| = 11 and |2b – 6| = 8, what is the minimum possible value  [#permalink]

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New post 28 Jul 2017, 19:30
Gnpth wrote:
If \(|a – 5| = 11\) and \(|2b – 6| = 8\), what is the minimum possible value of \(\frac{a}{b}\)?

(A) \(\frac{-6}{7}\)
(B) 6
(C) \(\frac{16}{7}\)
(D) -6
(E) -16



Hi,

Ian has beautifully explained what an absolute modulus means on a number line.

You can find the value of variables by opening the MOD..
1) |a-5|=11..
a-5=11.... a=16
-(a-5)=11... a=-6
2)|2b-6|=8
2b-6=8... b=7
-(2b-6)=8... b=-1

For Min value of a/b..

If a/b is NEGATIVE, Max value of a and min value of b, AS we want max NUMERIC value and ensuring one value is NEGATIVE and one POSITIVE
If a/b is positive, min value of a and Max value of b, AS we want min NUMERIC value.

Here we have negative values too, so a/B should be NEGATIVE.

Ans = -16/1=-16
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Re: If |a – 5| = 11 and |2b – 6| = 8, what is the minimum possible value   [#permalink] 28 Jul 2017, 19:30
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If |a – 5| = 11 and |2b – 6| = 8, what is the minimum possible value

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