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# Inequality - Absolute value

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Inequality - Absolute value [#permalink]  10 Apr 2011, 21:20
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40% (01:57) correct 59% (00:43) wrong based on 1 sessions
If x is a number such that –2 ≤ x ≤ 2, which of the following has the largest possible absolute value?

3x – 1
x2 + 1
3 – x
x – 3
x2– x

Can this be done Algebrically / graphically.

I got the answer using substitution but found that time consuming.
[Reveal] Spoiler: OA

Last edited by IEsailor on 11 Apr 2011, 10:27, edited 1 time in total.
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Re: Inequality - Absolute value [#permalink]  10 Apr 2011, 22:02
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Why not A?

3x – 1=3*(-2)-1=-7, absolute 7.
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Re: Inequality - Absolute value [#permalink]  11 Apr 2011, 14:17
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IEsailor wrote:
If x is a number such that –2 ≤ x ≤ 2, which of the following has the largest possible absolute value?

3x – 1
x2 + 1
3 – x
x – 3
x2– x

Can this be done Algebrically / graphically.

I got the answer using substitution but found that time consuming.

If you know how to plot graphs, this is very straight forward, most of the expressions are very simple to plot.

In case you want to do algebraically, here are a couple of simple rules you need to know:
(a) For linear expressions, the maxima or minima will always be at the extreme ends of a range
(b) For quadratic expressions, the maxima or minima will always either be the extreme end of a range of the global max/min of the function

3x – 1 : At ends, value is 5 and -7
x2 + 1 : At ends, value is 5 and global min value is at x=0, where it is 1
3 – x : At ends, value is 1, 5
x – 3 : At ends, value is -5,-1
x2– x : At ends, value is 2,6. Global min value is at x=0.5, where it is -0.25

(Its easy to tell what the global min for these simple quadratic functions is knowing where there roots are, it is half way between the roots)

Clearly, answer is A
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Re: Inequality - Absolute value [#permalink]  11 Apr 2011, 23:47
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gmat1220 wrote:
hello shrouded1
I don't understand your last statement. Pls can you elaborate.

(Its easy to tell what the global min for these simple quadratic functions is knowing where there roots are, it is half way between the roots)

On way to get the global min is applying the calculus. -
y = x^2+1
dy/dx = 0 or 2x = 0. So the min occurs at x=0

Similarly y = x^2-x
dy/dx = 0 or 2x - 1 = 0 . So the min occurs at x = 1/2 = 0.5

Thats implicitly what I did. But I used a little trick so I dont have to do the calculations.
For any quadratic function with real roots, the minima/maxima (depending on wether the coefficient of x^2 is positive or negative) will occur at the mean of the roots. (This is very easy to prove, all I am saying is that the minima is -b/2a, which you can show by differentiation).

So for x^2-x = x(x-1) the roots are 0,1 and the minima will be at 0.5

For x^2+1, the roots are not real, but this function is simply a vertically translated version of x^2=0, hence it also minimizes/maximizes at the same level which is x=0.

Generalizing, you can always use the fornula x=(-b/2a) for the min or max valuation
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Re: Inequality - Absolute value [#permalink]  10 Apr 2011, 21:39
I don't know how to do graphically, but the idea is that we have to maximize the values of both the terms in the expression. In that case, the last option is the biggest one with -2 as the value :

x2– x = (-2)^2 -(-2) = 4 + 2 = 6

For the other options :

3x – 1 -> Max value in the given range is 3*2 - 1 = 5
x2 + 1 -> Max value in the given range is (2)^2 + 1 = 5
3 – x -> Max value in the given range is 3 - (-2) = 5
x – 3 -> Max value in the given range is 2 - 3 = -1

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Re: Inequality - Absolute value [#permalink]  10 Apr 2011, 21:40
Answer is E. As you see E increases when x is negative. So the highest value occurs at -2

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Re: Inequality - Absolute value [#permalink]  10 Apr 2011, 22:26
You are right.

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Re: Inequality - Absolute value [#permalink]  10 Apr 2011, 22:34
for those of you who havent already understood, options B and E are
X^2 + 1
X^2 - X

Also guys any other way apart from substitution ?
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Re: Inequality - Absolute value [#permalink]  10 Apr 2011, 23:15
There is trick used by designers. more often when you have rhe choice of negative and positive values and the argument is dependent on the sign of the number- negative number will be decisive. The people will overlook the negative value more than the positive the designer is going to use this trap. Since we should look in the opposite way first exhaust all the negative values to ensure there is no trap. And you are good!
IEsailor wrote:
for those of you who havent already understood, options B and E are
X^2 + 1
X^2 - X

Also guys any other way apart from substitution ?

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Re: Inequality - Absolute value [#permalink]  11 Apr 2011, 03:57
And most importantly read the question well, I had missed the absolute term first time i read the problem.
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Re: Inequality - Absolute value [#permalink]  11 Apr 2011, 10:05
Guys the OA is A. By substitution of (-2) in A we get -7 and hence an absolute value of 7

Last edited by IEsailor on 11 Apr 2011, 13:34, edited 1 time in total.
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Re: Inequality - Absolute value [#permalink]  11 Apr 2011, 12:01
The question is asking us to find the largest possible absolute value.

Hence by plugging in - 2 in the expression 3x -1 gives the largest possible absolute value 7.

So, Ans A
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Re: Inequality - Absolute value [#permalink]  11 Apr 2011, 13:56
The maximum absolute value is 7. Hence the answe is A
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Re: Inequality - Absolute value [#permalink]  11 Apr 2011, 23:40
hello shrouded1
I don't understand your last statement. Pls can you elaborate.

(Its easy to tell what the global min for these simple quadratic functions is knowing where there roots are, it is half way between the roots)

On way to get the global min is applying the calculus. -
y = x^2+1
dy/dx = 0 or 2x = 0. So the min occurs at x=0

Similarly y = x^2-x
dy/dx = 0 or 2x - 1 = 0 . So the min occurs at x = 1/2 = 0.5
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Re: Inequality - Absolute value [#permalink]  12 Apr 2011, 06:10
Thanks for the wonderful explanation Shrouded1. Kudos for you
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Re: Inequality - Absolute value [#permalink]  30 Apr 2011, 20:33
|3x-1| = |3* (-2) -1 | = 7
Rest all have values at max 5.
Hence A.
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Re: Inequality - Absolute value   [#permalink] 30 Apr 2011, 20:33
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