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Absolute Values Dilemma

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Absolute Values Dilemma

by rainmaker » Wed Apr 14, 2010 8:04 am
Hi Guys,

I'm little confused regarding the modulus operator:

In equations such as x|x| < 2^x. If x<0, why is the left side negative.
For x<0, shouldn't |x| be -ve and because -ve * -ve = +ve, the left side should be positive.

Whenever we solve equations with absolute values, for x>0, we consider the value +ve and for x<0 -ve.

For eg: |5+2x| = 3

If x>0, 5+2x = 3. So x = -1

If x<0, -(5+2x) = 3, So x = -4

Thanks
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by Stuart@KaplanGMAT » Wed Apr 14, 2010 10:17 am
Hi,

absolute value is a measurement of distance; as such, it's always non-negative (i.e. it can be 0 or positive).

So, once you "do" the absolute value operation, you always end up with a non-negative result, regardless of the sign of the term inside the brackets.

Accordingly, if we have an expression such as:

x|x|,

the term inside the brackets is never negative after performing the operation.

if x = 2, we'd solve as:

2 * |2| = 2 * 2 = 4

if x = -2, we'd solve as:

-2 * |-2| = -2 * 2 = -4

You're dead on with how we solve absolute value equations; we look at the positive and negative cases because regardless of the sign inside the brackets, the result is positive. When you solve, therefore, you're figuring out "what could I put inside the brackets to generate the desired result".

As an aside, another way to solve is by squaring both sides of the equation - that often alleviates confusion. For example:

|5+2x| = 3

(|5 + 2x|)^2 = 3^2

(5 + 2x)(5+2x) = 9

25 + 20x + 4x^2 = 9

rearranging to a quadratic:

4x^2 + 20x + 16 = 0

4(x^2 + 5x + 4) = 0

4(x+4)(x+1) = 0

x = -4 or x = -1
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by eaakbari » Wed Apr 14, 2010 10:21 am
rainmaker wrote:Hi Guys,

I'm little confused regarding the modulus operator:

In equations such as x|x| < 2^x. If x<0, why is the left side negative.
For x<0, shouldn't |x| be -ve and because -ve * -ve = +ve, the left side should be positive.

Whenever we solve equations with absolute values, for x>0, we consider the value +ve and for x<0 -ve.

For eg: |5+2x| = 3

If x>0, 5+2x = 3. So x = -1

If x<0, -(5+2x) = 3, So x = -4

Thanks
Remember that the modulus operator gives nothing but the absolute value. If we have -2, |-2| = 2
Basically we say that if
x>0 then |x|=x
but of x <0 in order to get a positive absolute value, we will have to multiply x by -ve as -ve *-ve will be +ve
For eg If its -2
for -2 to become absolute 2
-(-2)

So basically when x<0, |x| = -x

HTH
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by rainmaker » Fri Apr 16, 2010 10:25 am
Thanks Stuart for the awesome explanation.

Thanks eaakbari, for the helpful insight!
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