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
Absolute Values Dilemma
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- Stuart@KaplanGMAT
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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
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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- eaakbari
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Remember that the modulus operator gives nothing but the absolute value. If we have -2, |-2| = 2rainmaker 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
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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