sinhap07 wrote:

Bunuel wrote:

Nai222 wrote:

hey,

Can you please explain why it is negative? I thought when the two number are enclosed within an absolute value bracket, it becomes positive.

\(-xy\) is not negative it's positive. \(xy\) is negative, thus \(-xy=-(negative)=positive\).THEORY:

. . .

What function does exactly the same thing [as the square root sign]? The absolute value function: \(|x|=x\), if \(x\geq{0}\) and \(|x|=-x\), if \(x<0\)...

sinhap07 How is mod of -x equal to -x? Shouldnt it be x? sinhap07 , I assume you refer to

**Quote:**

\(|x|= -x\) , if \(x<0\)

For me, the rule can be counterintuitive. I've seen a few ways to think about the rule. Maybe they will help.

If x is negative, then |x| = -x

Possible ways to think about the rule:

1. "The negative of a negative is positive."

Think of the negative sign as signifying "opposite."

That is, the negative sign functions as "the negative of a negative number." And the negative of a negative number is positive. See

Bunuel above in bold.

Let x = -3. Per |x| = -x:

|-3| = 3, and +3 is the opposite of -3, thus +3 = -(-3)

2. OR think: "The negative sign on RHS means (-1) multiplied by a negative number \(x\)," thus

|x| = -x

|x| = (-1)(x)

|x| = (-1)(negative #)

|x| = a positive number

3. OR think (similar to #1): "in this rule there is a hidden minus sign."

With a number, the "two negatives" are easy to see

|-3| = 3

|-3| =

-(-3)BUT: |x| = -(x) = -x

With variable \(x\), it is easy to forget that there ARE two negative signs.

With the variable, there is only one minus sign on RHS... because the negative variable \(x\) already "contains" a minus sign.

We just don't (can't) write the minus sign twice with the variable.

|-3| = -(-3) = 3

|x| = -(x) = -x

Those two equations are functionally equivalent.

4. Summary - use any negative number, substituted for x, to see that, if x < 0 , then |x| = -x. Reasons:

|-3| = 3, where +3 is the opposite of -3 [+3 = -(-3)]; RHS is the negative of a negative number

|-3| = 3 = (-1)(-3)

|-3| = -(-3) = 3

The absolute value IS positive (or nonnegative). The sign of a negative variable can obscure that fact.

Hope that helps.

_________________

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