mynamesmurph
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View attachment 4843
Can you please explain the reasoning and significance of the statement
No, x is not always positive. For instance, if x = -3, then x is not positive.Can you please explain the reasoning and significance of the statement
"Note that x2 is non-negative for any real number x, so 2 + x2 is positive. Then ........ is negative.
I see that any quantity of x will be positive, but how does it change the value to 2 + x^2?
No, x is not always positive. For instance, if x = -3, then x is not positive.
The text actually says that, regardless of the value of x, the square of that value will always be positive, or at least non-negative. (Strictly speaking, if x = 0, then x2 = 0, which is not "positive", but is certainly not negative.)
I'm not sure what you mean by "changing" "the value" "to" "2 + x2"...? But look at how they use this value in the example, relating 2 + x2 = -2 - x2 = -1(2 + x2), along with the definition of "absolute value": If you take the absolute value of a negative, then you evaluate by taking the bars off and putting a "minus" sign on whatever was inside. If you have a variable expression for which you don't know the value of the variable, you may not be able to tell if the expression is positive or negative, so you may not be able to tell whether or not you need a "minus" sign when you remove the bars.
However, in your specific case, since 2 + x2 must be positive, no matter what x might be, then -(2 + x2) must be negative. And this tells you that you can remove the bars, because you do know what sign you have on the insides.
View attachment 4843
Here is a screenshot of the example. Can you please explain the reasoning and significance of the statement
Note that x2 is nonnegative for any real number x, so 2 x2 is positive. Then ........ is negative.
I see that any quantity of x will be positive, but how does it change the value to 2 + x^2?