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In the real numbers ##5 > 1##. But in this statement ##5 < 1##. Therefore no such ##x## exists.

While this may be a single domain restriction that is true based on the inequalities found, it is quite the waste of time, considering the question asks to find where the function is increasing and decreasing - not just for a single domain restriction for where it is only increasing. The...

Rather than using the ##>## symbol, I like to find the critical points, where ##f'(x)=0##. So rather than using ##f'(x)>0## we can use ##f'(x)=0## to get ##x = 0## and ##x = 2## and ##x = -1##. Then try test points in between these critical points, to see if the function is increasing or...

One thing that may be helpful to ponder is how the gradient, ##m##, affects the line ##y=mx##. When ##m=1## the line will be on a ##45°## angle. However, when the gradient is doubled to ##m=2##, the angle does not double with it to ##90°##, rather the angle changes to ##63°##. When ##m=3##, the...

The ##I## should trigger the phrase "##I##dentity Matrix". It was the reason ##I## was chosen rather than some other letter in the first place.

Thanks, I've edited.

Yes, I think we can do exactly that. Consider the distance from ##x## to ##0## where ##x<0##. ##|-x|=|x|<c## If ##|-x|<c## then ##-x<c## and this implies ##x>-c##. Consider the distance from ##x## to ##0## where ##x>0##. ##|x|<c## If ##|x|<c## then ##x<c##. Combining we have that if...

Well if we are focussing in on language, you should say "n times" or "n factors" etc. but not "n terms" as this expression has one term, ##x^n## and each ##x## is a factor not a term. When we take the ##||## of ##x## we are taking the distance of ##x## from zero. Distance is always a positive...

Hello! Yes, keep learning the language of maths. Formulating questions can often lead to solving the questions you are formulating. As for the absolute value, you really can think about it as "length" or "distance", because the distance from me to you is the same distance as you to me. So if I...

Where exactly are you having trouble? It seems to me you are asking why two distinct yet equivalent statements compute the same result. Is this question akin to asking, "Is ##x## the distance from ##0## to ##x##, or from ##x## to ##0##"?

There might be, but I don't know of it. With three unknowns ## A, B ## and ## C ## we usually need three equations to find their values. The only way to extract three equations from the given equation is to equate the three coefficients.

Yes you can solve it that way. By subtracting the ## (1)x^2 ## from both sides, we have ## (0)x^2 ## on the RHS. Now equating coefficients we get: ## -2A - 1 = 0 ## ## -2A = 1 ## ## A = -1/2 ## Just as in the given solution

I read the three dots to mean, "Keeping following this pattern from my left, and you'll get this next thing to my right". My guess is this is not the official definition. 😀

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I thought you might be able to do this: ## 0=2x - 1 + \sqrt{2-x}## ## 0=(2x - 1 + \sqrt{2-x})(2x - 1 - \sqrt{2-x}) ## ## 0=4x^2 - 4x + 1 - (2-x) ## ## 0=4x^2 - 3x - 1 ## ## 0=(x-1)(4x+1) ## But by multiplying both sides by the conjugate we are also adding extraneous solutions as it is...