Questions tagged [absolute-value]

For questions about or involving the absolute value function also known as modulus function.

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Removal of absolute signs

I'm a high school student and I'm currently studying differential equations. I encountered this one question: An object is dropped from a cliff. The object leaves with zero speed, and t seconds later ...
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Question about the proof that if x is a real number then there exists a postive integer n such that -n < x < n. [closed]

One of my homework questions of real analysis went like this: Prove that if $x$ is a real number then there exists a postive integer $n$ such that $-n < x < n$. My answer was this: First notice ...
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Help with ODE - Cannot get to computed solution - Possible problem with absolute values?

I'm struggling with the following ODE, which computed gives a different answer than the one I got doing it analytically. Maybe I have made a silly mistake or I do not get how to work with the absolute ...
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what does m<|x| imply?

I know $|x|\le M \ \implies -M\le x \le M$ but what does $m\le|x|$ imply? Can I just plainly interpret this as $m\le x$? and if I say $m\le x \le M$ does this still mean $-M \le x \le M$?
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Solving an absolute value inequality with fractions

I'm having a hard time figuring out this inequality: $\bigg|\dfrac{x-4}{x+5}\bigg| \le 4$ I use one of the absolute value properties, which results in: $-4 \leq \dfrac{x-4}{x+5} \leq 4$. From there, I ...
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How would I find the critical values of an absolute value function? $y=|x+1| + |x-1|$ over $[3, -2]$

Would I just have to graph this to find the critical points? Because I tried the method where you rewrite the expressions in the absolute brackets as $\sqrt{(x+1)^2}$ and $\sqrt{(x-1)^2}$ and take the ...
In the middle of problem i've got inequality: Find all $z\in\mathbb{C}$ such as: $$||z|-1|<|z+1|$$ So i start to elaborate on that for $z=x+iy$: $$|\sqrt{x^2+y^2}-1|<\sqrt{(x+1)^2+y^2}$$ Now ...