# Questions tagged [absolute-value]

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

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### Values of $m$ such that $|x^2-2x+m| + 2x + 1$ has $3$ extrema

I was given the following question: Find the values of $m$ for which the curve $y=|x^2-2x+m| + 2x + 1$ has $3$ extrema. My teacher suggested that we should use the quadratic formula $(b^2-4ac)$ and ...
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### Simplifying the expression $\frac{1}{3} \ln (x+2)^3+\frac{1}{2}\left[\ln x-\ln \left(x^2+3 x+2\right)^2\right]$

Express as a single logarithm. Simplify. $$\frac{1}{3} \ln (x+2)^3+\frac{1}{2}\left[\ln x-\ln \left(x^2+3 x+2\right)^2\right]$$ So I am posting the question, how I solved it and then how the TA ...
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### Evauate this limits without using $l´hopitals: \lim_\limits{x \to 0}\frac{x^2}{|x|}$

$$\lim_\limits{x \to 0} \frac{x^2}{|x|}$$ my argument is that we are looking for values near zero, not in zero hence we can get rid of the x denominator with one in the numerator then evaluating the ...
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1 vote
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### Algebraic rules of absolute value

In evaluating elementary $\epsilon$, $\delta$ proofs of limits, one often sees the following sort of move: $$\left|2x - 8\right| = \left|2(x-4)\right| = 2\left|x - 4\right| \dots$$ (See e.g. here (14:...
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### Why does the absolute value of difference between two consecutive integers in a finite string of positive reals always end with 0s?

Suppose I take a finite string of positive reals 1 4 19 3 In the first step, I find the absolute difference between consecutive numbers, the above string becomes (4-1) (19-4) (19-3) (3-1) ⟹ 3 15 16 2 ...
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### Can I replace modulus inequalities with rooted square arguements?

Suppose I want to show $|x-5|<|x+1|$. One way (and the way my lecturer shows) to do it is look at the negative and positive regions and solve the inequality. But with the definition $\sqrt{x}\geq0$,...
1 vote
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### How to solve $xy'=3y-6x^2$ using integrating factors?

I am trying to solve $xy'=3y-6x^2$ using integrating factors. I am facing 2 issues when doing so. In order to find $P(x)$ to be used in $e^{\int P(x) dx}$, I am dividing by $x$ which, it seems, ...
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### for $x \in \mathbb{R}$ find the number of solutions in $3x^2 + 4|x^2 - 1| + x - 1 = 0$. Why we are considering negative values here?

I was solving some practice problems, from a booklet of math, and one of the problems is like this for $x \in \mathbb{R}$ find the number of roots in the eq . $$3x^2 + 4|x^2 - 1| + x - 1 = 0$$ I know ...
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### Finding the derivative of a function with two absolute values within it, using a piecewise function

I'm trying to solve some a problem relating to absolute values. I found online the strategy for solving similar functions from here: https://www.youtube.com/watch?v=eIHtq67nh7w&list=...
1 vote
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### The mode minimizes the $l_0$ norm?

Suppose we have a set $S$ of $N$ real numbers. Show that $$\sum_{s_i\in S}|s_i-x|^0$$ is minimal if $x$ is equal to the mode of S. I'm a bit confused about that, because assuming $0^0 = 1$ the whole ...
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### Archimedean absolute value on the set of real rational fractions, whose induced absolute value in $\mathbb{R}$ is the trivial one.

Archimedean absolute value on the set of real rational fractions, whose induced absolute value in $\mathbb{R}$ is the trivial one. Can this be done? So, my idea is to see it in $\mathbb{C}(t)$ so I ...
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### Collapsed terminology for simple logarithmic procedure

During some computation, I had to rescale a value $x$ by taking the $\log_{10}(x)$. If $x$ was positive, I merely took the log, but if $x$ was negative i used -$\log_{10}(|x|)$. Is there a shorter ...
1 vote
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### When is the absolute value of logarithm equal to the input?

So does the following equation have any closed form solution(s), or if not, can the solution(s) be expressed as functions of well known math constants (e.g., $\pi,e$)? Also, what is the relevance of ...
In Ginzburg Landau equation, there is a term, $|A|^2A$ and $A$ is space and time dependent function or $A(x,t)$ Why do we have norm or absolute value under square? Is square not enough? My guess is ...