# Questions tagged [absolute-value]

For questions about or involving the absolute value function.

1,990 questions
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### Triangle Inequality and absolute value

I'm curious if the triangle inequality (and reverse triangle inequality) still hold if we only take the absolute value of one term. For example: $$||a| - b| \le |a - b|$$ If $b \ge 0$, then $|b|$ is ...
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### The Median Minimizes the Sum of Absolute Deviations (The ${L}_{1}$ Norm)

Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x|$$ is minimal if $x$ is equal to the median. This is a sample exam question of one of the exams that I need to take and ...
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### Redefinition of a Constant Leading to Nullification of Absolute Value

I am currently taking a calculus 3 class in college, and my teacher did an intriguing problem about a tsunami in class which took up about 4 full white boards. Anyways, at a certain point in the ...
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### bound for $a,b,c$ in $|ax^2+bx+c| \leq 1\;\forall x\in \left[0,1\right]$

Consider $a,b,c\in\mathbb{R}$ such that $|ax^2+bx+c|\leq 1\;\forall x\in \left[0,1\right]$. Prove that $|a|\leq 8\;\;,|b| \leq 8$ and $|c| \leq 1$. My Attempt: Set $x = 0$ in $|ax^2+bx+c|\leq 1$ to ...
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### solving equation involving absolute values [closed]

Given the equation $|x+2| + |3x+6| = 8,$ how can I find the sum of all its roots?
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### How can I solve this absolute value equation?

This is the equation: $|\sqrt{x-1} - 2| + |\sqrt{x-1} - 3| = 1$ Any help would be appreciated. Thanks!
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### Number of solutions of an equality with absolute value operator

Consider: $$\left|\left|\left|x-1\right|-2\right|-4\right|=4$$ What is the number of solutions for this equation? This one was particularly easy to me. If first observed that if this inequality ...
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### Suppose $|z|\ge 2$, Prove $|z^8+135|\ge121$.

Suppose $|z|\ge 2$, Prove $|z^8+135|\ge121$. My work: $|z^8+135|=\sqrt{(z^8+135)(\bar{z}^8+135)}=\sqrt{|z|^{16}+135(z^8+\bar{z}^8)+135^2}\ge\sqrt{2^{16}+135(z^8+\bar{z}^8)+135}$ For the last term, ...
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Let $F$ be a field, and $\| \cdot\|$ be an absolute value (or norm). I want to prove that with this norm, multiplication is continuous map on $F$ in the sense: for $x_0,y_0\in F$, and any $\... 3answers 42 views ### How can$|n_1 - n_2| + |n_3 - n_4|$be equal two different formulas I have a formula to be calculated such as;$|n_1 - n_2| + |n_3 - n_4|$. I want to calculate it with two tuples of values such as;$(n_1,n_2)$and$(n_3,n_4)$When I try this method, it is equivalent;... 2answers 43 views ### Prove using the triangular inequality that:$|a+b| \geq |a| - |b|$How can I prove using the triangular inequality that: $$|a+b| \geq |a| - |b|$$ I already proved it by considering all 8 possible scenarios (like a>b and b=0 ... etc) However I couldn’t manage to find ... 3answers 54 views ### Is it valid to take$|- \infty | = \infty$? Is it valid to take$|- \infty | = \infty$? or is the absolute value e.g. not defined for infinity? Particularly, if one wishes to argue that operator$f(x)=x$is not bounded below on$\mathbb{R}_{-...
I was calculating basic rational integrals and came up with this kind of problem. I have this expression: $$2\ln|x|$$ I can re-write it down like that: $$\ln{x^2}$$ and thus cancel the modulus. ...