Questions tagged [absolute-value]

For questions about or involving the absolute value function.

181 questions
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Reverse Triangle Inequality Proof

I've seen the full proof of the Triangle Inequality \begin{equation*} |x+y|\le|x|+|y|. \end{equation*} However, I haven't seen the proof of the reverse triangle inequality: \begin{equation*} ||x|-|...
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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 ...
124k views

Proof of triangle inequality

I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
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Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
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Significance of $\sqrt[n]{a^n}$?!

There is a formula given in my module: $$\sqrt[n]{a^n} = \begin{cases} \, a &\text{ if n is odd } \\ |a| &\text{ if n is even } \end{cases}$$ I don't really understand the ...
31k views

Proving square root of a square is the same as absolute value

Lets say I have a function defined as $f(x) = \sqrt {x^2}$. Common knowledge of square roots tells you to simplify to $f(x) = x$ (we'll call that $g(x)$) which may be the same problem, but it isn't ...
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Proving that $x_n\to L$ implies $|x_n|\to |L|$, and what about the converse?

Problem 3. Show that for a sequence $(x_n)$ the following are true: (i) $\lim x_n=0$ if and only if $\lim |x_n|=0$. (ii) $\lim x_n=L$ implies $\lim |x_n|=|L|$. Is the converse true? Prove or give ...
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Equality holds in triangle inequality iff both numbers are positive, both are negative or one is zero

How do we show that equality holds in the triangle inequality $|a+b|=|a|+|b|$ iff both numbers are positive, both are negative or one is zero? I already showed that equality holds when one of the ...
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LOGARITHMIC INEQUALITY - TWO HOURS

Source: Brilliant Solve the inequality $$\ x \log_{\log_{|x^2 - 3 | - 2 } (x^2 - 3|x| + 2) } \left( \dfrac{x^3 - |3x+2|}{x^3 - |3x-2|}\right) \geq 0$$ I have tried this many times, but I keep ...
23k views

Prove that the absolute value of a product is the product of the absolute values of factors.

Theorem. $|a||b|=|ab|$ Proof. Applying the definition of absolute value, the left hand side of the equation could be either $a\times(-b)$ or $(-a)\times(b)$ or $a\times b$ or $(-a)\times(-b)$. For ...
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Proving an identity relating to the complex modulus: $z\bar{a}+\bar{z}a \leq 2|a||z|$
Show that $z\bar{a}+\bar{z}a \leq 2|a||z|$ I was able to check this using some examples which is not ideal for a mathematician way of proving a problem/case/theorem. I couldn't generalize it (prove ...
How to solve equations involving modulus function of the type $|x+1| - |1-x|=2$ and $|x-1|=|x|+a$?
I am able to solve equation of the type $|5x+1|=|11-2x|$. I square both the side and my equation becomes $(5x+1)^2=(11-2x)^2$ further simplification gives me $(5x+1)=\pm (11-2x)$. I get have ...