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Questions tagged [absolute-value]

For questions about or involving the absolute value function.

80
votes
4answers
124k views

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|-|...
85
votes
8answers
42k views

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 ...
38
votes
9answers
123k 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 :)
24
votes
6answers
8k views

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.
25
votes
2answers
3k views

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 ...
15
votes
4answers
30k 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 ...
4
votes
2answers
4k views

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 ...
8
votes
3answers
11k views

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 ...
5
votes
1answer
4k views

Prove variant of triangle inequality containing p-th power for 0 < p < 1

Sorry if this is a trivial question, but I am kind of stuck with proving the following inequality and have been searching for a while: $\rho \left( \sum\limits_i^n d_i \right) \leq \sum\limits_i^n \;\...
3
votes
4answers
4k views

Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$

Let $a,b,c$ real numbers. Prove the inequality $|a-b| \leq |a-c| + |c-b|$. Prove that equality holds if and only if $a \leq c \leq b$ or $b \leq c \leq a$. I've tried starting with just $a \leq c$ ...
7
votes
5answers
265 views

How is it, that $\sqrt{x^2}$ is not $ x$, but $|x|$?

As far as I see, $\sqrt{x^2}$ is not $x$, but $|x|$, meaning the "absolute". I totally get this, because $x^2$ is positive, if $x$ is negative, so $\sqrt{y}$, whether $y = 10^2$ or $y = -10^2$: $y$ is ...
2
votes
3answers
253 views

Is $\sqrt{x^2}=|x|$ or $=x$? Isn't $(x^2)^\frac12=x?$ [duplicate]

$|x|=\sqrt{x^2}$ as Wolfram|Alpha shows. But, as $(x^2)^\frac12=x$, I can't understand where am I wrong interpreting Square-root.
10
votes
2answers
9k views

How to use triangle inequality to establish Reverse triangle inequality

I need to use $|a+b| \leq |a|+|b|$ to show that $||a|-|b|| \leq |a-b|$ . I have tried to represent $||a|-|b||$ as $||a|+(-|b|)|$ , and then get $||a|+(-|b|)| \leq |a|+|-|b||$ , but that isn't ...
1
vote
6answers
1k views

How could we solve $x$, in $|x+1|-|1-x|=2$?

How could we solve $x$, in $|x+1|-|1-x|=2$? Please suggest a analytical way that I could use in other problems too like this $ |x+1|+|1-x|=2$ and of this genre. Thank you,
6
votes
5answers
35k views

How to solve inequalities with absolute values on both sides?

If you have an inequality that has two absolute value bars like $|4x+1|<|3x|$, how do you go about doing this? I know that if $4x+1<3x$, then those $x$'s will work but what else do I do? I think ...
17
votes
4answers
1k views

How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$

let $a,b,c,d,e\in R$,and such $$a^2+b^2+c^2+d^2+e^2=1$$ find this value $$A=\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$$ I use computer have this $$A=\dfrac{2}{\sqrt{10}}$$ ...
14
votes
1answer
11k views

The absolute value of a Riemann integrable function is Riemann integrable.

This is an exercise in Bartle & Sherbert's Introduction to Real Analysis second edition. They ask to show that if $I=[a,b]$ is a closed bounded interval and that $f:I\to\mathbb{R}$ is (Riemann) ...
2
votes
3answers
246 views

Use the triangle inequality to show that $|a|+|b| \leq |a+b|+|a-b|$

How would I go about proving that $$|a|+|b| \leq |a+b|+|a-b|$$ Using the triangle inequality? I tried squaring both sides, yielding: $$|a|^2 +|b|^2 +2|a||b| \leq 2|a|^2+2|b|^2+2||a|^2-|b|^2|$$ Is ...
11
votes
2answers
54k views

Square root of a number squared is equal to the absolute value of that number [duplicate]

Possible Duplicate: Significance of $\displaystyle\sqrt[n]{a^n} $? The square root of a number squared is equal to the absolute value of that number. Why is $\sqrt{x^2} = |x|$? Why not just $x$? ...
0
votes
2answers
310 views

Derivative of $f(x)=|x|$

Okay, so $\displaystyle \frac{d}{dx} |x| = \frac{|x|}{x}$. But I have trouble seeing why. Here's what I've tried: $$\frac{d}{dx}|x|=\begin{cases} \frac{d}{dx}x & \text{if }x > 0 \\ \frac{d}{dx}...
33
votes
1answer
50k views

Why is the absolute value function not differentiable at $x=0$?

They say that the right and left limits do not approach the same value hence it does not satisfy the definition of derivative. But what does it mean verbally in terms of rate of change?
10
votes
7answers
36k views

Converting absolute value program into linear program

I have the generic optimization problem: $$ \max c^T|x|$$ $$ \text{s.t. } Ax \le b $$ $x$ is unrestricted How do I convert it into a linear programming problem? Online I read something about ...
6
votes
1answer
4k views

Continuity of absolute value of a function

Let $f(x)$ be a continuous function. Prove that $\left|f(x)\right|$ is also continuous. Is it correct to say that, by the reverse triangle inequality, $\left|f(x)-f(c)\right| \geq \left|f(x)\right|-\...
6
votes
3answers
561 views

Inequality involving rearrangement: $ \sum_{i=1}^n |x_i - y_{\sigma(i)}| \ge \sum_{i=1}^n |x_i - y_i|. $

If $x_1 \ge x_2 \ge \cdots \ge x_n$ and $y_1 \ge y_2 \ge \cdots \ge y_n$ are real numbers, and $\sigma$ is any permutation, then $$ \sum_{i=1}^n |x_i - y_{\sigma(i)}| \ge \sum_{i=1}^n |x_i - y_i|. $$ ...
4
votes
2answers
11k views

General Proof for the triangle inequality

I am trying to prove: $P(n): |x_1| + \cdots + |x_n| \leq |x_1 + \cdots +x_n|$ for all natural numbers $n$. The $x_i$ are real numbers. Base: Let $n =1$: we have $|x_1| \leq |x_1|$ which is clearly ...
4
votes
2answers
6k views

When does the equality hold in the triangle inequality? [duplicate]

Hi guys could you please help me on this question I'm confused. question: when does the equality hold in the triangle inequality: my attempt : $|x + y| \leq |x| + |y|$ this implies $(|x+y|)^2 = (|...
2
votes
2answers
447 views

Solving $|x-2| + |x-5|=3$ [duplicate]

Possible Duplicate: How could we solve $x$, in $|x+1|-|1-x|=2$? How should I solve: $|x-2| + |x-5|=3$ Please suggest a way that I could use in other problems of this genre too Any help to ...
0
votes
1answer
311 views

Is there a number whose absolute value is negative?

I've recently started to think about this, and I'm sure a couple of you out there have, too. In Algebra, we learned that $|x|\geq0$, no matter what number you plug in for $x$. For example: $$|-5|=5\...
7
votes
4answers
318 views

Why does $\sqrt{x^2}$ seem to equal $x$ and not $|x|$ when you multiply the exponents?

I understand that $\sqrt{x^2} = |x|$ because the principal square root is positive. But since $\sqrt x = x^{\frac{1}{2}}$ shouldn't $\sqrt{x^2} = (x^2)^{\frac{1}{2}} = x^{\frac{2}{2}} = x$ because of ...
4
votes
9answers
162 views

Show that $y = \frac{2x}{x^2 +1}$ lies between $-1$ and $1$ inclusive.

Prove, using an algebraic method,that $y=\frac{2x}{x^2 +1}$ lies between $-1$ and $1$ inclusive. Hence, determine the minimum and maximum points $y=\frac{2x}{x^2 +1}$ . What I tried: Firstly, I ...
7
votes
4answers
10k views

Show $\max{\{a,b\}}=\frac1{2}(a+b+|a-b|)$

I am tasked with showing that If $a,b\in \mathbb{R}$, show that $\max{\{a,b\}}=\frac1{2}(a+b+|a-b|)$ I think I can say "without loss of generality, let $a<b$." Then $b-a>0$ But also, $$\...
10
votes
7answers
9k views

what does $|x-2| < 1$ mean?

I am studying some inequality properties of absolute values and I bumped into some expressions like $|x-2| < 1$ that I just can't get the meaning of them. Lets say I have this expression $$ |x|&...
6
votes
1answer
4k views

Equality of triangle inequality in complex numbers

$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$. I approached this by letting $z=a+ib$, and $w=c+id$, and kinda ...
25
votes
7answers
5k views

My teacher describes absolute value confusingly: $|x|=\pm x,\quad \text{if}\enspace x>0 $.

He says (direct quote): "In higher mathematics the absolute value of a number, $|x|$, is equal to positive and negative $x$, if $x$ is a positive number." Then he wrote: $|x|=\pm x,\quad \text{if}\...
3
votes
2answers
810 views

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 ...
3
votes
3answers
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 ...
4
votes
3answers
1k views

Proof for Integral Inequality $|\int f| \le \int |f|$ - is it sufficient enough?

Claim: If f is integrable, $\left|\int_a^bf(x)dx\right|\le\int_a^b|f(x)|dx$ Proof (attempt): We know $-|f|\le f \le|f|$, so $\int-|f| \le \int f \le \int|f|$.* Since, if $-b<a<b$, we say $|...
2
votes
2answers
720 views

Does every non-Archimedean absolute value satisfy the ultrametric inequality?

The Archimedean property occurs in various areas of mathematics; for instance it is defined for ordered groups, ordered fields, partially ordered vector spaces and normed fields. In each of these ...
1
vote
1answer
2k views

Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere?

Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere? I found this question here, the question seems much interesting but for obvious reason it is closed there, I was ...
1
vote
3answers
3k views

Rotate the graph of a function?

How do I rotate a graph of a function around a point, and show it in the related equation? An example could be $f(x)=\lvert x\rvert$ (absolute Value) and $f(x)=x^2$
0
votes
3answers
3k views

Proof for absolute value inequality of three variables: $|x-z| \leq |x-y|+|y-z|$ [duplicate]

$|x-z| \leq |x-y|+|y-z|$ We know that both LHS and RHS are non negatives. So, I thought of proving this by comparing the squares of both sides but can't advance beyond that step. Any help would be ...
20
votes
1answer
20k views

Integral Inequality Absolute Value: $\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$

Suppose we are given the following: $$\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$$ How would we prove this? Does this follow from Cauchy Schwarz? Intuitively ...
6
votes
3answers
819 views

How prove this inequality: $\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|$? [duplicate]

Let $x_{1},x_{2},\cdots,x_{n}$ be real numbers. Show that $$\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|.$$ I think this problem may be solved using nice methods, but I can't find them ...
3
votes
3answers
12k views

Proving absolute value inequalities

I need help proving the last two cases for the following inequality: $\bigl|\lvert x\rvert-\lvert y\rvert\bigr| \le \lvert x-y\rvert$. Case 1: $x > 0$ and $y > 0$: the inequality simplifies ...
1
vote
1answer
5k views

Proving the reverse triangle inequality of the complex numbers

I'm having trouble understanding this proof for the reverse triangle inequality of the complex numbers. Suppose for any $z, w \in \mathbb{C}$, we have $|z + w| \leq |z| + |w|$ (the triangle ...
7
votes
2answers
6k views

How does one DERIVE the formula for the maximum of two numbers [duplicate]

I want to derive (not prove that this is true) the formula $\max (x,y) = \dfrac{x + y + |y-x|}{2}$ I was reading a proof (which they have the result ahead of time already) that we do cases and then ...
7
votes
4answers
46k views

Is the absolute value function a linear function?

I'm inclined to say yes, as it doesn't involve exponentiation, roots, logarithmic or trigonometric functions, but I watched a video where the teacher said that the absolute value function is "clearly ...
6
votes
4answers
877 views

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 ...
6
votes
5answers
899 views

Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$

Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$ example1: $|x-1|+|x-2|\geq 1$ my solution:(substitution) $x-1=t,x-2=t-1,|t|+|t-1|\geq 1,|t-1|\geq 1-|t|,$ square, $t^2-2t+1\geq 1-2|t|+t^2,\text{...
3
votes
2answers
11k views

Solving inequality with two absolute values

Hey, ! In my pre-calculus class the teacher showed the solution of the following example: \begin{align} \vert x-3 \vert \lt \vert x - 4 \vert + x \end{align} He started by stated the domains ...