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Questions tagged [absolute-value]

For questions about or involving the absolute value function.

82
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8answers
41k 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 ...
80
votes
4answers
122k 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|-|...
38
votes
9answers
121k 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 :)
32
votes
1answer
49k 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?
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 ...
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}\...
25
votes
1answer
354 views

Is there an easy way to see that this simple recurrence is 9-periodic? [duplicate]

In a colloquium talk yesterday, Robert Bryant pointed out that for all initial values $a_0, a_1 \in \mathbb{R}$, the sequence generated by the recurrence relation $$ a_{n+1} = |a_n| - a_{n-1} $$ turns ...
24
votes
6answers
7k 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.
23
votes
6answers
2k views

Where is the absolute value when computing antiderivatives?

Here is a typical second-semester single-variable calculus question: $$ \int \frac{1}{\sqrt{1-x^2}} \, dx $$ Students are probably taught to just memorize the result of this since the derivative of $...
19
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 ...
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 $$A=\dfrac{2}{\sqrt{10}}$$ ...
17
votes
3answers
297 views

Quadratic inequality puzzle: Prove$ |cx^2 + bx + a| ≤ 2$ given $|ax^2+bx+c| ≤ 1$

I came across this problem as part of a recreational mathematics challenge on university: Suppose $a, b, c$ are real numbers where for all $ -1 \le x \le 1 $ we have $|ax^2 + bx + c| \le 1$. Prove ...
16
votes
5answers
2k views

When I was teaching absolute function properties, I suddenly made this question …

I was teaching absolute function properties in a K-12 class. I made this question in my mind. Suppose $f(x)$ is a one-to-one function, and its definition is $f(x)=max\left \{ x,3x\right \}=ax+b|x|+c$...
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 ...
14
votes
2answers
278 views

Finding all solutions to the equation $|||||x|-1|-1|-1|-1|=0$

I was presented this question by a student I was tutoring: Suppose $x \in \mathbb{R}$. Find all solutions of the equation $$|||||x|-1|-1|-1|-1|=0.$$ What I explained to the student: Given $|||||...
14
votes
1answer
10k 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) ...
12
votes
1answer
65k views

Derivatives of functions involving absolute value

I noticed that if the absolute value definition $\lvert{x}\rvert=\sqrt{x^2}$ is used, we can get derivatives of functions with absolute value, without having to redefine them as piece-wise. For ...
11
votes
9answers
2k views

Given $P(x)=x^{4}-4x^{3}+12x^{2}-24x+24,$ prove $P(x)=|P(x)|$ for all real $x$

$P(x)=x^{4}-4x^{3}+12x^{2}-24x+24$. Prove: $P(x)=|P(x)|$ I don't know where to begin. What would be the first step?
11
votes
3answers
167 views

Why $|x|$ is not rational expression?

I'm 9th grade student, and my teacher said that $|x|$ is not rational expression ( expression like $\frac{p(x)}{q(x)}$ s.t $p(x)$ and $q(x)\neq 0$ are polynomial) but he didn't have convincing reason. ...
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 $x$? ...
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|&...
10
votes
6answers
3k views

Absolute Value of A Real VS. Complex Number

I am came across a problem that asked me to find the magnitude of a complex number: $$|4 + 3i|$$ After reading online and the textbook, I figured out the question could be solved by finding the ...
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 ...
10
votes
5answers
563 views

Is -5 bigger than -1?

In everyday language people often mix up "less than" and "smaller than" and in most situations it doesn't matter but when dealing with negative numbers this can lead to confusion. I am a mathematics ...
10
votes
6answers
24k views

Inequality with two absolute values

I'm new here, and I was wondering if any of you could help me out with this little problem that is already getting on my nerves since I've been trying to solve it for hours. Studying for my next ...
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 ...
10
votes
2answers
17k views

Question regarding usage of absolute value within natural log in solution of differential equation

The problem from the book. $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6 -y$ I understand the solution till this part. $\ln \vert 6 - y \vert = x + C$ The solution in the book is $6 - Ce^{-x}$ My ...
10
votes
2answers
786 views

How to determine all valuations of the field $\mathbb{Q(\sqrt[n]{2})}$?

This is an exercise from the book Algebraic Number Theory by Jurgen Neukirch, on page 166. And, after solving several previous exercises, I found this to be particularly difficult to solve. I am ...
9
votes
9answers
3k views

What's wrong with solving absolute value equations in this way?

Say I have $3x-2 = |x|$. Why can't I just do this: $3x - 2 = -x$ and $3x - 2 = x$ and then get two values for $x$: $1$ and $0.5$? I know the answer $0.5$ doesn't work if you plug this in. However, I ...
9
votes
5answers
1k views

If three complex numbers $z_k$ have modulus $1$, then $|z_1+z_2+z_3| = |\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|$

Our teacher gave us a hard question (according to her, it is pretty hard for our level): Given that $|z_1| = |z_2|= |z_3|=1,z \in\mathbb{C}$, prove that $|z_1+z_2+z_3| = |\frac{1}{z_1}+\frac{1}{z_2}...
9
votes
3answers
903 views

Absolute value graph sketching: $||x-1|-1|$

Where would you start if you were told to plot: $$||x-1|-1|$$ I looked at just $f(x) = |x-1|$ and noticed that the two equations are: $\pm (x-1)$ for $x \geq 1$ and $x < 1$. Extrapolating then: $\...
9
votes
2answers
1k views

Getting wrong answer for absolute value inequality and not sure why

The question: The function p is defined by $-2|x+4|+10$. Solve the equality $p(x) > -4$ Here were my steps to solving this: 1.) Subtract 10 from both sides -> $-2|x+4| > -14$ 2.) Divide ...
9
votes
6answers
1k views

How to calculate with absolute value.

Calculate:$$\frac{ \left| x \right| }{2}= \frac{1}{x^2+1}$$ How do I write the whole process so it will be correct? I need some suggestions. Thank you!
9
votes
3answers
8k views

Double absolute value proof: $||a|-|b||\le |a-b|$ [duplicate]

I guess, I know how to solve inequalities with absolute value, but I have problems with this one. $||a|-|b||\le |a-b|$ $a,b\in \mathbb{R}$ I tried to solve the inequation like this: case1: $a>0$...
9
votes
2answers
538 views

Could we invent a new number with $|p|=-1$?

We know that how a single definition $i^2=-1$ revolutionized our mathematics and solved many many problems. I wonder whether the definition $|p|=-1$ could have the potential of creating a new ...
9
votes
1answer
99k views

Integral of an absolute value function

How do I find the definite integral of an absolute value function? For instance: $f(x) = |-2x^3 + 24x|$ from $x=1$ to $x=4$
9
votes
3answers
1k views

How does this proof of Theorem 1 in Spivak's Calculus work?

Hello I just started Spivak's Calculus and I've come across something I don't really understand. What is trying to be proved is $$ |a+b|\leq|a|+|b|\,. $$ The proof is based on the observation that $|...
9
votes
1answer
3k views

Is there a lower-bound version of the triangle inequality for more than two terms?

The triangle inequality $|x+y|\leq|x|+|y|$ can be generalized by induction to $$|x_1+\ldots+ x_n|\leq|x_1|+\ldots+|x_n|.$$ Can we generalize the version $|x+y|\geq||x|-|y||$ to $n$ terms too? I need ...
9
votes
2answers
420 views

Modulus Equations

$$ |x + 1| + |x − 1| = x + 4$$ The only way I can solve this equation is to graph it...Through graphing, I get the following solutions: $$x = -\frac{4}{3}, 4$$ Is their a general algebraic method ...
8
votes
5answers
11k views

Absolute value and max/min function: why $a + b + |a - b|=2\max(a,b)$? [duplicate]

I am being told that $a + b + |a - b|$ is equal to $2\max(a,b)$. What is the reasoning behind this?
8
votes
3answers
6k views

Limit of the absolute value of a function

Say I have a real valued function $f(x)$, is it true that if $\lim_{x \rightarrow c} |f(x)| = 0$ then $\lim_{x \rightarrow c} f(x) = 0$?, where $c$ can be a real number or $\pm \infty$.
8
votes
2answers
14k views

How does the triangle inequality work for $|x-y|$?

I know that $|x+y|\leq |x|+|y|$... But is it similar for $|x-y|$? That is, is $|x-y|\leq |x|+|y|$? I ask because of the following: $x-y=x+(-y)$, so $|x+(-y)|\leq |x|+|-y|=|x|+|y|$ Is it possible ...
8
votes
5answers
643 views

Inequality for absolute values

How do you show either of the equivalent inequalities: $$2(|a|+|b|+|c|)\leq |a+b+c|+|a+b-c|+|a-b+c|+|a-b-c|$$ or $$|x+y|+|x+z|+|y+z|\leq |x|+|y|+|z|+|x+y+z|$$ Hold for complex numbers or in $n$ ...
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 ...
7
votes
8answers
2k views

Why is the absolute sign needed in the definition of a bounded function

A function $f$ is bounded if there exists a real number $M$ such that $|f(x)| \le M$ for all $x \in \operatorname{dom}(f)$. Why is the absolute sign needed?
7
votes
4answers
687 views

Why is the Absolute value / modulus function used?

Why is the absolute value function or modulus function $|x|$ used ? What are its uses? For example the square of a modulus number will always be positive, but why is it used when for example the ...
7
votes
4answers
691 views

What is $\frac{1}{|{x}|}-\frac{x^2}{|x|^3}$?

What's the result of: $$\frac{1}{|{x}|}-\frac{x^2}{|x|^3}$$ Is it $$\frac{1}{|{x}|}-\frac{x^2}{|x|x^2}=\frac{1}{|{x}|}-\frac{1}{|x|}=0$$ or $$\frac{1}{|{x}|}-\frac{x^2}{|x|^2x}=\frac{1}{|{x}|}-\frac{...
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, $$\...
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 ...
7
votes
4answers
3k views

Is $e^{|x|}$ differentiable?

My thoughts go as follows: For $x > 0$, $e^{|x|} = e^x $ For $x < 0$, $e^{|x|} = e^{-x}$ Both $e^x$ and $e^{-x}$ are differentiable at every point in their domains, so $e^{|x|}$ will be ...