For questions about or involving the absolute value function.

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18
votes
2answers
42k 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*} ...
9
votes
5answers
6k views

The median minimizes the sum of absolute deviations

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 ...
13
votes
4answers
40k 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 :)
19
votes
2answers
2k views

Significance of $\displaystyle\sqrt[n]{a^n} $?

There is a formula given in my module: $$ \sqrt[n]{a^n} = a \text{ if $n$ is odd } $$ $$ \sqrt[n]{a^n} = |a| \text{ if $n$ is even } $$ I don't really understand the differences between them, ...
19
votes
6answers
3k 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.
6
votes
4answers
9k 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 ...
7
votes
2answers
6k 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 ...
3
votes
3answers
5k 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 ...
2
votes
2answers
747 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 ...
1
vote
6answers
666 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,
16
votes
4answers
794 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 $$A=\dfrac{2}{\sqrt{10}}$$ ...
4
votes
1answer
569 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 ...
3
votes
4answers
180 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.
3
votes
2answers
2k 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 = ...
5
votes
1answer
2k 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 ...
4
votes
5answers
158 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 ...
1
vote
4answers
425 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 ...
6
votes
3answers
752 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 ...
3
votes
5answers
13k 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 ...
5
votes
4answers
146 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
3answers
641 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
5answers
581 views

Prove the triangle inequality [duplicate]

I want to porve the triangle inequality: $x, y \in \mathbb{R} \text { Then } |x+y| \leq |x| + |y|$ I figured out that probably the cases: $x\geq0$ and $y \geq 0$ $x<0$ and $y < 0$ $x\geq0$ ...
2
votes
2answers
199 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 ...
2
votes
3answers
364 views

Question about solving absolute values.

I solved the following problem from by book, but the answer of this problem at the end of book is $x \leq 3$. Please tell me how I can get this answer.
10
votes
7answers
1k 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 $$ ...
7
votes
2answers
2k views

How does one DERIVE the formula for the maximum of two numbers

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 ...
1
vote
3answers
6k 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 ...
7
votes
2answers
167 views

Maximum of the difference

What is the maximum value of $f(… f(f(f(x_{1} – x_{2}) – x_{3})-x_{4}) … – x_{2012})$ where $x_{1}, x_{2}, … , x_{2012}$ are distinct integers in the set ${1, 2, 3, …, 2012}$ and $f$ is the absolute ...
4
votes
3answers
70 views

Proving or disproving that an inequality implies another inequality.

I am wondering if $|a| > |b|$ implies $|\frac{b+b^{2}}{a+a^{2}}| < 1$, where $a$ and $b$ are real numbers. I have tested numerically with many cases and I have found this to be true in all of my ...
4
votes
4answers
23k 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 ...
3
votes
1answer
156 views

If $|ax^2+bx+c|\le 1\ \forall |x|\le 1$, then what is the maximum possible value of $\frac 83a^2+2b^2$? [closed]

Let $f(x) = ax^2 + bx + c$ ; $a,b,c\in\mathbb R$ It is given that $|f(x)| \le 1$ $\forall |x| \le 1$ Q1) The possible value of $|a+c|$, if $\displaystyle \frac{8}{3} a^2 + 2b^2$ is maximum, is ...
4
votes
3answers
1k views

Proving two integral inequalities

Can anyone help me to prove that these integral inequalities hold? Here $x$ is a real value: $$ \left| \int_a^b\ f(x) dx \right| \leq \int_a^b\ |f(x)| dx $$ Here $z$ is a complex value: $$ \left| ...
3
votes
2answers
514 views

Prove the monotonicity of the expectation of a binary random variable function

Consider $R$ independent binary random variables $y^1, \ldots, y^R$ over the space $\{-1, +1\}$ such that $\Pr(y^j = 1) = p^j \geq 0.5$ and $\Pr(y^j = -1) = 1 - p^j$, $\forall j = 1,\ldots,R$. ...
3
votes
2answers
364 views

Why exactly can you take the absolute value of one side of this inequality and assume it is still true?

Exercise: Show that if $(b_n) \to b$, then the sequence of absolute values $\left| b_n \right|$ converges to $\left| b \right|$. Solution (partial): By the triangle inequality, ...
3
votes
2answers
25k 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 ...
3
votes
2answers
9k 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 ...
2
votes
0answers
38 views

Cases in which a certain inequality holds true.

I previously asked this question on this forum, and have been demonstrated counterexamples to the claim that $|a| > |b|$ implies $\big|\frac{b+b^{2}}{a+a^{2}}\big| < 1$, which I had previously ...
2
votes
2answers
353 views

Prove triangle inequality using the properties of absolute value

So I was given the task of proving the following variant of the triangle inequality using only the properties of the absolute value: $\vert\lvert x\rvert -\lvert y \rvert \rvert \leq \lvert ...
2
votes
2answers
4k 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 ...
2
votes
2answers
264 views

Why is the derivative of $\frac{|x|}{x}$ equal to $\emptyset$ at $x=0$?

I got a bit of a confusion here. If $\varphi(x)=\frac{|x|}{x}$, then $$ \varphi(x) = \left.\Bigg\{ \begin{array}{cc} 1 &if \ x>0\\ \emptyset & if \ x=0\\ -1 & if \ x <0 \end{array} ...
2
votes
1answer
197 views

Absolute value of a real number

My question is: Solve: $|x-4|< a$, where $a$ belongs to the real numbers. Solve this by considering various cases depending upon whether $a$ is negative, positive or zero. What I have tried ...
1
vote
3answers
66 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
1answer
40 views

Triangle inequality univalent

$|Z_1| = | \frac{v(1+\alpha)+ \sqrt{v^2(1+\alpha)^2-4\alpha}}{2}|$ I know that using triangle inequality method $|Z_1|$ is: $|Z_1|= |\frac{v(1+\alpha)}{2}| + ...
0
votes
1answer
34 views

Finding $\lim_{t\to 0}\frac{|t-2|}{t}$ and $\lim_{t\to \infty}\frac{|t-2|}{t}$

Find $$\lim_{t\to 0}\frac{|t-2|}{t}$$ and $$\lim_{t\to\infty}\frac{|t-2|}{t}$$ Usually I would simply the top and bottom but I'm not sure what to do for absolute values. Any help would be ...
0
votes
2answers
107 views

Absolute value question false solution

|x| = 3x – 2 Why does this statement eventually give you a solution that isn't valid. So this equation comes out: x = 3x - 2 2 = 2x x = 1 OR x = -3x + 2 4x = 2 x = 1/2 However 1/2 doesn't ...
0
votes
3answers
688 views

Integrating absolute value function

I'm working on a problem drawing phase plane diagrams in my applied mathematics course. I'm supposed to draw the phase line diagram of $x''+\vert x\vert=0.$ In the process, I get to the differential ...
-2
votes
1answer
870 views

Example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points [duplicate]

So I had an exam today and one of the questions were: Give an example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points. At first I had no idea how to do it ...
5
votes
5answers
13k 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 ...
7
votes
4answers
440 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 $n$ ...
3
votes
4answers
128 views

The Definition of the Absolute Value

The Absolute Value can be defined in many ways, but these are the two most common : 1. As a Piecewise Function $$ |x|= \begin{cases} -x&\text{if } x < 0\\ x&\text{if } x\geq 0 \end{cases} ...