For questions about or involving the absolute value function.

learn more… | top users | synonyms

3
votes
3answers
163 views

Finding the derivative of $|x|^4$ using the chain rule.

I am presented with the following task: Can you use the chain rule to find the derivatives of $|x|^4$ and $|x^4|$ in $x = 0$? Do the derivatives exist in $x = 0$? I solved the task in a rather ...
1
vote
1answer
65 views

Help with showing that $|x+y|=|x|+|y| \longleftrightarrow xy\ge0 $

I need to prove that $|x+y|=|x|+|y| \longleftrightarrow xy\ge0 $. I proved before that for any $x,y\in R$ holds $|x+y|\le|x+y|$ and I thought maybe it could help me with my arguments to show what I ...
2
votes
1answer
51 views

How to draw $|y|=|(x-2)^2-1|$?

This correspondence's domain and codomain is available for all real numbers. So the codomain and domain are the set of all real numbers on? And, this graph is true? (This graph was called ...
0
votes
0answers
29 views

Questions about $|f(1+a+bi)|<|f(1+a)|$

Let $a,b >0$ and $|*|$ denote the absolute value. Let $f(z)$ be a realvalued analytic function defined for $Re(z)>1.$ For any $a,b$ we have $|f(1+a+bi)|<|f(1+a)|$. Some questions : $1)$ If ...
0
votes
2answers
556 views

Absolute value fraction problem

Is there an easier way to (I am aware of my poor translation, but am not familiar with english terminology; however, I think you will understand.) "Reduce the following fraction:" $\dfrac{x | x-1 ...
1
vote
1answer
402 views

Finding domain of a rational function

Find the domain and graph: $$f(t)=\frac{-t}{|t|}$$ My book says to define it piecewise. My questions: $\mathbf{1)}$ Do all rational functions have to be defined piecewise, or just this ...
0
votes
1answer
70 views

A simpler proof for an equality regarding sums of absolute values

Question Let $I$ be a set of indices, and for all $i \in I$, assume $$0 \le {a_i},{b_i},{c_i},{u_i},{v_i} \le 1 \;,$$ with $u_i+v_i=1$. Assume that $L$ is an upper bound on both ${\sum\nolimits_i ...
3
votes
1answer
130 views

Indicating when $|x + y + z| = |x| + |y| + |z|$ holds

This is a problem from Spivak's Calculus $3^{rd}$ ed., Chapter I, Problem $12$(vii) Indicate when $|x + y + z| = |x| + |y| + |z|$ holds, and prove your statement. My attempt: Clearly, $|x + y + ...
0
votes
2answers
372 views

Absolute value of polynomial

I don't seem to grasp why $$|1+6ωi-9ω^2| = 1+9ω^2, ω\gt 0$$ Where does the $6ω$ go? I'm thinking of $|6ωi|=6ω$.
2
votes
2answers
116 views

An Inequality Involving $\min(x, y)$

The following problem is from Spivak's Calculus (4th ed., pg. 18): Prove that if: $|x-x_0| < \min(\frac{\epsilon}{2(|y_0|+1)}, 1)$ and $|y - y_0| < \frac{\epsilon}{2(|x_0|+1)}$ then $|xy-x_0 ...
1
vote
1answer
347 views

Maximum of an absolute value complex function

I'm working my way through Marsden's Basic Complex Analysis book and I can't solve this problem. It's problem 23 of section 1.2 if that helps. Let $a$ be a complex number, find the maximum of ...
3
votes
2answers
274 views

Complex number with z to the power of 4

I have to find all $z\in C$ for which BOTH of the following is true: 1) $|z|=1$ 2) $|z^4+1| = 1$ I understand that the 1) is a unit circle, but I can't find out what would be the 2). Calculating ...
0
votes
1answer
86 views

Matrix integral of absolute exponential item

If $A=(a_{ij})$ is an $n\times n$ symmetric positive matrix, is it possible to calculate the following matrix integral? $$\int_{0}^{\infty}\left | e^{-A(t+1))}-e^{-At)} \right |\mathrm dt,$$ where ...
1
vote
1answer
3k views

Normal distribution with absolute value

I am new to the normal distribution topic. While I have understood and solved various different kind of questions, the normal distribution questions with absolute value, are the ones I have no idea ...
0
votes
2answers
103 views

How to solve $|2x +1|< 1/4$?

How do you solve $$|2x +1|< \frac{1}4$$
1
vote
3answers
197 views

Solve the equation : $x^2 − 6 |x − 2| − 28 = 0$

The following is an absolute value quadratic equation that I want to solve: $$x^2 − 6 |x − 2| − 28 = 0$$ Here is what I did : $x^2 − 6 |x − 2| − 28 = 0$ $x^2 − 6 |x − 2| − 28 = 0$ ...
0
votes
2answers
86 views

Prove $|x+1|\leq 4$ implies that $-4\leq x\leq 2$.

How do I prove that if $x$ is a real number, then $\lvert x+1 \rvert\leq 3$ implies that $-4\leq x\leq 2$. EDIT: $\lvert x+1 \rvert\leq 4$ should be $\lvert x+1 \rvert\leq 3$
2
votes
4answers
161 views

Prove that $||x|-|y|| \leq |x-y|$ [duplicate]

$||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ In Principles of MA(Rudin), the author said one sees easily that $||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ (p.88, Rudin) from the triangle ...
0
votes
1answer
122 views

Determine all the values of the parameter $a$ for which the inequality $3-|x-a|>x^2$ is satisfied by at least one negative $x$.

I wanted to know, how can I determine all the values of the parameter $a$ for which the inequality $3 - |x-a| > x^2$ is satisfied by at least one negative $x$. I tried for $x<a, |x-a|=-(x-a)$ ...
2
votes
3answers
146 views

Inequalities - Absolute Value $|2x-1| \leq |x-3|$

$$|2x-1| \leq |x-3|$$ Answer is $$-2 \leq x \leq \frac43$$ My Question is HOW?
1
vote
2answers
65 views

Finding $x$ from inequality: $\left | \frac{3^n + 2}{3^n + 1} - 1 \right | \le \frac{1}{28}$

Find $x$ in $\mathbb{Z}$ satisfying this inequality: $$\left | \frac{3^n + 2}{3^n + 1} - 1 \right | \le \frac{1}{28}.$$ I tried something, but I don't think it's correct. $$-\frac{1}{28} ...
-1
votes
1answer
628 views

Describe the set of points on the complex plane…

Describe the set of points on the complex plane for which $|z-2| + |z+2|=4$... So, I know you can solve this instantly, just by using definition, but I want to do it the long way.. So, $$|x- i*y ...
2
votes
5answers
280 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 $y < 0$ $x\geq0$ ...
1
vote
2answers
1k views

Proof the maximum function $\max(x,y) = \frac {x +y +|x-y|} {2}$ [duplicate]

I want to prove the maximum function max: $\mathbb{R} \rightarrow \mathbb{R}$, which is defined by $$\max(x,y) = \begin{cases}x, \text { if } x \geq y , \\ y, \text { if } x < y \end{cases}$$ ...
1
vote
2answers
208 views

Graphing - Absolute Value and Circle

The diagram Shows The Graphs of $y = |x + 2|$ and $y = \sqrt{4 - x^2}$ Write down the solution for $\sqrt{4 - x^2}$ is equal to or less than $y = |x + 2|$.
2
votes
1answer
45 views

Given certain conditions for $\delta$, how do I show that an inequality relating delta to x is true?

This is a problem out of a textbook (though there's no answer to this one in the back). If   $0 < \delta < 1$ and $|x-4| < \delta$ show:   $|\sqrt{x}-2| < ...
1
vote
2answers
804 views

How to find minimum of sum of mod functions?

How to find minimum value of $$|x-1| + |x-2| + |x-31| + |x-24| + |x-5| + |x-6| + |x-17| + |x-8| + \\|x-9| + |x-10| + |x-11| + |x-12|$$ and also where it occurs ? I know the procedure for find answer ...
1
vote
1answer
56 views

What is the minimum of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ over all complex numbers?

Find the Least value of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ My try:: Let $A(2,2)$ and $B(-1,5)$ and $C(6,-2)$ and $P(x,y)$ be a point Here $A,B$ and $C$ are the point ...
0
votes
1answer
120 views

Find the value of this logarithmic expression involving fifth root of unity.

Let $\alpha$ be the fifth root of unity. We then want to evaluate the expression $$\log |1 + \alpha + \alpha^2 + \alpha^3 - 1/\alpha |$$ Thanks in anticipation for your help in solving this!
10
votes
7answers
980 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 $$ ...
1
vote
3answers
110 views

Absolute value and roots

I've been trying to solve this problem and I always get 1, but the answer is $1 - 2x$. If $x<\frac12$ then what is $\left|x-\sqrt{(x-1)^2}\right|$ ?
4
votes
5answers
344 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 ...
3
votes
0answers
89 views

Phrase and symbol for “geometric absolute value”$ e^{|\ln(x)|}?$

I'm calculate the median fractional difference between two vectors (to characterise the error in a quantity with a high dynamic range). If $a/b = 0.1$, the fractional difference is $10$, and if $a/b ...
4
votes
2answers
160 views

How do we know that $|i!| = \sqrt{\pi \operatorname{csch} \pi}$?

(Source: Wolfram Alpha) Or, to write it out in full, $$|i!| = \sqrt{\frac{2\pi e^\pi}{e^{2\pi} - 1}}$$ How is this identity derived? Also, knowing this, could we find the exact values for the real ...
3
votes
2answers
85 views

How prove this $|x_{p}-y_{q}|>0$

let $$x_{1}=\dfrac{1}{8},x_{n+1}=x_{n}+x^2_{n},y_{1}=\dfrac{1}{10},y_{n+1}=y_{n}+y^2_{n}$$ show that: for any $p,q\in N^{+}$ we have $$|x_{p}-y_{q}|>0$$
3
votes
4answers
5k 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 ...
15
votes
6answers
1k views

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.

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.
4
votes
1answer
342 views

intersection of two graph

i would like to clarify some questions from GRE,which at first seems a little difficult to understand,suppose that we have some function $f(x)=|2*x|+4$ and graph of this function is given ...
1
vote
2answers
669 views

Express $y=|-x^2+1|$ as a piecewise function.

I'm unsure of how to start this problem. Any help would be greatly appreciated.
3
votes
3answers
411 views

Solving two greatest integer function equations

If $$x\lfloor x\rfloor =39 \quad \text{and}\quad y\lfloor y \rfloor=68.$$ What is the value of: $$\lfloor x\rfloor+\lfloor y \rfloor $$ I don't know how to solve such problems. I would appreciate ...
4
votes
1answer
59 views

Does the triangle inequality follow from the rest of the properties of a subfield-valued absolute value?

(This is a much more specific version of my earlier question from over a year ago.) Let $F$ be a field, let $E$ be an ordered subfield of $F$, and let $\;\; |\hspace{-0.03 in}\cdot\hspace{-0.03 in}| ...
1
vote
4answers
306 views

Please help me to prove this inequality: $|x|+|y|≥|x+y|$

Please help me to prove the following inequality: $|x|+|y|\geq|x+y|$ in which $x$ and $y$ are real numbers. Any help or hint would be appreciated. Thanks :)
1
vote
2answers
46 views

Question involving absolute function

I saw this interesting problem in a math puzzle forum:- Find all integral values of $t$ such that the equation $|s-1| - 3|s+1| + |s+2| = t $ has no solutions. How does one approach these kind of ...
5
votes
1answer
111 views

Solving equation with absolute value signs

Can someone see why there is only get one solution when solving following equation in this way: The equation $|x+1|+|2x-3|=|x-5| $ $$|x+1|+|2x-3|=|x-5| $$ $$\pm (x+1) \pm(2x-3)=\pm(x-5)$$ $$\pm x ...
1
vote
1answer
226 views

Intersection of a point and absolute value function contained within a circle

I'm attempting some crazy ideas while programming a game and ran into the following math problem that has been bugging me for a few days: Given a unit circle and a random point $P$ within the circle, ...
0
votes
1answer
157 views

Absolute value of infinite sum smaller than infinite sum of absolute values

A question emerging from an exercise in Ok, E. A. (2007). Real Analysis with Economic Applications. Princeton University Press. The exercise consists in showing that if $\sum_{i=1}^\infty x_i$ ...
3
votes
2answers
307 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
1answer
152 views

Question based on Triangle Inequality $\displaystyle |x+y|\leq |x|+|y|$

If $x,y,z\in \mathbb{R}-\left\{0\right\}$. Then prove that $\displaystyle 1\leq \frac{|x+y|}{|x|+|y|}+\frac{| y+z|}{| y |+| z |}+\frac{| z+x|}{| z |+| x |}\leq 3$ My Try:: Using Triangle Inequality ...
2
votes
1answer
68 views

Simplifying $\left|\left|\sqrt{-x^2}-1\right|-2\right|$

How do we simplify the expression $\left|\left|\sqrt{-x^2}-1\right|-2\right|$? This is very confusing. Do they cancel out and become just simply $\sqrt{-x^2}-1-2$?
0
votes
0answers
104 views

Calculation of the sub gradient of the first norm of a matrix

Lets say I have a matrix X and its first norm $||X||_1$. How do I calculate the subgradient of this norm with respect to matrix X itself.