# Tagged Questions

The absolute-value tag has no wiki summary.

1answer
28 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 $\sqrt{-x^2}-1-2$?
0answers
34 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.
1answer
19 views

### How to graph an absolute value equation?

How would you graph: $|x+y|=1$ ? I can do the normal $y=|x+1|$ and all that. But how would you do a question with two of these unknowns in the absolute value? Any help would be greatly appreciated, ...
1answer
36 views

### Is any norm on $\mathbb R^n$ invariant with respect to componentwise absolute value?

Given $\mathbf{x}=(x_1,...,x_n) \in \mathbb{R}^n$ , define $\mathbf{x}'=(|x_1|,...,|x_n|)$ . Then, is it $||\mathbf{x}'|| = ||\mathbf{x}||$ for every norm on $\mathbb{R}^n$ ? NB: The answer ...
1answer
19 views

### How to linearize the following LP

I want to minimize $|d_1-d_2|+e1+e2+e3$ where $d_1,d_2,e_1,e_2,e_3>=0$ and $|.|$ denotes the absolute value, for some linear constraints. Is there any way I can linearize the objective function?
1answer
59 views

### Absolute values in $\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$

in my math class we were given a list of indefinite integrals, and one of them was: $$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$$ My working: \int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}=\int ...
1answer
20 views

### how to find absolute value for complex fraction

I have a Fourier transfer equation $H(jw) = \frac{jwL}{(jw)^2LC+jw\frac{L}{R}+1}$, and I need to find frequency to make $|H(jw)|$ is max. I know I should take the derivative of $|H(jw)|$ then find ...
2answers
50 views

### Prove That $|a +b| = |a| +|b|$ if $a$ and $b$ Have Same Signs, And $|a +b| < |a| + |b|$ if $a$ and $b$ Have Opposite Signs (Proved Differently) [duplicate]

My Proof: This problem has mainly four cases, they are as follows: 1) $a, b > 0$ 2) $a, b < 0$ 3) $a > 0 > b$ 4) $a < 0 < b$ Let suppose that the sum of the real numbers \$a ...
1answer
41 views

### What is the modulus of a number?

What is the exact definition of the modulus of a number? As far as I know, it is the distance between the origin and the point associated with this number. So if \$z=a+bi \in \Bbb ...
5answers
76 views

### Prove That $|a +b| = |a| +|b|$ if $a$ and $b$ Have Same Signs, And $|a +b| < |a| + |b|$ if $a$ and $b$ Have Opposite Signs

My Proof: $|a +b| = |a| +|b|$ ..... $(i)$ $|a +b| < |a| + |b|$ ..... $(i)$ If $'a'$ and $'b'$ have same signs: Let $a$ and $b$ be equal to $-x$. Replacing $a$ and $b$ with $-x$ in the equation ...
2answers
72 views

### Absolute Value of $|-3 -2|$

$|-3 -2|$ is the distance between the points $-3$ and $-2$. If we solve it further then, in one way I get $|-5| = 5$. But $5$ is the distance between $0$ and $-5$ in this case. In other way, \$2 ...
2answers
33 views

### Absolute Value Problem, Solution and Method

Please check my method and also if I have solved the following problem correctly: Problem: $f(x) = |x - \frac12| + |x + \frac12|$ If $x = -1$, then: $f(-1) = |-1 - \frac12| + |-1 + \frac12|$ From ...
3answers
36 views

### Question about absolute value in inequalities

My book presents the following: $$7 \le x \le 9$$ so $$-1 \le x - 8 \le 1$$ and $$|x-8| \le 1$$ I usually get confused with the way that taking the absolute value of an expression works. Could ...
3answers
41 views

### finding values for absolute convergence

Find all values of real number p or which the series converges: $$\sum \limits_{k=2}^{\infty} \frac{1}{\sqrt{k} (k^{p} - 1)}$$ I tried using the root test and the ratio test, but I got stuck on ...
1answer
43 views

### Absolute of a trig function

Consider the function $$f(x) = 1\dfrac{1}{2} - 3\sin \left(\dfrac{1}{2}x \right).$$ I need to find the absolute of this function, which to my eye would just be  f(x) = 1\dfrac{1}{2} + 3\sin ...
2answers
43 views

### Absolute Convergence of a Series

Find all values of real number p for which the series converges absolutely $$\sum_{k=2}^{\infty} \frac{1}{k\, (\log{k})^p}$$
2answers
74 views

### Calculating the integral $\int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$

How do we calculate the following integral: $$\int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$$
2answers
86 views

### How to evaluate the inequality $|x+1|<-1$?

Okay perhaps the title isn't specific enough, I didn't know how to word it exactly. I'm finding the interval of convergence for a power series and i know the answer to be (-2,0] I end up with the ...
2answers
43 views

### Can summations distribute across absolute values?

Can I distribute a summation as follows? $$k\sum_{x \in X} \left| x - b \right| = \left| \left(k\sum_{x \in X}x \right) - \left( k\sum_{x \in X}b \right) \right|$$
0answers
76 views

### Continuous, differentiable, continuously differentiable

I came across the following problem: Let $\alpha \in \mathbb R$. Where is the function continuous, differentiable, continuously differentiable? f(x) = \begin{cases} x|x|^\alpha & ...
2answers
43 views

### Finding absolute max and min values of function

Function given as $f(x,y) = 3x^2 + 2xy^2$. If $(x,y)$ lies in the region inside including edges of the triangle in the first quadrant given by $x\ge0, y\ge0, y\le2-x$. Reduce $f$ to a single variable ...
1answer
82 views

### For what $\alpha \in \mathbb R$ is $|x|^\alpha$ differentiable in $x=0$?

I came across the following question: For what $\alpha \in \mathbb R$ is $|x|^\alpha$ differentiable in $x=0$? What I have tried: Since for $\alpha = 1$ is clearly non-differentiable in ...
1answer
26 views

### Is this (or when) does this equality hold?

Let $a,b,c,d \in \mathbb{R}$ and $x,y$ are variables which are also real numbers $$|ax + by|^2 + |cx + dy|^2 + 2|ax + by||cx + dy| = (ax + by)^2 + (cx + dy)^2 + 2(ax + by)(cx + dy)$$ Is this always ...
2answers
59 views

### proving $|x - 1| < {1\over4} \Rightarrow |2x - 1| \geq {1\over 2}$

I tried solving the above, consider that: ($x \in R)$, I know it's not a complicated problem to solve though I struggle getting on with this question, What I've done far is: \$|x-1|<{1\over4} ...
3answers
55 views

### Find $x$ for absolute value inequalities

I'm trying to figure out this inequality: $|x+1| + |x| \leq x^2$ I thought about trying it with two cases: $(x = -x)$ and $(x = +x)$ but I don't seem to find out how to go through from here, ...
1answer
54 views

### Finding The Contour Maps Of A Function Of Two Variables

I am given the function $f(x,y) = \ln|y-x^2|$, and am suppose to find the contour maps. Let $z = c = f(x,y)$. $c = \ln|y-x^2| \rightarrow e^c = e^{\ln|y-x^2|} \rightarrow e^c = |y-x^2|$ I know I ...
3answers
67 views

### Solving $2|x+1|>|x+4|$

I'm trying to solve the following equations and inequalities for $x\in\mathbb R$: $$2|x+1|>|x+4|$$ I know I'm supposed to consider the intervals $(-\infty,-4), [-4,-1]$ and $(-1,\infty)$ but ...
3answers
103 views

### Determine all solutions to $|x+12|+|x-5|=15$

Determine all solutions to the following. $$\lvert x+12\rvert +\lvert x-5\rvert =15.$$
1answer
146 views

### Finding the points of the curve where the tangent line is horizontal

The curve given is $\displaystyle y = \ln|x-2| + x + \frac{12}{x-2}$. Find the points of the curve where the tangent line is horizontal. My first stumbling block is the absolute value function. I ...
3answers
105 views

### How does one calculate the integral of the sum of two absolute values?

I know how to find the integral of just one absolute value, but this problem presents the integral of the sum of two absolute values. Help! I want to evaluate: $$\int_a^b{(|x-1| + |x+1|) dx}$$
2answers
55 views

### Prove $||a| - |b|| \leq |a - b|$ [duplicate]

I'm trying to prove that $||a| - |b|| \leq |a - b|$. So far, by using the triangle inequality, I've got: $$|a| = |\left(a - b\right) + b| \leq |a - b| + |b|$$ Subtracting $|b|$ from both sides yields, ...
2answers
86 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 $6 - Ce^{-x}$ ...
3answers
108 views

### Exposition On An Integral Of An Absolute Value Function

At the moment, I am trying to work on a simple integral, involving an absolute value function. However, I am not just trying to merely solve it; I am undertaking to write, in detail, of everything I ...
4answers
170 views

### Truth set of $-|x| \lt 2$?

An exercise in my Algebra I book (Pearson and Allen, 1970, p. 261) asks for the graph of the truth set for $-\left|x\right| \lt 2; x \in \mathbb{R}$. I've re-stated the inequality in the equivalent ...
3answers
179 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 :)
2answers
174 views

### How to find critical points of an absolute values function

I am asked to find How many critical points does the function $g(x) = |x^2 − 4|$ have? I know that the result is $3$ but I can only find $2$. What I do, is to equal the equation to $0$, so $x^2-4=0$ ...
1answer
38 views

### Integral of a mode squared [closed]

Hi could someone show me how to calculate $\psi_0$ out of the equation below? $$\int \limits^{}_{V} \big|\psi_0 \sin (\omega t - kx) \big|^2 \, \textrm{d} V = 1\\$$
2answers
43 views

### Absolute ratios

I'm curious about the following idea: suppose we have two values $P$ and $Q$, and the magnitude of the ratio $\frac{P}{Q}$ is between $0$ and $\infty$. If $P$ is smaller, then it's between $0$ and ...
3answers
106 views

### Adding equations in Triangle Inequality Proof

Inequality to prove: $|a+b|\leq |a| + |b|$ Proof: $-|a| \leq a \leq |a|$ $-|b| \leq b \leq |b|$ Add 1 and 2 together to get: $-(|a|+|b|)\leq a+b\leq|a|+|b|$ $|a+b|\leq|a|+|b|$ What is the ...
1answer
57 views

### Integration Involving the Absolute Function

How do I integrate the double integral of the form $|x^2-y|$ with the boundaries $-1\leq x\leq 1$ and $-1\leq y\leq 1$?
2answers
85 views

### Solving $| ax + b | \gt c$

Does $| a x + b | > c$ always result in two solutions, $x \gt \dfrac{c - b}{a}$, and $x \lt\dfrac{-c - b}{a}$? If I understand correctly, the first solution, $x > \dfrac{c - b}{a}$, is only ...
1answer
36 views

### minimum of absolute value

If we consider the following problem $$\mathbb{E}[(Y-y)^2 | X=x]$$ I can easily show that the minimum with respect to $y$ occurs at $$y=\mathbb{E}[Y |X=x]$$ How can I find the minimum of  ...
3answers
85 views

### Graphing Absolute Value Functions

Given: $y = -|2x + 1|-3$ I came up with the graph of... $1, -7$ $0, -5$ $-1, -3$ $-2, -1$ $-3, 1$ If you were to graph this, it would turn out to be an entirely straight line. This is an ...
2answers
68 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 ...
2answers
80 views

### Log laws and modulus

If you have the log of a modulus, (like after integration), how do the log laws work? So if you have $a\ln\left|2x-3\right|$ does it become: $\ln\left|(2x-3)^a\right|$ or $\ln(\left|2x-3\right|)^a$, ...
1answer
74 views

### Absolute function continuous implies function piecewise continuous?

I have a simple true/false question that I am not sure on how to prove it. If $|f(x)|$ is continuous in $]a,b[$ then $f(x)$ is piecewise continuous in $]a,b[$ Anyone that can point me in the ...
2answers
68 views

### True/false question: limit of absolute function

I have this true/false question that I think is true because I can not really find a counterexample but I find it hard to really prove it. I tried with the regular epsilon/delta definition of a limit ...
1answer
83 views

### double integral of an absolute function

I'm just a little unsure of how to tackle this one. I understand that typically you would separate the integral into two for where x is positive or negative, I'm just unsure of how to separate it for ...
1answer
59 views

### Separable first-order linear equation and absolute value removal

We can use the integral of $\frac{1}{x}$ in order to solve a separable first-order linear equation like this: $\frac{dy}{dt} + f(t) y = 0$ $ln |y| = \left(-\int f(t)\,dt\right) + C$ and then: ...
2answers
177 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} ...
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