For questions about or involving the absolute value function.

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5
votes
2answers
143 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, ...
6
votes
2answers
2k 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 $6 - Ce^{-x}$ ...
3
votes
3answers
309 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 ...
1
vote
4answers
223 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 ...
2
votes
3answers
4k 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 :)
2
votes
2answers
6k 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 $x^2-4=0$ ...
2
votes
3answers
203 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 ...
1
vote
3answers
292 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 ...
1
vote
1answer
104 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 $-1\leq y\leq 1$?
1
vote
2answers
115 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 ...
1
vote
1answer
47 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 $$ ...
0
votes
3answers
215 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 ...
5
votes
2answers
411 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 ...
1
vote
2answers
644 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 $\ln(\left|2x-3\right|)^a$, ...
0
votes
1answer
333 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 ...
0
votes
2answers
216 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 ...
1
vote
1answer
490 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 ...
2
votes
1answer
287 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: ...
2
votes
2answers
233 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} ...
0
votes
1answer
47 views

whats the absolute time interval?

For the time interval $1< \lvert t+1 \rvert \leq 3 $ I am trying to solve for t to get my range to plot a function. I know $\lvert t+1 \rvert \leq 3 \iff -3\leq t+1 \leq 3 \iff -4 \leq t \leq 2$ ...
3
votes
1answer
105 views

Integral of absolute e-function

I have to integrate the following function: $$\int e^{-|x|}$$ I tried this and I don't think, that this is right. So can you tell me, where my fault is? $$\int e^{-|x|} = ...
5
votes
1answer
150 views

Ring of integers in a field of fractions

Let $R$ be ring with complete non archimedian absolute value. Let $Q$ be the associated field of fractions with the extended absolute value. Does the ring $O_Q = \{x\in Q | |x|\leq 1\}$ is complete ...
5
votes
2answers
652 views

Why is the absolute value needed with the scaling property of fourier tranforms?

I understand how to prove the scaling property of Fourier Transforms, except the use of the absolute value: If I transform $f(at)$ then I get $F\{f(at)\}(w) = \int f(at) e^{-jwt} dt$ where I can ...
1
vote
2answers
320 views

How to prove this simple statement: $\max\{a,b\}=\frac{1}{2}(a+b+|a-b|)$ [duplicate]

I am trying to prove this statement. for any $a,b \in \mathbb{R}$, $$\max\{a,b\}=\frac{1}{2}\big(a+b+|a-b|\big)$$ and $$\min\{a,b\}=\frac{1}{2}\big(a+b-|a-b|\big)$$ I am eating myself not knowing ...
1
vote
3answers
2k 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 ...
5
votes
1answer
5k views

Derivatives of functions involving absolute value

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

Constructing new numbers from negative absolute value

Before the construction of the complex numbers, people thought you couldn't take the square root of a negative number. Then came along of the definition of the imaginary unit $$i^2 = -1$$ and now ...
2
votes
4answers
201 views

double absolute values

I am having a little bit of problem with an inequality with nested absolute values: $$|z^2-1| \ge |z+|1-z^2||$$ I've tried solving it by making three cases, $z\ge1$, $z\le-1$ and $z$ between $1$ and ...
1
vote
2answers
199 views

When does a absolute value equation have one unique solution?

Find $m \in \mathbb R$ for which the equation $|x-1|+|x+1|=mx+1$ has only one unique solution. When does a absolute value equation have only 1 solution? I solved for $x$ in all 4 cases and got ...
1
vote
2answers
77 views

Confusion solving $\sqrt{4m^2-4m+1}+|1-2m|\leq2$, weird solution.

I am trying to solve $\sqrt{4m^2-4m+1}+|1-2m|\leq2$. Since i know $|1-2m| = \pm(1-2m)$ i tough solving $\sqrt{4m^2-4m+1}+1-2m\leq2$ and $\sqrt{4m^2-4m+1}-1+2m\leq2$. As solutions i get $0\leq2$ and ...
1
vote
2answers
97 views

How to prove this inequality $x,y\in\Bbb R$, $|x|<1,|y|<1$ show that $\bigg|\frac{x-y}{1-xy}\bigg| < 1$ (and similar ones)

I have to show that the inequality below is true, i tried some thing but got stuck, i tried to eliminate the absolute value $-1<\frac{x-y}{1-xy}<1$ and then solve for $x$ and $y$ with no ...
7
votes
2answers
166 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 ...
0
votes
1answer
72 views

CDF for random variable $X(\omega) = 2(1-|2\omega - 1|)$

I don't know how to calculate this cdf, the modulus is very annoying, because the cdf definition is $P(X< x)$ in my case $P(\omega < x)$. But in the modulus equality I get this $P(-\omega < ...
1
vote
2answers
196 views

Calculating the absolute value of a complex number - am I right?

To calculate the absolute value of a complex number u must use the following formular $(a^2+b^2)^½$=|a+bi| So for instance with -4-5i would have the absolute value ...
1
vote
2answers
393 views

Absolute value and sign of an elasticity

In my microeconomics book, I read that when we have $1+\dfrac{1}{\eta}$ where $\eta$ is an elasticity coefficient, we can write $1-\dfrac{ 1}{|\eta|}$ "to avoid ambiguities stemming from the negative ...
0
votes
1answer
58 views

how to prove if $a|b$ and $b\neq 0$, then $|a|\leq|b|$

where the conditions are: $a \neq 0$, $b \neq 0$ and $a$ and $b$ are integers. maybe i'm missing something very basic about the properties of an absolute values. My approach was to supposed, on the ...
4
votes
3answers
806 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| ...
0
votes
1answer
27 views

Find absolute value inequality describing the result of measurement

This is a problem from my homework where a sample of a quantity is $37.5\pm 1.2$ grams. And if the actual quantity is $x$, write the results as an absolute value inequality and solve for $x$. I ...
-1
votes
1answer
144 views

The maximum absolute value of DFT of window vector

Let x=[1, ⋯ ,1, 0, ⋯ ,0] be a window vector of length N, which consists of B consecutive 1s and the remaining N-B consecutive 0s. I took the N-point DFT on x and got X=[X_0, X_1, ⋯, X_(N-1)] which is ...
4
votes
3answers
168 views

Is there an alternate definition for $\{ z \in \mathbb{C} \colon \vert z \vert \leq 1 \} $.

Is there a method of constructing a subset of a reasonably arbitrary ring so that when the construction is applied the $\mathbb{C}$ the result is $B = \{ z \in \mathbb{C} \colon |z| \leq 1 \} $? My ...
6
votes
2answers
210 views

Sum of two absolute values in complex plane

I'm trying to find out all $z \in C$ that satisfy the following condition: $|z+1|+|z-i|=3$ I understand that $|z|=r$ represents a circle with a radius of $r$. I also understand that $|z+1|=r$ can ...
0
votes
2answers
1k views

Distribution and Absolute Value

I have a question about distribution and absolute values. I was solving a problem and was wondering if it would be okay to distribute a number into an absolute value with two terms. For example ...
5
votes
1answer
124 views

An absolute value problem

Let $a$ and $b$ in $\mathbb{R}$ 1) Show that $||a|-|b||\leq|a+b|\leq|a|+|b|$. 2) Prove that the one or the other of the two inequalities is an equality. It's fine whit the 1st question but i can't ...
2
votes
4answers
440 views

Double module in a inequality

Can somebody explain to me (or link me a site which does) how to solve this? $$ ||x+1| -1| \geq 3 $$ I have no idea how to work out this double absolute value sign.
0
votes
2answers
240 views

absolute value inequality limit

Could someone verify this proof? Prove $\lim\limits_{n\to\infty} C_n = \lim\limits_{n\to\infty} \dfrac{4n+3}{7n-5} = \dfrac{4}{7}$ Proof: Let $\epsilon > 0$ and take $N = \dfrac{41}{49\epsilon} + ...
3
votes
3answers
911 views

Absolute value $\epsilon - \delta$ limit definition

For some reason, I have trouble getting absolute value right. This is of a great importance in the definition of the limit. How do I solve the following inequality for $x$: $$|x -a| < \epsilon$$ ...
2
votes
3answers
5k views

Taking the square roots in inequalities

I have a question regarding taking square roots in inequalities. I have a problem asking: Suppose $3x^2+bx+7>0$ for every real number x. Show that $|b|<2\sqrt{21}$. In an earlier question it ...
1
vote
4answers
371 views

Is it possible to take the absolute value of both sides of an equation?

I have a problem that says: Suppose $3x^2+bx+7 > 0$ for every number $x$, Show that $|b|<2\sqrt21$. Since the quadratic is greater than 0, I assume that there are no real solutions since $y = ...
-3
votes
2answers
157 views

Help solving equation involving absolute value, $|x|+x=0$

I have seen the teacher solving this equation, but for me isn't explained. May someone solve that question. Are there another solutions. $$|x|+x=0$$ Solution $$x+(-x)=0$$ $$0=0. $$
3
votes
1answer
2k 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 ...