For questions about or involving the absolute value function.

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1
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3answers
105 views

Finding the limit: $\lim_{x\to3^+} \frac{\sqrt{x-3}}{|x-3|}$

Can anyone tell me how to properly solve this limit? $\displaystyle \lim_{x\to3^+} \frac{\sqrt{x-3}}{|x-3|}$ I know the answer is positive infinity, and I would know how to do the problem if $x$ was ...
0
votes
0answers
11 views

Using the negation of a statement to disprove original statement

Prove the following statement is false by first writing the negation, then proving the negation is true: For all sets, S, if S ⊆ ℕ, then there exists some t ∈ S such that |t| ≥ 1. So far, I've ...
2
votes
3answers
34 views

How to solve $\lim _{k\rightarrow 1}\dfrac {1+\ln k}{\left| \ln \left( \ln k\right) \right| }$

How to solve $\lim _{k\rightarrow 1}\dfrac {1+\ln k}{\left| \ln \left( \ln k\right) \right| }$ I stucked at the denominator.
6
votes
5answers
122 views

How to find $\int|\cos x|\,dx$?

How do I find closed form for $\int|\cos x|\,dx$ for all real $x$? It can be expressed as incomplete elliptic integral of the second kind: $$\int|\cos x|\,dx=\int\sqrt{1-1^2\sin^2x}\,dx=E(x,1)$$ ...
0
votes
2answers
24 views

Limit of |x-2| as x approaches -2

I believe that it equals -4. In the epsilon-delta definition, we can set delta equal epsilon and I become this satisfies the definition. The problem is I can't seem to prove based on this that 0 less ...
0
votes
2answers
22 views

How to simplify abs(x)/x

I've been trying to find a way to simplify $\frac{|x|}{x}$ if $x$ is real and $\neq{0}$. The two possible outcomes to this are $\pm{1}$ but I believe there is one required answer. I've noticed that if ...
3
votes
4answers
227 views

Finding the definite integral of a function that contains an absolute value

The integral in question is this: $\int_{-2\pi}^{2\pi}xe^{-|x|}$ My attempt: Since there is a modulus, we split it up into cases. I'm not really sure which cases to split it into, do I just ...
1
vote
5answers
56 views

For what real number $c$, this equation has exactly three solutions?

For what real number c does the equation $|x^2 + 12x + 34| = c$ has exactly three solutions?
1
vote
3answers
30 views

Finding the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$.

So I am trying to find the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$. I know the integral converges, and I know the answer as well, but I am confused on how to get the correct answer. My problem ...
1
vote
3answers
36 views

Limit-related inequalities with absolute values

Recently I decided to learn calculus on my own and I stumbled across something which I cannot figure why is correct. Let $f$ be some function for which you know only that if $0<|x-3|<1$, then ...
0
votes
1answer
27 views

Is an absolute value acting on complex numbers a linear operator?

I just have to prove that it isn't with O(A+B)=O(A)+O(B) and O(kA)=k(OA) where O is the linear operator (i.e the absolute value), A+B and A would be a complex number, and k is some real constant. I ...
0
votes
0answers
20 views

Definition of the absolute value of a polynomial

I am having a hard time verifying that this is the definition of the absolute value of a polynomial: Given a polynomial with (possibly) complex coefficients: $p(z) = a_0 + a_1 z + a_2 z^2 + ... + ...
3
votes
4answers
151 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.
4
votes
5answers
123 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 ...
-2
votes
2answers
72 views

What is $\sqrt{x^2}$ when $x<0$? [closed]

$x\in \mathbb{R}$\ $\{0\}$ $$\frac{\sqrt{x^2}}{|x|}+1 =?$$ What is the answer when $x \lt 0$? $2$ or $0$?
1
vote
1answer
243 views

How to express 2 absolute values as a piecewise function??

I understand how to solve 1 absolute value as a piecewise function. $f(x)=|x-1|$ $$ f(x)= \begin{cases} x-1& \text{if }x\ge1\\ 1-x&\text{if }x<1 \end{cases} $$ But when a function ...
1
vote
1answer
21 views

Absolute values and inequalities

So I've been trying to solve this one for a few hours and am now out of ideas on how to approach this problem. Here are the inequalities: $$\text{show that if}$$ $$z,w \in \Bbb C$$ $$|z| < ...
1
vote
3answers
145 views

Is there a function whose derivative is $|x|$?

Is there a function $y=f(x)$ such that $$\frac{df}{dx}|_{x=a} =|a|$$ for all $a\in \mathbb R$? I'm in a debate with my friend over it and we are stuck
0
votes
1answer
34 views

On complex numbers and absolute values

Exercise 1.31 of Analysis by Apostol states: Given three complex numbers $z_1,z_2,z_3$ such that $|z_1| = |z_2| = |z_3| = 1$ and $z_1 + z_2 +z_3 = 0$. Show that these numbers are vertices of an ...
1
vote
4answers
26 views

Showing an Absolute Value Inequality Problem Proof

I tried solving this question but it does not works for me. Q.) Show that $\left|x + \frac1{x}\right| \ge 2$ for all $x \ne 0$ There are two ways to do. One is squaring and other is to use absolute ...
2
votes
4answers
1k 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 ...
2
votes
2answers
114 views

How to prove this max absolute value equation?

How to prove this equation? $$\max(|x_1-x_2|,|y_1-y_2|) = \frac{\left|x_1+y_1-x_2-y_2\right|+\left|x_1-y_1-(x_2-y_2)\right|}{2}$$
2
votes
2answers
51 views

Finding the limit of $F(x)=\frac{x^2-4}{|x+2|}$

Let $F(x)=\dfrac{x^2-4}{|x+2|}$ and find the following limits $(a) \; \; \lim_{x \to -2^-}F(x)=$ $(b) \; \; \lim_{x \to -2^+}F(x)=-4$ $(c) \; \; \lim_{x \to -2}F(x)=DNE$ I substituted $-2$ to find ...
1
vote
3answers
38 views

What are the steps to solving |3x + 1| > |2x - 7| with the given answer as $(-∞,-8)\cup(6/5,∞)$?

What are the steps to solving $|3x + 1| > |2x - 7|$ with the given answer as $(-∞,-8)\cup(6/5,∞)$? I am having difficulty with understanding inequalities with absolute value functions on both ...
2
votes
3answers
136 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?
3
votes
2answers
327 views

Why do definitions of distinct conic sections produce a single equation?

I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
-2
votes
1answer
28 views

Can one apply absolute value to both sides of an inequality?

Can I claim this: $$c>b \implies |c|>|b|$$ with $c>0$? I ask because I want to use it in a proof but I am not sure. Thank You.
4
votes
2answers
53 views

Piecewise linear function and absolute value

While writing a solution to homeworks for my students, I had to write the function $$f(x)=\left\{\begin{array}{ll} \frac{x+2}{2}, & x\leqslant -4\\ \frac{x}{4}, & -4\leqslant x\leqslant 4 \\ ...
0
votes
1answer
31 views

Maclaurin series for $\frac{1}{|1+x|}$

I believe that there is no Maclaurin Series for $\frac{1}{|1+x|}$ as the latter is not differentiable at $x=-1$. However, would it be appropriate for me to refer $\frac{1}{|1+x|}$ as 'not a smooth' ...
2
votes
6answers
79 views

Adding $2$ absolute values together: $|x+2| + |x-3| =5.$ [duplicate]

I came across a very basic absolute value question $|x+2| + |x-3| =5.$ Initially, I thought the answer was $x=-2$ and $x=3$ because I let each absolute values be either positive and negative and ...
0
votes
5answers
131 views

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

Determine all solutions to the following. $$ \lvert x+12\rvert +\lvert x-5\rvert =15.$$ How can I solve problems like this? Should I try graphing the function? Should I somehow consider various ...
12
votes
2answers
99 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 ...
0
votes
2answers
48 views

How to find roots for $y = ||x^2-x-20|-8|$

$$ y = ||x^2-x-20|-8| $$ After I set $y = 0$, I do not know how to deal with multiple absolute values.
9
votes
9answers
2k 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 ...
1
vote
1answer
65 views

Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which says what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
-3
votes
1answer
472 views

What properties of the absolute value function should one remember? [closed]

When one begins to study real analysis, the absolute value function quickly enters and a large number of exercises involve manipulations with it. What are the basic properties of absolute value that ...
1
vote
1answer
90 views

Inequality: $\left|x^3-y^3\right|<|x|^3+|y|^3$

Could anyone show me why $$\left|x^3-y^3\right|<|x|^3+|y|^3$$ for all real numbers (x,y) except 0? I'm thinking of whether of how to remove the modulus sign on the left hand side of the ...
0
votes
2answers
78 views

Expressing absolute value equations as piecewise functions

I'm not sure how to express this function in piecewise form without using absolute values: $$ f(x) = 3|x-2| - |x+1|$$ I know how to do it when there is just one absolute value, such as: $$g(x) = ...
3
votes
2answers
97 views

Does my proof of $|x+y| \le |x| + |y|$ make sense? How do I conclude a proof?

Thank you for reading it. I know I made a lot of mistakes. This is my first ever proof that I have attempted. Another note is that I only have been studying proofs for about a week. Any advice will be ...
2
votes
2answers
29 views

Taking root from absolute expression

Why is the following true? (Where all terms are positive) $$|x-y| < \epsilon^2 \implies |\sqrt x - \sqrt y| < \epsilon$$
3
votes
1answer
65 views

Geometric idea behind equations of the form $|x-a|\pm|x-b|=c$

So let's say I want to solve $$|x-a|\pm|x-b|=c$$ Using the classic multiple cases approach, one can show that the solutions are given by $$x=\frac{a+b\pm c}2 $$ But how can one make sense of this ...
4
votes
2answers
105 views

Calculating the best match between two sets

I’m a PHP developer and I have a problem calculating the perfect match between two different data sets. I have data sets from companies, where each company defines the requirements for a specific ...
-1
votes
3answers
137 views

An inequality with absolute value and a parameter: $|x-4|>a$

Solve : $|x-4|>a$. Case 1: $a>0$; Case 2: $a<0$ Progress I am getting answers which look similar in both cases: Let $a>0$ so $x>4+a$ or $x<4-a$ , Let $a<0$ so ...
3
votes
0answers
36 views

Nested absolute operations

The question is: are the following two functions equivalent? And if yes, what properties of the absolute value should I use to prove it? $f_1(x,y,z)$ = $|\, x + |y+z| \,|$ $f_2(x,y,z)$ = $| \,|x+y| ...
3
votes
1answer
53 views

Why is the value of $\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|\,dx$ independent of integer parameter $n$?

I am not able to find an easy solution for the following formula $$\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|dx=4+\frac{4}{3}\pi\sqrt{3}.$$ Please help me prove it. Why it does not depend on the (positive) ...
-3
votes
1answer
69 views

Someone can solve this limit? [closed]

$$f(x) = \frac{9-2\sqrt{\left\vert\,x\,\right\vert}}{3\sqrt{-x}}$$ $$\lim_{x\to-\infty} f(x) = l$$ I need the method of calculate l and solve this: (proving limit using epsilon-delta definition) ...
3
votes
0answers
82 views

Dealing with absolute values after trigonometric substitution in $\int \frac{\sqrt{1+x^2}}{x} \text{ d}x$.

I was doing this integral and wondered if the signum function would be a viable method for approaching such an integral. I can't seem to find any other way to help integrate the $|\sec \theta|$ term ...
0
votes
2answers
40 views

Graphs for mod functions

Can someone please teach me how to obtain graphs for the following types of functions: $2+3|x-1|$ $|x-1|+|x|+|x+1|$ $|x-1|-|x|-|x+1|$ $|x-1|^2$ Thanks.
0
votes
1answer
19 views

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$, where $aj+b$ is a complex number, and $|f(x)|$ is the modulus function. In the past I've been calculating $|(aj+b)^{-1}|$ by multiplying the numerator and ...
4
votes
3answers
93 views

Is it always true? $\left|A-B\right| \le \left|A\right| + \left|B\right|$

Is it always right to claim that: $$\left|A - B\right| \le \left|A\right| + \left|B\right|$$ where $A, B \in \mathbb{R}$ ?