For questions about or involving the absolute value function.

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0
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1answer
64 views

absolute value inequality with complex number

Strangely, I don't find easily on the internet sources about inequalities with complex numbers. In this moment, I am interested to absolute value inequalities with complex numbers but would be good ...
1
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3answers
65 views

Different way to solve $|x-3|<|2x|$

I know two ways I can solve $|x-3|<|2x|$ By squaring both sides By interpreting the inequality as a statement about distances on the real line. Question: How can I solve this inequality ...
2
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1answer
59 views

Solve the equation within 'floor function'

I added my solution, but I'm not sure I've got it right. I'd like to know what you think. The question: Solve the equation: $$\lfloor |x+1|-|x| \rfloor \ge x^2.$$ the left and right symbols ...
0
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3answers
56 views

Is it possible to find the absolute value of an integer using only elementary arithmetic?

Using only addition, subtraction, multiplication, division, and "remainder" (modulo), can the absolute value of any integer be calculated? To be explicit, I am hoping to find a method that does not ...
3
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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 ...
1
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0answers
29 views

Is this a misprint or am I missing something?

What I'm given is this: Evaluate: x = 5, |x| -2 I'm thinking they probably mean |x|=-2, in which case the evaluation would be false. But then again I second ...
2
votes
3answers
56 views

Definition of limit with$ f(x)=|x^3|$

Using the definition of the limit I tried to find the derivative of $f(x)=|x^3|$. I came up with: $$f'(x)=\frac{3x^5}{|x^3|}$$ Question: Why is the derivative (according to this answer) not defined ...
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0answers
25 views

Absolute value with inequalities

Can we solve the following $ |f(x)| + |g(x) | < b$ by taking the intersection of the solutions for $f(x) + g(x) < b$ $-f(x) - g(x) < b$ $f(x) - g(x) < b$ $-f(x) + g(x) < ...
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0answers
31 views

Show that $|ax^2+bx+c|<1$ gives us $|c|<1$

$a,b,c \in \mathbb{R}$ and for all $-1<x<1$ Show that if $|ax^2+bx+c|=<1$ So : 1) $|c|=<1$ 2) $|a+c|=<1$ 3) $a^2+b^2+c^2=<5$ For the first one ; if I choose x=0 So |c|=<1 ...
0
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0answers
18 views

Graphing an Absolute Value Equation

How would I graph the equation: abs(x)+abs(y)=1+abs(xy). I have tried to consider cases, but am not sure if I need to graph it by considering a piecewise-defined function.
0
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1answer
25 views

how to rearrange for y for: $\ln{|y|}= \frac{x^3}{3}$?

Is it $y= \pm e^{\frac{x^3}{3}}$? I am not sure about whether the plus minus sign is correct. Just here to look for confirmation or correction! Thanks!
9
votes
2answers
210 views

Modulus Equations

$$ |x + 1| + |x − 1| = x + 4$$ The only way I can solve this equation is to graph it...Through graphing, I get the following solutions: $$x = -\frac{4}{3}, 4$$ Is their a general algebraic method ...
3
votes
3answers
90 views

How to solve this absolute value inequality? $ |x| + |x - 2| \gt 5 $

I'm not sure how to solve this inequality. Can someone please explain step-by-step? Thanks! $ |x| + |x - 2| \gt 5 $
1
vote
2answers
29 views

Finding a Real Number using epslion

Fix a real number $x$ and $\epsilon>0$. If $|x-1|\le \epsilon$ show $|2-x|\ge 1- \epsilon$ I think we were supposed to use the triangle inequality to show this. If we use the triangle inequality ...
1
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0answers
47 views

Compare absolute values of two expressions!

I have two sets of numbers as follows: $$X = \{x_1, x_2, ..., x_n\}\\ Y = \{y_1, y_2, ..., y_n\}$$ And a number $r$. Let $x^\ast$ and $y^\ast$ is average values of set $X$ and $Y$ respectively. ...
0
votes
1answer
59 views

Max function is a metric?

I was wondering if the max function is a metric or if, in particular, $\max(|x + y|, 1)$ is equal or less than $\max(|x|, 1) + \max(|y|, 1)$ with $x$ and $y$ belonging to $R$.
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0answers
24 views

Turning points of an absolute value equation

Given the equation $$\left| x\; -\; c \right|\; +\; \left| x\; +\; c \right|\; -\; \left| y \right|\; -\; \left| \left| x\; -\; c \right|\; -\; \left| y \right| \right|\; = b \text{,} -c, c, b \in ...
0
votes
1answer
107 views

Absolute Value equations in 2 variables (both x and y ) (a relation based on absolute)

I came across equations in a math book that contains absolute of both x and y. I have done many complicated absolute equations and inequalities, but with only the x being in absolute bars. I don't ...
0
votes
1answer
41 views

Solving absolute value equations with two variables

I have an absolute value term in equation, which looks like this $$||x| - |y||\text{.}$$ If I remember correct, this term will have 4 variations, depending on values of $x$ and $y$. $x > 0\wedge ...
5
votes
1answer
30 views

Does $|x^*|=|x|$ in a star ring with an absolute value?

Let $R$ be a star ring with an absolute value. Is it true that $|x^*|=|x|$ for all $x\in R$? Here a star ring is a ring with a function $*:R\to R$ called conjugation such that $(x+y)^*=x^*+y^*$ ...
1
vote
1answer
26 views

$|x^{-1}-y^{-1}|=|x-y|/|x||y|$ in a normed ring

I hit a slight snag when trying to prove that the inverse function $x\mapsto x^{-1}$ on the unit group is continuous in a ring with an absolute value, so I'd like some confirmation that the theorem is ...
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3answers
26 views

Absolute Value Inequality Including Itself

Given a real number $a$. Will it be correct to use the following inequality in the proof: $$-a\le|a|\le a$$ Although "less" and "greater" parts never actually happen, the whole equation will always ...
0
votes
2answers
85 views

How to solve inequalities with more than one absolute value expression

There are two parts to this question. 1. I'm seeing the correct method to solve these types of inequalities as something to do with "transition points". I don't quite understand this method. How do we ...
1
vote
1answer
42 views

Absolute value of product is less than product of absolute values: $|(1+a_1)(1+a_2)\dots (1+a_n)-1|\leq (1+|a_1|)(1+|a_2|)\dots (1+|a_n|)-1$

For a sequence $a_n\in\mathbb{C}$ I want to show that $$|(1+a_1)(1+a_2)\dots (1+a_n)-1|\leq (1+|a_1|)(1+|a_2|)\dots (1+|a_n|)-1$$ I think I should show this by induction on $n$. For the base case I'm ...
0
votes
1answer
35 views

Prove that $0 \leq \frac{x+|x|}{2} \leq |x|$

$\frac{x+|x|}{2}$ is superior or equal to $0$ but inferior or equal to $|x|$ where the $x$ is a real number. I must prove this by the method of proof by cases. I have no idea one how to begin ...
1
vote
3answers
101 views

Question about an inequality on a proof

I'm stuck on a proof. There's a step that says: $$ \left| \Im\left(\frac{1-e^{i(N+1)x}}{1-e^{ix}}\right)\right| \leq \left| (\frac{1-e^{i(N+1)x}}{1-e^{ix}}) \right|,\quad \text{with } N \in ...
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2answers
23 views

Module properties

I've got stuck on this problem Let $a$, $b$, $c> 0$ and $x$ a real number such that $$|ax - b| \leq c,$$ $$|bx - c| \leq a$$ and $$ |cx - a| \leq b.$$ Prove that $0 \leq x \leq 2$. What ...
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1answer
32 views

Anita Tabacco include zero to N? [closed]

I have two problems actually, first one: Proffesor always grumbled that Anita Tabacco include zero to natural numbers, and proffesor dont include zero to the natural numbers when explain somethink or ...
0
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3answers
79 views

Some confusion with absolute value

Today at a math lecture, I solved the equation $|x+1|+|x-1|+|x|=4$ by using elementary arithmetic. But my professor did it a little bit differently: I didn`t pay attention to the teacher's ...
1
vote
1answer
66 views

Solve absolute value inequality

I have to show the inequality $$ \left|\frac{1}{2 + a}\right| < 1. $$ How do I do this? I know that a fraction is less than 1 when the denominator is greater than the numerator, but I cannot ...
0
votes
2answers
69 views

Finding the max absolute value of this analytic function

On the line segment from $z=R, R>0,$ to $z=R+i2\pi$, I want to find the maximum of: $\lvert e^{3z}/(1+e^z) \rvert$. If $z=x+iy$, this is equal to: $$ \lvert e^{3R}e^{i3y} / (1+e^Re^{iy}) \rvert ...
0
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2answers
35 views

Absolute Value derivative

What would the following derivative be? $$ \frac{\partial}{\partial x_k} |x_i - x_j| =?$$ Where $$ \frac{\partial x_i}{\partial x_k} = \delta_{ik}$$ For context, what I'm actually trying to do is ...
0
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3answers
120 views

The set of all real numbers $x$ such that $\sqrt{x^2}=-x$

The questions goes as The set of all real numbers $x$ such that $\sqrt{x^2} = -x$ consists of a. Zero only b. Nonpositive real numbers only c. positive real numbers only d. all real ...
0
votes
1answer
20 views

why it is not continuous for a absolute value division?

the question is : is y=|x-1|/(x-1)continuous on (-infi, +infi): I am wondering why this equation is not continuous when x = 1 I think when x=1, y will be 1
0
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2answers
48 views

Amplitude for sum of sinusoids

I am trying to plot the following function: $$ \max_t \left| \frac{1}{1-r^2} \big( \sin(r \omega_n t) - r \sin(\omega_n t)\big) \right| $$ By inspection, I have determined that the amplitude of the ...
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1answer
38 views

Equation with logarithms and absolute values

I have this equation and I want to solve it for $x<0$. $$\frac{\ln|x|}{|x|}=\frac{\ln|x|}{x}$$ According to WolframAlpha, the solution is $x=-1$ but I don't know how to get that. My approach: ...
0
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1answer
24 views

How are absolute value of a field $F$ and norm of vector space $F/F$ related?

So I think it's true that a field $F$ and its respective vector space $F/F$ are isomorphic, since they consist of the same elements, and the operations of $F$ (addition,mult.) and $F/F$ (vector ...
0
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0answers
17 views

Question related to $F(x) = |x-a_1| + |x-a_2|+ … + |x-a_N|?$

Suppose $a_1 < a_2 < \cdots < a_N $. So $F(x) = S_N - 2S_i + (2i-N)x$ if $a_i < x < a_{i+1}$ with $S_i = a_1 + ... + a_i$. Assume $ a_i < u < a_{i+1} < v $, we have: $F(u) = ...
-1
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1answer
23 views

How do you calculate the absolute value of trigonometric functions? [closed]

How do you calculate: $|\tan t|= \sqrt{3}$ The answer to this is required to be in the interval $[0,2\pi]$
3
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3answers
110 views

Why does the integral $\int\frac{1}{x+i}dx$ not require the absolute value in the logarithm?

Going through some old calculus exams, I find a solution to an integral via partial fraction decomposition. The solution manual does not perform full decomposition to avoid complex numbers, however; I ...
0
votes
1answer
43 views

How to prove: if $x\in [-3,4]$ then $5\leq |x-3|+|x+2| \leq 7$

So far I have that if $-3 \leq x \leq 4$ then we have that: $-6 \leq x -3 \leq 1$ and $-1 \leq x+2 \leq 6$ So $|x-3| \leq 6$ and $|x+2| \leq 6$ but I'm not sure how to continue the proof.
0
votes
1answer
80 views

Complex Analysis Questions - $|z + 2| + |z - 2| = \sqrt{10}$

I'm just starting in a Complex Analysis course, and I am stuck on a couple questions. The questions are as follows: If $z = a + bi$ is a point on the curve $|z + 2| + |z - 2| = \sqrt{10},$ find ...
-2
votes
4answers
40 views

Solving $|x+2| = 4 + |x-7|$ [closed]

How would one solve the following equation over $\mathbb R$? $$|x+2| = 4 + |x-7|$$
0
votes
2answers
27 views

question about absolute value inequalities

We know that $|a|<b$ implies $-b<a<b$. Would that still hold if $-|a|<b$? That is, would that imply $-b<-a<b$? Thanks
1
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4answers
114 views

Compute $\int_{a}^{b}\left|x\right|\mathrm{d}x$

The reason I ask this is because $$\int_{x=a}^{x=b}\left|x\right|\mathrm{d}x$$ gives exactly the same result as ...
0
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0answers
29 views

Absolute Value Inequality Proof - Hint needed

I am having difficulty on a problem. If someone could explain where I should start or what I can use to help solve the proof it would help. Prove that if $|(x+2)| \lt 1$ then $|(x^2 +2)| \gt 3$
0
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0answers
20 views

Functional Derivative of Complex Absolutes

It is said, that the lowest order complex amplitude equation shows potential dynamics with the functional \begin{align} V[A] = \int\limits_{a}^{b} \text{d}X\left[- \vert A \vert^2 + \frac{1}{2} \vert ...
2
votes
0answers
36 views

absolute values and integals

I have the following integral $$\int_{- \infty}^\infty e^{-|x|} dx$$ and the following two questions (1) Since the preimages $x$ determine the the images $e^{-|x|}$ for nonnegative and negative ...
2
votes
2answers
54 views

Absolute value inequality $3 > |x + 4| \geq 1$

I've just started with absolute value equations and I have a real hard time understanding how to solve this. I got the following question, and I can't make heads or tails out of it. Assume that $x, ...
2
votes
2answers
46 views

How to solve complex equation with same variable on two sides

What is the analytic solution of X for the equation below? $$conjugate(X)= \frac{-2\times A}{B\times X} $$ A, B and X are complex numbers. Would the magnitude of X be given by this? $$ |X| =\sqrt ...