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Questions tagged [absolute-value]

For questions about or involving the absolute value function.

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1answer
23 views

Gaussian distribution with absolute value

I am doing my homework about continuous random variable and Im struggling with this problem : Given a Gaussian random variable $T(85,10)$, find $c$ satisfying $\mathbb{P}[|T| < c] = 0.9$. Could ...
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0answers
23 views

Show a complex number equality using parallelogram identity [on hold]

Show that $$\vert z\vert + \vert w \vert = \left| \frac{z+w}{2} - \sqrt{zw}\right| + \left| \frac{z+w}{2} + \sqrt{zw} \right|$$ Hint: it can be useful the paralleogram identity. I was given this ...
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0answers
38 views

proving |x| + |y| + |z| − |x + y| − |y + z| − |z + x| + |x + y + z| ≥ 0. [duplicate]

how is it possible to prove the following inequality: |x| + |y| + |z| − |x + y| − |y + z| − |z + x| + |x + y + z| ≥ 0. What I first tried to do is move all the negatives to the right side to get |x| ...
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0answers
8 views

How to find values for these variables?

I have four unknowns and one equation. Is there a way to assign non-trivial values to them? The variables are: $$q_0, q_1 \in [0, 1] \\ e_0, e_1 \in [{z \in \mathcal{C} : \lvert z \rvert \le 1}]$$ ...
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4answers
59 views

What is the minimum value of $|x| + |2x+1|+|3x+2|+\cdots+|99x+98|$?

What is the minimum value of the following? $$A = |x| + |2x+1|+|3x+2|+\cdots+|99x+98|$$ What I've tried so far: Since $|x| = |-x| $ it is clear that $|3x + 2|$ = $|-3x - 2|$, $|5x + 4| = |-5x-4|$ ...
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1answer
24 views

Graphing absolute value inside absolute value equation

How would I graph an equation with an absolute value inside an absolute value? For example, $\left|-\left|x\right|+1\right|+\left|y-2\right|=3$ I tried graphing this on Desmos and it gave me some ...
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3answers
50 views

$\log_{0.1}(x^4) - 4 \geq 0$ - Solution varification

My question: is this steps ok? Be more precise is it step with taking four rooth ok? $$\log_{0.1}(x^4) - 4 \geq 0$$ $$\iff$$ $$\log_{0.1}(x^4) \geq 4$$ $$\iff$$ $$\log_{0.1}(x^4) \geq \log_{0.1}\left(\...
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2answers
1k views

Getting wrong answer for absolute value inequality and not sure why

The question: The function p is defined by $-2|x+4|+10$. Solve the equality $p(x) > -4$ Here were my steps to solving this: 1.) Subtract 10 from both sides -> $-2|x+4| > -14$ 2.) Divide ...
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5answers
62 views

Why am I getting two solutions for this absolute value equation?

The question is "State with a reason whether there are any solutions to |12-5x| = -2x+3" I can clearly see there are no solutions when I graph it but I've learned to solve these questions doing the ...
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3answers
43 views

Single-Line Equation for Equilateral Triangle

Is it possible to come up with a single-line equation in rectangular coordinates for an equilateral triangle with circumradius $R$, positioned symmetrical about the $y$-axis, as shown in the diagram ...
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3answers
41 views

Computing the product $xy$

$$\dfrac{|x|}{2}=y$$ $$|y|-x=4$$ Compute $xy$ Here I would have been able to solve this question in such case there was only one unknown. I, however, cannot solve this type of ...
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2answers
34 views

$||x|-|y||\leq|x-y|\Rightarrow |x-y|\geq|x|-|y|$ and $|x+y|\geq|x|-|y|$.

I think I can prove the inequality but in order to do so I Need to understand whether if $|a|>|b|$ then $|a|> b$ and $|a| > - b (*)$ My proof would be then $||x|-|y||\leq|x-y|\Rightarrow |...
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3answers
41 views

Cancelling with absolute values

I am trying to understand how cancellation works in integrals when there are absolute value expressions involved. For example: $$\int\frac{sinx}{\sqrt{1-cos^2x}}dx = \int \frac{sinx}{\lvert sin(x) \...
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2answers
49 views

Why and When do we use the absolute value?

I looked at a video from Khan Academy on limits at infinity and in the video, the instructor said $\sqrt {x^2}$ is 'essentially taking the absolute value of x; |x|'. I couldn't understand why is the ...
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1answer
48 views

Absolute value of a complex expression

I've been stuck on this problem for a while: $$\vert 1-e^{-i2\pi f}+.5e^{-i2\pi f\cdot 2}\vert^2,$$ where $i =$ the imaginary unit, $(2\pi f) =$ a real value, and $(2\pi f\cdot 2)=$ a real value. I ...
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2answers
44 views

How does one solve the following system of equations?

$\left\{ \begin{aligned} |x| + |y| + z &= a\\ |x| + y + |z| &= a + 1\\ x + |y| + |z| &= a + 2\end{aligned} \right.$ The domain is $\mathbb{R}$, $a > 1$.
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1answer
82 views

$ \int \bigl| a\sin(x)+b\cos(x) \bigr| {\rm d}x = ? $

My friend evaluated this to be $$ \int \bigl| a\sin(x)+b\cos(x) \bigr| {\rm d}x \\ = \sqrt{a^2+b^2} \left( \sin(x-\phi)\text{sign}(\cos(x-\phi)) +\frac{2}{\pi} \bigl(x-\arctan(\tan(x-\phi)) \bigr) \...
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1answer
48 views

Inequalities with two absolute values (check by different methods)

I have to teach mathematics to one of the kid in my family. Now, it has been a long time I have not done such exercises so could you help me out ? The exercise requires to solve the following ...
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1answer
76 views

The expectation of absolute value of the sum of n i.i.d. random variables

Let $\varphi_i$ be a Gaussian random variable such that $$\varphi_i \sim N(0,\sigma^2), \quad i = 1,2,\ldots,n.$$ What's the expectation: $$E\left(\left | \sum_{i=1}^n e^{j \varphi_i} \right |\right) $...
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0answers
41 views

Need Help with Absolute Value Equation involving Trig Function

I am asked to show that |sin(x)|=|AC|, where x=theta and -pi/2< x< pi/2 and AC corresponds to the length of the vertical line that falls from the point (cosx,sinx) on the unit circle to the ...
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2answers
63 views

If $x\in[a,b],$ then $|x|\leq \max\{|a|,|b|\}$

The following seems to be quite known: If $x\in[a,b],$ then $|x|\leq \max\{|a|,|b|\}$ I do use it while treating problems on series of functions but how does it come about? Any hint please?
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4answers
658 views

Why is the Absolute value / modulus function used?

Why is the absolute value function or modulus function $|x|$ used ? What are its uses? For example the square of a modulus number will always be positive, but why is it used when for example the ...
4
votes
2answers
145 views

Absolute value in indefinite integral

I have to show that $$ \int \left(\frac{dx}{x^2 \sqrt{x^2+4}}\right) = \left(\frac{-\sqrt{x^2+4}}{4x}\right) + c$$ I used the substitution $ \frac{x}{2} = \tan u$, and I got: $$\frac{1}{4}\int \...
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1answer
28 views

Show that $|x-y|-|x|=-y[I(x>0)-I(x<0)]+2\int_0^y[I(x\le s)-I(x\le0)]ds$.

Is the equation below true? How to prove it? $$|x-y|-|x|=-y[I(x>0)-I(x<0)]+2\int_0^y[I(x\le s)-I(x\le0)]ds.$$
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2answers
22 views

Absolute Value and Exponents

In my homework I've been accustomed to assuming that $|x|^a = |x^a|$ Recently however, I've begun to doubt that. Take the following example: $$ \begin{equation*} \begin{split} |\sqrt{-|x|} | &= ...
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2answers
29 views

Absolute Value Inequalities Analytical Approach

For $|x-3|-|2x+1|<0$, I considered adding $|2x+1|$ on both sides and solving it. $$|x-3|<|2x+1|$$ $$x-3< |2x+1|$$ $$x-3 < 2x+1$$ However, I keep getting $x>-4$ and $x<2/3$ instead ...
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1answer
51 views

Is there a bound for $|a+b|$ of the form $f(a,b) \leq |a+b|$

$a,b$ are integers and the $f(a,b)$ is a function of $a$ and $b$. I know that $|a+b| \leq |a|+|b|$. But what about $(? \leq |a+b|)$
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3answers
41 views

What's the different between necessary, sufficient, necessary and sufficient condition?

1)The range of values of "$a$", such that $|x-2|< a$ is a necessary condition for $x^2-3x-10<0$ 2)The range of values of "$a$", such that $|x-2| < a$ is a sufficient condition for $...
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1answer
51 views

Inverse of absolute value function?

I am graphing a square with the following equation: $$|y|=1-|x|$$ However, I need the equation in terms of y. That is, the form y=f(x) as opposed to the current <...
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1answer
25 views

How to find absolute mininum/maximum of a function on a set?

I have this function : $f(x,y)=x²+y²-2y-x$ I want to find the absolute min/max of the function on the set given in the figure below (where L3 is a piece of a circle with center (0, 0). I've already ...
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3answers
47 views

Idempotence of absolute value: how to show $\big| |a| \big| = |a|$? [closed]

How to prove that $\big| |a| \big| = |a|$? I mean it is somehow obvious as squaring makes numbers positive and the square root is defined as a positive number, but I would appreciate a (long) answer. ...
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1answer
23 views

Let $f (z) = u+iv$ be an analytic function, then show that $ (∂^2/∂x^2 + ∂^2/∂y^2)|f(z)|^2 = 4|f'(z)|^2 $.

Let $f (z) = u+iv$ be an analytic function, then show that $$(∂^2/∂x^2 + ∂^2/∂y^2)|f(z)|^2 = 4|f'(z)|^2\,.$$ $f(z) = u + iv $ $ϕ = |f(z)|^2 = u^2 + v^2 $ $f'(z) = ∂u/∂x + i∂v/∂x $ $|f'(z)|^2 = (∂u/...
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3answers
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Absolute value less than some value [closed]

This is a noob question. If, $$\biggm| \frac{1}{2} - e \biggm | \le n$$ Then how do I get the following? $$e \le \frac{1}{2} + n$$
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1answer
33 views

Valuation of the p adic logarithm

I'm stuck in some propertie about the $p$-adic logarithm. The propertie comes from a proposition in a Book by Dwork which I'm studying. The proposition says: If $v_{p}(x)>\frac{1}{p-1}$, then $v_{...
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2answers
40 views

Integral absolute value proof

Can anyone prove this? I can understand this intuitively, but can't prove it mathematically. Please help me. $$ \left|\int_a^b f(x)\,dx\,\right| \le \int_a^b \lvert f(x)\rvert \,dx $$
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2answers
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How to show that $\left| \Gamma \left(x + iy \right) \right|^{2} \approx (\pi y^{(2x - 1)}) /(\cosh(\pi y))$?

I found the approximation: $$\left| \Gamma \left(x + iy \right) \right|^{2} \approx \frac{\pi y^{(2x - 1)}}{\cosh(\pi y)} $$ for $y \gt 2$, within an answer for another question, but I could not ...
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0answers
78 views

What's the better proof for $|a| - |b|$ $\leq$ $|a - b|$

Theorem. $|a| - |b| \leq |a - b|$. Are the following two proofs equivalent? Proof I. $|a| - |b|$ $\leq$ $|a| + |b|$ by the triangle inequality. This is equal to $$|a| - |a| - |b|\leq|a| - |a| +...
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1answer
19 views

transforming from absolute sign to plus minus sign

I have recently encountered the following algebra transformation from an absolute sign to plus minus sign. I am unable to get my head around on how it really works? What is the underlying principle ...
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0answers
69 views

On an Inequality for the Riemann Zeta Function

Okay, firstly a bit of background to set the scene. My question comes from the approaches made by R. Spira in his paper, "An inequality for the riemann zeta function," regarding the initial steps he ...
0
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1answer
25 views

Let $a,b,c,d\in\Bbb{R}$ such that $|c|\ne|d|$. Prove that $\frac{a+b}{c+d}\le\frac{|a|+|b|}{||c|-|d||}$ [closed]

I can figure that $a+b\le|a+b|\le|a|+|b|$ but I do not know how to deal with $||c|-|d||$
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1answer
39 views

For every real number $x$, $ \lvert 2x - 6 \rvert \gt x \iff \lvert x-4 \rvert \gt 2 $

I was hoping to get a bit of feedback on a proof I've done involving absolute values. This problem is taken from D. Velleman's How to Prove it (#3.5.11). I've only written on one side of the ...
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1answer
15 views

If $\beta^n + \alpha_1 \beta^{n-1} + \dots + \alpha_n = 0$, $|\alpha_i | \leq 1$, then $|\beta| \leq 1$, with $|\cdot|$ ultrametric absolute value.

Let $|\cdot|$ be an ultrametric absolute value (i.e it is a function from a field $\mathcal{K}\rightarrow [0,\infty) $ that satisfies that $|\alpha \beta| = |\alpha||\beta|$ and $|\alpha + \beta| \leq ...
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2answers
23 views

Find the coordinates for the absolute maximum and minimum values of the function on the given interval

I am trying to learn how to find the coordinates for the absolute maximum and minimum values of the function on the given interval. $$f(t) = 2-|t|, -1 ≤ t ≤ 3$$ The answer in my textbook says the ...
0
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1answer
32 views

Absolute Value inside an integral

So I have that $|f(x) - h(x)| \le |f(x) - g(x)| + |g(x) - h(x)|$. What I'm wondering is if this is the same as saying $$\int_a^b |f(x) - h(x)|{\rm d}x \le \int_a^b |f(x) - g(x)|{\rm d}x + \int_a^b |...
7
votes
4answers
679 views

What is $\frac{1}{|{x}|}-\frac{x^2}{|x|^3}$?

What's the result of: $$\frac{1}{|{x}|}-\frac{x^2}{|x|^3}$$ Is it $$\frac{1}{|{x}|}-\frac{x^2}{|x|x^2}=\frac{1}{|{x}|}-\frac{1}{|x|}=0$$ or $$\frac{1}{|{x}|}-\frac{x^2}{|x|^2x}=\frac{1}{|{x}|}-\frac{...
1
vote
1answer
44 views

Proving that $|x \, g(y) - y \, g(x)| \leq C$ for $x,y \in [-1,1]$, where $|g''(t)| \leq C$

Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $f(x,y) = x \, g(y) - y \, g(x)$, where $g:\mathbb{R} \rightarrow \mathbb{R}$ such that $g \in C^2$, $g(0)=0$ and $|g''(x)| \leq C $ $\forall ...
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votes
1answer
38 views

how to simplify and solve this equation :

Solve this equation for $x$: $$\left|x \sqrt{1-x^2} +x \right| = \sqrt{1+x^2}$$ I'm having a problem getting rid of the square root!
0
votes
1answer
43 views

Power on absolute value expressions

How to deal with absolute value, when raised to odd number? Like here: $$\left|\log_2\left(\frac{x}{2}\right)\right|^3 + |\log_2 (2x)|^3 = 28$$
1
vote
3answers
85 views

Solving $ z \cdot |z| = 4 $

I have to solve $ (z^{*})^2 \cdot z^6 = 256 $ After some transformations I have: $$ |z|\cdot z = 4 \vee |z|\cdot z = -4 \vee |z|\cdot z = 4i \vee |z|\cdot z = -4i $$ How can I solve $ |z|\cdot z = 4$ ?...
0
votes
0answers
36 views

Solving a complicated inequality

I have got the following inequality $|(a_{12}+a_{21}) x_1 - i (a_{12}-a_{21}) x_2 + (a_{11}-a_{22}) x_3 + (a_{11}+a_{21}) X| \le \sqrt{a_{11}^2 + |a_{21}|^2} + \sqrt{a_{22}^2 + |a_{12}|^2}$, where $...