For questions about or involving the absolute value function.

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1answer
2k views

Normal distribution with absolute value

I am new to the normal distribution topic. While I have understood and solved various different kind of questions, the normal distribution questions with absolute value, are the ones I have no idea ...
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2answers
101 views

How to solve $|2x +1|< 1/4$?

How do you solve $$|2x +1|< \frac{1}4$$
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3answers
159 views

Solve the equation : $x^2 − 6 |x − 2| − 28 = 0$

The following is an absolute value quadratic equation that I want to solve: $$x^2 − 6 |x − 2| − 28 = 0$$ Here is what I did : $x^2 − 6 |x − 2| − 28 = 0$ $x^2 − 6 |x − 2| − 28 = 0$ ...
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2answers
82 views

Prove $|x+1|\leq 4$ implies that $-4\leq x\leq 2$.

How do I prove that if $x$ is a real number, then $\lvert x+1 \rvert\leq 3$ implies that $-4\leq x\leq 2$. EDIT: $\lvert x+1 \rvert\leq 4$ should be $\lvert x+1 \rvert\leq 3$
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4answers
140 views

Prove that $||x|-|y|| \leq |x-y|$ [duplicate]

$||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ In Principles of MA(Rudin), the author said one sees easily that $||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ (p.88, Rudin) from the triangle ...
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1answer
101 views

Determine all the values of the parameter $a$ for which the inequality $3-|x-a|>x^2$ is satisfied by at least one negative $x$.

I wanted to know, how can I determine all the values of the parameter $a$ for which the inequality $3 - |x-a| > x^2$ is satisfied by at least one negative $x$. I tried for $x<a, |x-a|=-(x-a)$ ...
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3answers
122 views

Inequalities - Absolute Value

$$|2x-1| \leq |x-3|$$ Answer is $$-2 \leq x \leq \frac43$$ My Question is HOW?
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2answers
56 views

Finding $x$ from inequality: $\left | \frac{3^n + 2}{3^n + 1} - 1 \right | \le \frac{1}{28}$

Find $x$ in $\mathbb{Z}$ satisfying this inequality: $$\left | \frac{3^n + 2}{3^n + 1} - 1 \right | \le \frac{1}{28}.$$ I tried something, but I don't think it's correct. $$-\frac{1}{28} ...
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1answer
480 views

Describe the set of points on the complex plane…

Describe the set of points on the complex plane for which $|z-2| + |z+2|=4$... So, I know you can solve this instantly, just by using definition, but I want to do it the long way.. So, $$|x- i*y ...
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5answers
182 views

Prove the triangle inequality [duplicate]

I want to porve the triangle inequality: $x, y \in \mathbb{R} \text { Then } |x+y| \leq |x| + |y|$ I figured out that probably the cases: $x\geq0$ and $y \geq 0$ $x<0$ and $y < 0$ $x\geq0$ ...
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2answers
788 views

Proof the maximum function $\max(x,y) = \frac {x +y +|x-y|} {2}$ [duplicate]

I want to prove the maximum function max: $\mathbb{R} \rightarrow \mathbb{R}$, which is defined by $$\max(x,y) = \begin{cases}x, \text { if } x \geq y , \\ y, \text { if } x < y \end{cases}$$ ...
1
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2answers
157 views

Graphing - Absolute Value and Circle

The diagram Shows The Graphs of $y = |x + 2|$ and $y = \sqrt{4 - x^2}$ Write down the solution for $\sqrt{4 - x^2}$ is equal to or less than $y = |x + 2|$.
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1answer
43 views

Given certain conditions for $\delta$, how do I show that an inequality relating delta to x is true?

This is a problem out of a textbook (though there's no answer to this one in the back). If   $0 < \delta < 1$ and $|x-4| < \delta$ show:   $|\sqrt{x}-2| < ...
1
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2answers
482 views

How to find minimum of sum of mod functions?

How to find minimum value of $$|x-1| + |x-2| + |x-31| + |x-24| + |x-5| + |x-6| + |x-17| + |x-8| + \\|x-9| + |x-10| + |x-11| + |x-12|$$ and also where it occurs ? I know the procedure for find answer ...
0
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1answer
101 views

Find the value of this logarithmic expression involving fifth root of unity.

Let $\alpha$ be the fifth root of unity. We then want to evaluate the expression $$\log |1 + \alpha + \alpha^2 + \alpha^3 - 1/\alpha |$$ Thanks in anticipation for your help in solving this!
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7answers
929 views

what does $|x-2| < 1$ mean?

I am studying some inequality properties of absolute values and I bumped into some expressions like $|x-2| < 1$ that I just can't get the meaning of them. Lets say I have this expression $$ ...
1
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3answers
98 views

Absolute value and roots

I've been trying to solve this problem and I always get 1, but the answer is $1 - 2x$. If $x<\frac12$ then what is $\left|x-\sqrt{(x-1)^2}\right|$ ?
4
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5answers
339 views

Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$

Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$ example1: $|x-1|+|x-2|\geq 1$ my solution:(substitution) $x-1=t,x-2=t-1,|t|+|t-1|\geq 1,|t-1|\geq 1-|t|,$ square, $t^2-2t+1\geq ...
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0answers
84 views

Phrase and symbol for “geometric absolute value”$ e^{|\ln(x)|}?$

I'm calculate the median fractional difference between two vectors (to characterise the error in a quantity with a high dynamic range). If $a/b = 0.1$, the fractional difference is $10$, and if $a/b ...
4
votes
2answers
156 views

How do we know that $|i!| = \sqrt{\pi \operatorname{csch} \pi}$?

(Source: Wolfram Alpha) Or, to write it out in full, $$|i!| = \sqrt{\frac{2\pi e^\pi}{e^{2\pi} - 1}}$$ How is this identity derived? Also, knowing this, could we find the exact values for the real ...
3
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2answers
80 views

How prove this $|x_{p}-y_{q}|>0$

let $$x_{1}=\dfrac{1}{8},x_{n+1}=x_{n}+x^2_{n},y_{1}=\dfrac{1}{10},y_{n+1}=y_{n}+y^2_{n}$$ show that: for any $p,q\in N^{+}$ we have $$|x_{p}-y_{q}|>0$$
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4answers
2k views

Converting absolute value program into linear program

I have the generic optimization problem: $$ \max c^T|x|$$ $$ \text{s.t. } Ax \le b $$ $x$ is unrestricted How do I convert it into a linear programming problem? Online I read something about ...
13
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6answers
906 views

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
4
votes
1answer
243 views

intersection of two graph

i would like to clarify some questions from GRE,which at first seems a little difficult to understand,suppose that we have some function $f(x)=|2*x|+4$ and graph of this function is given ...
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2answers
443 views

Express $y=|-x^2+1|$ as a piecewise function.

I'm unsure of how to start this problem. Any help would be greatly appreciated.
3
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2answers
313 views

Solving two greatest integer function equations

If $$x\lfloor x\rfloor =39 \quad \text{and}\quad y\lfloor y \rfloor=68.$$ What is the value of: $$\lfloor x\rfloor+\lfloor y \rfloor $$ I don't know how to solve such problems. I would appreciate ...
4
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1answer
57 views

Does the triangle inequality follow from the rest of the properties of a subfield-valued absolute value?

(This is a much more specific version of my earlier question from over a year ago.) Let $F$ be a field, let $E$ be an ordered subfield of $F$, and let $\;\; |\hspace{-0.03 in}\cdot\hspace{-0.03 in}| ...
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4answers
279 views

Please help me to prove this inequality: $|x|+|y|≥|x+y|$

Please help me to prove the following inequality: $|x|+|y|\geq|x+y|$ in which $x$ and $y$ are real numbers. Any help or hint would be appreciated. Thanks :)
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2answers
44 views

Question involving absolute function

I saw this interesting problem in a math puzzle forum:- Find all integral values of $t$ such that the equation $|s-1| - 3|s+1| + |s+2| = t $ has no solutions. How does one approach these kind of ...
5
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1answer
91 views

Solving equation with absolute value signs

Can someone see why there is only get one solution when solving following equation in this way: The equation $|x+1|+|2x-3|=|x-5| $ $$|x+1|+|2x-3|=|x-5| $$ $$\pm (x+1) \pm(2x-3)=\pm(x-5)$$ $$\pm x ...
1
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1answer
171 views

Intersection of a point and absolute value function contained within a circle

I'm attempting some crazy ideas while programming a game and ran into the following math problem that has been bugging me for a few days: Given a unit circle and a random point $P$ within the circle, ...
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1answer
103 views

Absolute value of infinite sum smaller than infinite sum of absolute values

A question emerging from an exercise in Ok, E. A. (2007). Real Analysis with Economic Applications. Princeton University Press. The exercise consists in showing that if $\sum_{i=1}^\infty x_i$ ...
3
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2answers
229 views

Prove the monotonicity of the expectation of a binary random variable function

Consider $R$ independent binary random variables $y^1, \ldots, y^R$ over the space $\{-1, +1\}$ such that $\Pr(y^j = 1) = p^j \geq 0.5$ and $\Pr(y^j = -1) = 1 - p^j$, $\forall j = 1,\ldots,R$. ...
3
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1answer
125 views

Question based on Triangle Inequality $\displaystyle |x+y|\leq |x|+|y|$

If $x,y,z\in \mathbb{R}-\left\{0\right\}$. Then prove that $\displaystyle 1\leq \frac{|x+y|}{|x|+|y|}+\frac{| y+z|}{| y |+| z |}+\frac{| z+x|}{| z |+| x |}\leq 3$ My Try:: Using Triangle Inequality ...
2
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1answer
65 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 $\sqrt{-x^2}-1-2$?
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0answers
85 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.
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1answer
242 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, ...
2
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1answer
64 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 ...
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1answer
103 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?
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2answers
187 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 ...
1
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1answer
369 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 ...
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2answers
361 views

How does one DERIVE the formula for the maximum of two numbers

I want to derive (not prove that this is true) the formula $\max (x,y) = \dfrac{x + y + |y-x|}{2}$ I was reading a proof (which they have the result ahead of time already) that we do cases and then ...
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2answers
67 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 ...
1
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1answer
83 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 ...
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5answers
110 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 ...
1
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2answers
92 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 ...
0
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2answers
76 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 ...
2
votes
3answers
97 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 ...
0
votes
3answers
60 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 ...
0
votes
1answer
286 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 ...