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Questions tagged [absolute-value]

For questions about or involving the absolute value function.

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3answers
22 views

Absolute Value Rational Inequalities Help Please

I have been read plenty of questions on questions like this, but I still dont quite get it. For example, this question: $$ \left| \frac{2x+1}{x-3} \right| \ge 2 $$ How would I go about solving this?...
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1answer
16 views

Real Analysis introductory absolute value proof

let $x \in R$ Prove that $\vert x\vert \leq 2$ implies $\vert x^2 -4 \vert \leq 4 \vert x-2 \vert$ Here is my work: $\vert x-2 \vert \vert x + 2 \vert \leq 4 \vert x-2 \vert$ By the first ...
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1answer
40 views

Triangle Inequality and absolute value

I'm curious if the triangle inequality (and reverse triangle inequality) still hold if we only take the absolute value of one term. For example: $$||a| - b| \le |a - b|$$ If $b \ge 0$, then $|b|$ is ...
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0answers
41 views

Integrating $\left|f(x)\right|$ by pulling out $\mathrm{sgn}(f(x))$ from the integral

I tried doing the following integral: $\int_{0}^{\pi/4}\sqrt{1-\sin2x}\mathrm dx$. Firstly I completed the square by rewriting $1$ as $\sin^2x+\cos^2x$ to get the integral revised to this form: $$I=\...
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1answer
27 views

Graphing $f(x)=\left|x^2-2x \right|-x$.

I'm dealing with absolute value inequalities. I've realized it would be so much easier to confirm the solution sets I'm obtaining with a curve of the function. Currently I'm doing $\left|x^2-2x\...
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2answers
34 views

Proving an inequality involving absolute value; how do I justify using a conjunction (and) instead of a disjunction (or)?

I'm putting together the following the proof, and I have a question about one of the final steps. Definition of absolute value: $\forall x \in \mathbb{R}, (x \geq 0 \Rightarrow |x| = x) \...
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0answers
31 views

Uncoditional formula for unsigned area under a linesegment w.r.t. $y=0$

I'm interested in finding the unsigned area under a line segment with respect to $y=0$. The line segment is defined by start point $(s_x, s_y)$ and end point $(e_x, e_y)$ Without loss of generality, ...
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0answers
14 views

abs value equation with 3 solution, find value of constant in equation [closed]

If $||4x+5|-b|=6$ is an equation in $x$ with 3 distinct solutions, find the value (s) of rational number $b$
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1answer
23 views

Linear equation with absolute value.

Find $x$ if $|x-|3x+1||=4$ I got 4 values of $x$ out of which 2 are obsolete... Why so??
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2answers
38 views

How to plot absolute value graphs?

I have to plot graph of $$f(x)=|x|+2|x-1|+|x-4|$$ See I know graphs of individual $|x|,2|x-1|,|x-4|$ But how can I draw their sum. I have to find minimum value of the sum using graph.
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3answers
31 views

Inequality involving absolute values.

I want to ask is whether there is a method to solve following inequality more easily and compactly or it is the only method. $$|x-2|+|x-8|\le x-2$$ What I know is taking $x<2,8>x>2,x>...
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2answers
26 views

Log Functions Inside Absolute Value

Is the function below always positive for $0< x <1$? (I am determining if the function requires the modulus sign or not.) $$\frac{1}{2}\log\left|\frac{1+\log(x)}{1-\log(x)}\right|$$ My first ...
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0answers
31 views

Rephrasing what this proof is asking

I am new to proofs and am still struggling to parse them. I am not looking for a proof to the following statement; just guidance as to where to start or what the shape of a proof for it looks like. ...
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0answers
36 views

Absolute Value of a Integral, separation

This is a homework exercise, and ''Thomas early trascendentals'' book vol.4 says in property that : Def: To integrate an absolute value function, we have to look for specific cases when ; $ v(t)\...
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3answers
61 views

How to solve $|a+b|+|a-b|=c$?

It is intuitive that $a=\pm \frac{c}{2}$, with $-\frac{c}{2}\leq b\leq \frac{c}{2}$ or vice-versa are solutions to the problem. Can I get to these solutions without dividing the expression in all the ...
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5answers
53 views

Sum of two absolute values equal to a whole number

The following is the equation: $|x+1|+|x+2|=3$ How can I solve this problem? Do I have to reformat it to $|x+1|=3-|x-2|$? I would like a simple answer that by no means uses set theory. The answer ...
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1answer
14 views

Redefinition of a Constant Leading to Nullification of Absolute Value

I am currently taking a calculus 3 class in college, and my teacher did an intriguing problem about a tsunami in class which took up about 4 full white boards. Anyways, at a certain point in the ...
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1answer
38 views

solving equation involving absolute values [closed]

Given the equation $|x+2| + |3x+6| = 8,$ how can I find the sum of all its roots?
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2answers
78 views

How can I solve this absolute value equation?

This is the equation: $|\sqrt{x-1} - 2| + |\sqrt{x-1} - 3| = 1$ Any help would be appreciated. Thanks!
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1answer
21 views

Find the solution of $1 \le \left\lfloor \left\lvert \frac{x+1}{x-3}\right\rvert +x\right\rfloor \lt 2$

Find the solution of $$1 \le \left\lfloor \left\lvert \frac{x+1}{x-3}\right\rvert +x\right\rfloor \lt 2$$ My try: The only thing i know is that $$\left\lfloor \left\lvert \frac{x+1}{x-3}\right\rvert +...
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1answer
20 views

Solve for $x$: $3+|x-9|<\frac{2|x-1|}{x}$

I moved one part of the inequality to the other to create the following: $\frac{2|x-1|}{x}-3-|x-9|>0$ Eventually, I get to a case where I have 2 inequalities after opening 1 of the absolute ...
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4answers
53 views

If $|x|>|x-y|$ prove that $xy>0$

So the title says it all. I think using the square formula, not sure if that is legit. Thank you for your attention.
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1answer
47 views

Taking the absolute value in inequalities.

If I've a expression: $-4<3$ and take the absolute value $|-4|<|3|\implies 4<3$ which is false. So I though that maybe the inequality sign would change. But $|-2|<|3| \implies 2>3$ ...
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0answers
38 views

Writing Absolute Value For Equations With Inequalities

Write the absolute value equations in the form $|x−b|=c$ (where b is a number and c can be either number or an expression) that have the following solution sets: All numbers such that x≥5. ...
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1answer
37 views

Sketch the region in the plane consisting of all points $(x,y)$ such that $|x-y|+|x|-|y| \leq 2$

Sketch the region in the plane consisting of all points $(x,y)$ such that $|x-y|+|x|-|y| \leq 2$ I could consider the eight parts the plane gets divided into by the $x$-axis, $y$-axis, $y=x$ and $y=-...
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2answers
26 views

Do the $|$ around $|\langle u,v\rangle|$ refer to absolute value in the inner product version of the Cauchy-Schwarz inequality?

The full inequality is: $|\langle u,v\rangle| \leq ||u|| ||v||$ I understand that $||$ around the vectors $u$ and $v$ signifies the taking of their norm, but what do the single | around $\langle ...
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1answer
23 views

Minimization of multiple absolute sums

I know that on fist glance this seems as already explained problem on many internet places, but I haven't found solution anywhere. Anyway, here is my function: $$F=\sum_{i=0}^N |S_i|$$ where $$S_i=\...
2
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2answers
28 views

Number of solutions of an equality with absolute value operator

Consider: $$ \left|\left|\left|x-1\right|-2\right|-4\right|=4 $$ What is the number of solutions for this equation? This one was particularly easy to me. If first observed that if this inequality ...
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4answers
79 views

Suppose $|z|\ge 2$, Prove $|z^8+135|\ge121$.

Suppose $|z|\ge 2$, Prove $|z^8+135|\ge121$. My work: $|z^8+135|=\sqrt{(z^8+135)(\bar{z}^8+135)}=\sqrt{|z|^{16}+135(z^8+\bar{z}^8)+135^2}\ge\sqrt{2^{16}+135(z^8+\bar{z}^8)+135}$ For the last term, ...
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1answer
17 views

Continuity of product in absolute value

Let $F$ be a field, and $\| \cdot\|$ be an absolute value (or norm). I want to prove that with this norm, multiplication is continuous map on $F$ in the sense: for $x_0,y_0\in F$, and any $\...
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3answers
42 views

How can $|n_1 - n_2| + |n_3 - n_4|$ be equal two different formulas

I have a formula to be calculated such as; $|n_1 - n_2| + |n_3 - n_4|$. I want to calculate it with two tuples of values such as; $(n_1,n_2)$ and $(n_3,n_4)$ When I try this method, it is equivalent;...
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2answers
44 views

Prove using the triangular inequality that: $|a+b| \geq |a| - |b|$

How can I prove using the triangular inequality that: $$|a+b| \geq |a| - |b|$$ I already proved it by considering all 8 possible scenarios (like a>b and b=0 ... etc) However I couldn’t manage to find ...
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3answers
54 views

Is it valid to take $|- \infty | = \infty$?

Is it valid to take $|- \infty | = \infty$? or is the absolute value e.g. not defined for infinity? Particularly, if one wishes to argue that operator $f(x)=x$ is not bounded below on $\mathbb{R}_{-...
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2answers
45 views

Natural logarithm with absolute value: Can I cancel the absolute value?

I was calculating basic rational integrals and came up with this kind of problem. I have this expression: $$2\ln|x|$$ I can re-write it down like that: $$\ln{x^2}$$ and thus cancel the modulus. ...
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0answers
30 views

Rewriting a $\max\left\{0,\dots\right\}$ function in order to integrate the function more properly

Yesterday I asked a question about a certain integral. In the integral is the term: $$\max\left\{0,\left|\text{n}\cdot\sin\left(2\pi\cdot x\cdot t-\frac{\pi}{2}\right)\right|-2\cdot\text{z}\right\}\...
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1answer
60 views

Find a set of points in the given complex plane

Here's the Question: Find a set of points in the complex plane that satisfies: $$|z-i|+|z+i| = 1$$ Now from triangle inequality I found: $$|z+z+i-i|=|2z|\geq1 $$ Which refers that there's no ...
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1answer
66 views

Is it possible to give a purely syntactic proof of : $|x| <N$ is equivalent to $ x>-N \text{ AND } x<N$?

In the document accessible at the following link [1]: https://i.stack.imgur.com/P2AvU.jpg, I've tried to explain through purely logical means ( mainly DeMorgan's law) why a $<$ absolute value ...
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1answer
32 views

Max and Min Inequality

I have to show that for $a,b,c,d \in \mathbb{R}$ $$|a\vee b - c\vee d| \leq |a-c|\vee |b-d|$$ I know this can be showed using cases, but I need help with a proof that doesn't involve cases. I found ...
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3answers
104 views

Prove that $\neg \exists x$ such that $ \lvert x-1\rvert+\lvert x+1 \rvert <1$ [duplicate]

In Spivak Calculus, chapter 1 question 11 vi. asks the reader to find all numbers $x$ for which $\lvert x-1\rvert+\lvert x+1 \rvert <1$. Intuitively speaking, it is quite obvious that there is no ...
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2answers
48 views

Need help with a proof about absolute value

Prove $||x| - |y|| \le |x-y|$ Here's my attempt of the proof: Since $x-y = |x-y|$ or $x-y = -|x-y|$, then $-|x-y| \le x-y \le |x-y|$. Also, $|x| = |-x|$ and $|y| =|-y|$, so we have that $|x|-|y| \...
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1answer
11 views

Difference of any two elements of a sequence being less than some number?

Considering a finite sequence of real numbers $x_0, x_1, ..., x_n$, which has no particular order, could I write a single statement that says the difference between any two elements of the sequence is ...
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1answer
17 views

Unsure How to Proceed with Proof Related to Algebraic Limit Theorem

I'm independently studying Stephen Abbott's Understanding Analysis and am trying to follow the proof for $(b_n) \rightarrow b$ implies $(\frac{1}{b_n}) \rightarrow (\frac{1}{b})$ Specifically, I'm ...
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3answers
59 views

Is $|z^2 + 1| > |z|^2 - 1$?

I am often confused while using complex number formula involving comparisons. It is known that $|z^2 - a^2| > |z|^2 - a^2$. But is $|z^2 + 1| > |z|^2 - 1$? Where $z$ is a complex number. Also,...
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0answers
19 views

How Absolute changes power rules?

I have a question about power rules, we have f(m,n) : $\ a^{(m-n)}= \frac{a^m}{a^n}$ which is separable. What about: $\ a^{|m-n|}= ?$ Is it separable? I want f(m,n)=f(n,m) Thanks.
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0answers
24 views

Splitting expected value of absolute value

The following expected value is part of a limit I'm trying to evaluation (for the Lyapunov CLT). My question is essentially is this a valid approach, I have the expected value of the absolute value of ...
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1answer
40 views

1st order differential linear equation, question on absolute value

I'm trying to find the general solution to this equation: $$x \frac{dy}{dx}+3(y+x^2)=\frac{\sin(x)}{x} $$ Standard form puts it like this: $$\frac{dy}{dx}+\frac{3}{x}y=\frac{\sin(x)-3x^3}{x^2} $$ To ...
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3answers
33 views

Prove or disprove $|a| \geq 2|b| \implies |a+b| \geq |b|$

I conjectured that for all real $a, b$, that $|a| \geq 2|b| \implies |a+b| \geq |b|$. I'm using this to try to prove a limit where the denominator is $x^2+xy+y^2$ since I know $|x^2+y^2| \geq 2|xy|$. ...
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2answers
48 views

Maximum value of absolute sum in a polynomial

I have a great problem from Qvant magazine I can’t solve. Please help! Suppose that for any $-1\leq x\leq 1$, $|ax^2+bx+c|\leq 1$. Find the maximum possible value of $|a|+|b|+|c|$. My attempts: It ...
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3answers
72 views

Why $f(x)=|\sin{x}-1|=1-\sin{x}$? What if $\sin{x}-1$ is zero?

$g(x)=|\sin{x}-1|+|3-\cos{x}-\sin{x}|+2\sin{x}$ Answer: Above equality is simplified to $$1-\sin{x}+3-\cos{x}-\sin{x}+2\sin{x}=4-\cos{x}$$ $$-1 \le\sin{x}\le1$$ So , I know that $f(x)=|\sin{x}-...
2
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3answers
54 views

Inequalities, but working around an absolute value

So what I want to prove is $$ |xy+xz+yz- 2(x+y+z) + 3| \leq |x^2+y^2+z^2-2(x+y+z)+3| $$ for $x,y,z\in \mathbb{R}$, and I'm aware that the RHS is just $|(x-1)^2+(y-1)^2+(z-1)^2|$. Now I'm able to ...