What is Absolute value: Definition and 367 Discussions

In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.
Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

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  1. E

    Inequality and absolute value proof

    Homework Statement prove that llal-lbll\leqla-bl Homework Equations Triangle inequality lx+yl\leqlxl+lyl The Attempt at a Solution Let a=(a-b)+b By using the triangle inequality we get lal-lbl\leqla-bl Then from here I am not sure what I can do. I would like to say on the left...
  2. T

    Area between curves, with an absolute value function

    Homework Statement Then find the area S of the region. (between the curves) Homework Equations A = \int_a^b f(x)-g(x) dxThe Attempt at a Solution First I plotted both equations, determined the top and the bottom functions and found where they intersected (calculator). Then I set up the...
  3. L

    Absolute Value Inequality: How to Solve and Graph

    Homework Statement l [3/(x-1)] - 5l < 4Homework Equations The Attempt at a Solution My 1st step was to make the inequality like this. -4 < 3/(x-1) - 5 < 4 and then i multiplied (x-1) to both left and right side and as well as to the 5. but in the end, my result turns out to be...
  4. C

    Y= l sin x l < absolute value of sinx

    Homework Statement y= l sin x l < absolute value of sinx Homework Equations The Attempt at a Solution y= l sin x l= sinx, if x>0 -sinx, if x< 0 0, if x=0 I get that part, but when i draw the graph I don't get it
  5. P

    How to Solve Rational Inequalities?

    Homework Statement Solve (x+2)/(x+1) -2 < 0 Homework Equations LCD: x + 1 The Attempt at a Solution (x+2-2x-2)/(x+1) < 0 (-x )/(x+1) < 0 But somehow, I got the wrong answer. The answer is (4, 7]. But I don't know where I went wrong.
  6. D

    Proofs using absolute value with Triangle/AGM Inequality

    Homework Statement *Sorry I could not get the math symbols to work properly so I did it by hand. I hope this isn't too much trouble. Prove: | sqrt( x ) - sqrt( y ) | <= | sqrt ( x - y ) | for x, y >= 0 Hint: Treat the cases x >= y and x <= y separately. I am new to proofs and we can't use...
  7. V

    Absolute Value of e(i): Why is it Equal to 1?

    Why does |e^i| = 1 ?
  8. A

    What values satisfy this equation: ((2x-7)/(x3+3))<(9/x)<=x²-5x+9?

    Excuse me for my english; Decide for all the x so this works out; ((2x-7)/(x3+3))<(9/x)<=x²-5x+9 where "<=" is the same as "and =". Having some troubles getting the right values between (9/x)<=x²-5x+9, but otherwise it's fine.. hope you can help me!
  9. Z

    Absolute value on both side of equation

    I solved this absolute value equation and I am confuse about the answer given in the book for this question, the answer in the book is x > = 0 or x = -2/3 please tell me how can I get this answer from the following solution. Please tell the method which should be true for all such type of...
  10. Z

    Solving Absolute Value Problem: x ≤ 3

    Suppose we have this absolute value question | x-3 | = 3 – x If we solve this question we break it as X - 3 = 3 – x or -(x - 3) = 3 - x Now if we solve it we come to know that the part on right is true for all real numbers And the part on the left is true for only 3 I also have read that...
  11. S

    Absolute Value Inequalities: Solving and Graphing

    Homework Statement Solve the inequality and sketch the graph of the solution on the real number line. Homework Equations |x - a|< or equal to b, b > 0 Let us imagine that the ">" and "<" signs also include "equal to" except for the condition, b > 0, in order to solve this...
  12. K

    Expectation of an absolute value

    Homework Statement I have E(a) = 0, E(b) = x but E(|a+b|)=?? where E is the expectations operator and x is a known constant which is greater than zero. Homework Equations Any one know how I would go about determining E(|a+b|)?
  13. I

    Infinity symbol inside of an absolute value Tattoo

    I've been thinking about getting a tattoo. I was considering getting the infinity symbol inside of an absolute value. Or the absolute value of infinity. Whichever you prefer. How mathematically correct is that? The absolute value of infinity simplified is infinity is it not? Will I completely...
  14. K

    Absolute Value Proof: Showing |a|=sqrt(a^2) and |a/b|=|a|/|b|

    Homework Statement If a,b are real numbers and b does not equal zero show that |a|=sqrt(a^2) and |a/b|=|a|/|b|. Homework Equations I know that |ab|=|a||b| and a^2 = |a|^2 The Attempt at a Solution Attempt at showing that |a|=sqrt(a^2): |a|=sqrt(a^2) |a|^2=(sqrt(a^2))^2...
  15. B

    Is Using Absolute Value for Infinity Common Practice in Limits?

    Hi, I came across a book which looks at a problem like \lim_{x \to 0}\frac{1}{x} So you approach from 0-, and get -∞, approach from 0+, get ∞ Then it would write the answer as \lim_{x \to 0}\frac{1}{x} = \left| \infty \right| It looks bizarre to me. How do you parse this? Is...
  16. majormuss

    Absolute Value Question: Solving |2x - 1| = x^2 with Step-by-Step Guide

    Homework Statement how do I solve this? I am confused... |2x - 1| = x^2 Homework Equations The Attempt at a Solution when i tried it I ended up with a solution set of (1,-1). But the official answer is quite different so I am confused!
  17. L

    A basic function question - with a strange absolute value placement

    Homework Statement If f(x) is equal to... x+9 if x<-3 -2x if |x|\leq 3 -6 if x > 32. The attempt at a solution The first and third "segments" of when the function is defined as being x+9 and -6 are pretty straightforward to me, however I am unaware of the significance of the placement of the...
  18. I

    Uniform Distribution, Absolute Value

    Homework Statement Find the CDF of |X|, given that X is a random variable, uniformly distributed over (-1,3). Is |X| uniformly distributed? If yes, over what interval?Homework Equations The Attempt at a Solution I found so far that: Setting Y=|X| Then: Y \in (1,3) F_{Y}(y)=P\left\{Y\leq...
  19. S

    How to Split Absolute Value in an Integral

    Homework Statement How do split \int^1_{-1}\left| \frac{1}{2}+xt\right|dt Homework Equations The Attempt at a Solution \int_{-1}^0 -\frac{1}{2}-xt dt+\int_0^1 \frac{1}{2}+xt dt Im not sure if this is right, and if it is... i still don't understand how to split the absolute...
  20. A

    How Do You Integrate sqrt(|x|) for Different Values of x?

    How we can integrate this (without integration limits) sqrt (|x|)
  21. K

    Splitting an integral with an absolute value

    This is a physics problem, but I only need help with the calculus portion of it. I was having trouble figuring out how to split the integral to properly integrate. Homework Statement Homework Equations \int\stackrel{\infty}{-\infty}(x/x_0)e-2|x|/x_0dx where x_o is a constant...
  22. C

    Question related to Absolute Value

    Hi everyone; I've got this question concerning absolute value, although I understand the concept of absolute value, I can't quite understand why option B is bigger than option A. I would really appreciate some explanation on the following question: I am trying to post it in Latex, but in case...
  23. Z

    Factoring something with absolute value

    Homework Statement I need to find where these intersect: y = |x| 3y - x = 8 Homework Equations The Attempt at a Solution I equate them: |x| = (1/3)x + (8/3) |x| - (1/3)x - (8/3) = 0 |3x| - x - 8 = 0 |2x| - 8 = 0 |x| = 4 Is that right? How do I get the other solution?
  24. N

    Absolute value and differentiability

    Homework Statement Hi all I have f(x)=|x|. This I write as f(x) = -x for x<0 f(x) = x for x>0 f(x) = 0 for x=0 If I want to show that f(x) is not differentiable at x=0, then is it enough to show that f'(x) = -1 for x<0 f'(x) = 1 for x>0 and from this conclude that it is...
  25. F

    Solving two absolute value variables.

    Homework Statement Given: |A|+A+B=15 and A+|B|-B=13. What is A+B equal to? Give all possibilities. Homework Equations The Attempt at a Solution I solve for both absolute variables. So, A=15-A-B or A=A+B-15 and B=13-A+B or B=A-B-13 Firstly, I solve for A. A+A=15-B...
  26. Saladsamurai

    Spivak Error? Definition: Absolute Value

    Homework Statement In Spivak's Calculus 4e, he defines absolute value as: |a|= a,\qquad a\ge 0 \qquad \text{ and } \qquad -a,\qquad a \le 0 Did he really mean to include the '\le ' and not just '<' ? I know it does not affect the answer, but I didn't think that you could...
  27. C

    Induction: Absolute value of complex sequence

    Homework Statement Define sequence \left[ {z_n} \right]} _{n=1}^{\infty} of points in the complex plane with z_1 = 1 and z_{n+1} = (4+3i)z_n - 1. Show that |z_n| \ge 4^{n-1} for all n \ge 1.Homework Equations The Attempt at a Solution The base case of induction is true, since |z_1| \ge 4^{1-1}...
  28. F

    How Do You Simplify This Complex Absolute Value Expression?

    Homework Statement I need to simplify the expression below. The absolute value is throwing me off \left[\left|(\alpha + k)^{2}e^{-2i \alpha a} - (\alpha - k)^{2}e^{2i \alpha a}\right|\right]^{2} Homework Equations I know \left|e^{ix}\right| = 1 The Attempt at a Solution I...
  29. H

    Inequalities and absolute value

    Homework Statement 1) x^5 > x^2 2) 7| x + 2 | + 5 > 4 3) 3 - | 2x + 4 | <= 1 Homework Equations The Attempt at a Solution 1) x5 - x2 > 0 x2(x3 - 1) > 0 x2(x - 1)(x2 + x + 1) > 0 Im not too sure what to do next. I can't factor it any further, at least I don't think so. Which...
  30. S

    Derivative with Absolute Value

    Homework Statement Find f'(x), if f(x) = [x^2 * (3x + 2)^(1/3)] / [(2x - 3)3] Where the absolute value symbol surrounds the entire function. Homework Equations N/AThe Attempt at a Solution My attempts don't account for the absolute value of the function. Otherwise, I can still take the...
  31. K

    First order linear ODE-integrating factor has absolute value in it

    First order linear ODE-integrating factor with absolute value?! Homework Statement Solve the ODE y' + (3/t) y = t3. 2. Homework Equations /concepts 1st order linear ODE The Attempt at a Solution Integrating factor =exp ∫(3/t)dt =exp (3ln|t| + k) =exp (ln|t|3) (take constant of...
  32. A

    What is this? Absolute value derivative? Magnitude?

    I'm looking at this problem below: http://apthtml.com/images/deriv.png The first part I understand - a derivative of some vectors, not a big deal - but the second part is where I'm confused. I can't even come up with a name for it. First I thought magnitude, but I don't really see how...
  33. M

    Quadratic Absolute Value Inequality

    [b]1. Solve: l2x^2+7x-15l<10 [b]3. I split it up into two cases and got: 2x^2+7x-25<0 and 2x^2+7x-5>0 With the quadratic formula I got: x= [-7+/- (249)^(1/2)]/4 and x= [-7+/- (89)^(1/2)]/4 These clearly are the answers, but I am confused about the inequality from this...
  34. M

    Absolute value of acceleration Help

    Absolute value of acceleration Help! Homework Statement I am getting a value of -1.94 m/s^2 for my acceleraton. An it wants to know the absolute value of acceleration what do I put down. It also state that the plus or minus is just to indicate direction. But the way the problems is computed...
  35. A

    How can I get absolute value of impedance?

    Here's the thing. I have 4,7 nF capacitor and AC: 1 kHz < f < 10 MHz. I measure real and imaginary part of capacitance C*=C'+iC" (C'=C'(f) and C"=C"(f)). I did 2 measurements - in first capacitor is directly attached to measuring device, in second I use 1 m coaxial cables to attach it. I...
  36. S

    Quadratic equation and absolute value .

    Quadratic equation and absolute value... Consider equation |x2-2x-3|=m, m belongs to R If the given equation has no solution then find the interval in which the m lies. 2. Homework Equations -Given in question 3. Since this equation is in modulus there would be two equations...
  37. T

    Infinite limit of absolute value

    Homework Statement i just want to know one value that i can't find anywhere, and would love some help Homework Equations \lim_{x\rightarrow\ -\infty}|x|} The Attempt at a Solution thanks
  38. X

    How Do You Integrate Absolute Values with Complex Exponentials?

    Homework Statement \int_{-3}^{3}|t|e^{-jwt}dt The Attempt at a Solution I am not sure if I need to break this into two regions due to the abs value...
  39. C

    Solving Absolute Value Inequalities without Changing Directions

    Homework Statement |1/(x-1)|<1 The Attempt at a Solution is that the same as this -1<1/(x-1)<1 can i do each side by it self then take the values at which they intersect so i subtracted the 1 then got (-x+2)/(x-1) then made a sign chart with 2 and 1 on it then took the less than...
  40. D

    Confused about Absolute Value of Inverse Trigonometric Functions?

    Homework Statement Find \frac{d}{dx} \sin^{-1}(\sin x) Homework Equations The Attempt at a Solution Now, I know that the above expression simplifies directly to \frac{d}{dx} x = 1, but I attempted it the long way. Here it is: \frac{d}{dx} \sin^{-1}(\sin x) = \frac{\cos...
  41. W

    Sketching Absolute value graphs

    Homework Statement I previously left some absolute value questions which contained a few simple equations and equalities. i have a further question when it comes to slightly more complicated Absolute statements. Sketch the graph of y = 2|x-1| - 3|x+1| + 3x + 1 Homework Equations...
  42. B

    Inequality with absolute value of a complex integral

    I'm stuck trying to prove a step inside a lemma from Serre; given is 0<a<b 0<x To prove: |\int_{a}^{b}e^{-tx}e^{-tiy}dt|\leq\int_{a}^{b}e^{-tx}dt I've tried using Cauchy-Schwartz for integrals, but this step is too big (using Mathematica, it lead to a contradiction); something...
  43. V

    Absolute value of complex numbers

    When is it true that is |a|=|b|, then either a=b or a=b*, where a and b are complex numbers and b* is the complex conjugate of b?
  44. B

    Cauchy Integral with absolute value in simple curve

    Homework Statement Integrate using Cauchy Formula or Cauchy Theorem: I=\oint_{|z+1|=2}\frac{z^2}{4-z^2} dz (From Complex Variables, Stephen Fisher (2nd Edition), Exercise 2.3.3) Homework Equations \frac{1}{2\pi i}\oint_{\gamma} \frac{f(s)}{s-z} ds= \left\{ \begin{array}{lr}f(z) & : z \in...
  45. N

    Is the converse of the absolute value limit theorem true?

    Hi all, I'm wondering about this question I can prove that if lim_{n->inf} (a) = L then lim_{n->inf}abs(a) = abs(L) however.. is the converse true? thx
  46. S

    Dealing with absolute value functions

    In order to get an integral I need to find the difference between two functions, but I'm not sure how to deal with the absolute value... f(x) = \left|x-1-1\right| g(x) = x^2 + 2x g(x) - f(x) = (\left|x-1\right|-1) - (x^2 + 2x) =... I don't know if I can simplify it anymore... can I...
  47. D

    Exploring a function f(x) with absolute value

    when i am exploring a function eg. f(x)=|2x-6| can i treat it as two separate function all the way through, one to the left of x=3 and one to the right, and only at the very end, when i draw the graph connect them, ie draw a graph according to all the values i found from each side? will this...
  48. T

    Confused about the absolute value of a complex number

    Let z be the complex number: x+iy. Then |z|^2=x^2+y^2 according to my book. But according to the general definition of absolute value, |a|=(a^2)^.5. So letting z=a=x+iy. |z|^2=z^2=x^2+2ixy-y^2 This is not equal to x^2+y^2. I'm confused.
  49. U

    Limit of a function with absolute value of polynomial in a quotient

    Homework Statement Find: Lim | x2+x-12 |-8 / (x-4) x --> 4 Homework Equations The Attempt at a Solution My answer is 9. It it right ? or there is not a limit for F(x) when x --> 4
  50. W

    Integral of Absolute Value Function

    Homework Statement \int^{8}_{0}\left|x^{2} - 6x + 3\right|dx This is for a single variable AP Calculus AB class in which we are solving using substitution method. 2. The attempt at a solution I attempted it by just ignoring the abs value bars thinking that anything I am finding out...
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