Absolute value Definition and 368 Threads

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. 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...
  2. T

    Absolute value of a function integrable?

    this is the question, Prove that if f is continuous on (a,b] and if |f| is bounded on [a,b] then f is integrable on [a,b]. (note: it is not assumed that f is continuous at a.) I know you have to use the upper and lower bounds to prove this statement but i don't know where to start...
  3. T

    Absolute value of a function integrable?

    this is the question, Prove that if f is continuous on (a,b] and if |f| is bounded on [a,b] then f is integrable on [a,b]. (note: it is not assumed that f is continuous at a.) I know you have to use the upper and lower bounds to prove this statement but i don't know where to start? Thanks
  4. R

    How to Evaluate the Absolute Value Integration?

    1. Evaluate \int_{-1}^{3} \left|x^2 -4\right| dx 3. The Attempt at a Solution This is the first time I'm trying this type of question & I think I need to use the following theorem for such questions; f is integrable on a closed interval a to b. \int_{a}^{b}f(x)dx =...
  5. M

    Help, need to get down absolute value equations and inequalities.

    I'm taking an algebra & triginometry class at my college and my professor is kind of slow and unclear. I think I'm a fast learner and a good understander which is why I came here to get this info down. We're up to complex fractions or radical equations right now I think, forgot which. Something...
  6. T

    Proving Absolute Value Inequality: |a| ≤ b → -b ≤ a ≤ b (b≥0)

    Prove the following: if |a| \leq b then -b \leq a \leq b (where b \geq 0 ). So a \leq b and -a \leq b . Then -b \leq a so that -b \leq a \leq b . Suppose that -b \leq a \leq b . Then a \leq b and -a \leq b so that |a| \leq b . Is this a correct proof? You don't have to...
  7. G

    Expressions without absolute value signs

    Homework Statement Rewrite the following expressions without absolute value signs, treating various cases separately where neccesary Homework Equations a-Abs[(a-(abs)a)] the question is do i have 2 answers to this ?
  8. G

    A natural log inequality with absolute value

    Homework Statement F(x) = (8-12ln|x|)/(x^4) > 0 (a) For what values of x is the expression F(x) defined? Write your answer in interval notation. (b) At what value(s) of x is the expression F(x) equal to zero? If there is more than one answer separate them by commas. (c) The set of...
  9. H

    Calc 1 - derivative of absolute value

    Homework Statement Question is: how can you tell if there are any places you can't take the derivative of an equation that has an absolute value (using logic, not just graphing it) example equations 1. \left|x-5\right| 2. \left| x3+4x2+9x+17 \right| x2+1 3...
  10. N

    Why is the absolute value of x equal to -x for values under zero?

    Here, it says that for the limit f(x) = |x| / x, |x| = { x, x > 0 -x, x < 0 } What I don't undestand is why is |x| = -x for values under zero? Isn't the absolute value for negative values just x and not -x? thanks. EDIT: I don't want to start a new thread, but I got stuck on this...
  11. K

    Finding limits when there is an absolute value in the numerator

    Ok the problem is: lim x->-1 |x+1| / x2-1 (sorry i don't really know how to type the equation out) I think that you have to find the limit as x->-1 from both the left and right sides from right: so I got lim x->-1 = (x+1)/x2-1 = (x+1)/(x+1)(x-1) = 1/(x-1) =1/-1-1 =-1/2 How would I...
  12. J

    Absolute value theorem that I can't convince myself of

    While reading my text, I came across an inequality that I couldn't convince myself of... For real numbers a,b: \left|a+b|<= |a|+|b|. Is this something proven? Or is it an axiom or something?
  13. S

    Rigorous proof of basic absolute value theorem?

    Rigorous proof of basic "absolute value" theorem? Hey :) I'm working through a Real Analysis text, and I came across this theorem and "proof": http://img352.imageshack.us/img352/6725/proofbx2.png It kind of took me by surprise, because the author of the text is usually very careful about...
  14. R

    Properties of the Absolute Value

    Just wanted to say hi before I start my post! :smile: As you may know there is a property of the absolute value that states; for a, b \in R; |ab| = |a||b| Well, my friend asked me if I knew a proof for this... but I don't know... How can we prove this statement/property? I know there...
  15. P

    Finding Absolute Value of Complex Fractions

    Ok this is something i learned few years ago and I am a bit rusty. So i have to find the absolute value of: \frac{1 - 2i}{3 + 4i} + \frac{i - 4}{6i - 8} So first i add the two fractions and i get: \frac{(1 - 2i)(6i - 8) + (i - 4)(3 + 4i)}{(3 + 4i)(6i - 8)} Next i simplify and then...
  16. L

    Derivative of an absolute value

    I don't get why: \frac{d}{dx}[|u|]=\frac{u}{|u|}(u') Can someone give me an example to which this applies? Can you use any function in place of "u"?
  17. R

    Quicky on derivative of absolute value in exponential

    Hey folks, I'm looking for a little guidance in solving the derivative y'(x)of the following function containing an absolute in the exponent: y(x)=e^{a|x|} I'm pretty sure its not as simple as y'(x)=a e^{a|x|} Any suggestions??
  18. S

    Verifying the mean value theorum for absolute value function

    Homework Statement a) Find the average value of the function f(x)=|x-1| over [0,2]. b)Verify the mean value theorum for integrals for the function and interval in part (a). Homework Equations mean value theorum for integrals: \int_a^{b} f(x)dx=f(c)(b-a) mean value theorum for...
  19. S

    Typo in Real Analysis Study Page: Absolute Value of x for x<0?

    I found a study page which lists the absolute value of x for x<0 as -x. I think this has to be a typo. The study area is real analysis. Does anyone have better information? Maybe it is some special notation?
  20. S

    Equation and Inequation With Absolute Value [12th Grade]

    Homework Statement Resolve these equations and these inequations with the absolute values. Give the solutions in the form of interval : |2-x|< 4 |6-2x| = 3 |x+2| > 3 |4x²-12x+9| = 4 |3x+1|+|1-x|>3 |1-x²|=2x |x+2|<|x+3| |x^3-1|+pi[tex]>[tex]\sqrt{3} 3<|x+2|<4...
  21. P

    Solving an Absolute Value Inequality with Two Terms

    Homework Statement \left|x-5\right|\leq\left|x+3\right| i know how to slove abs inequalities when there is one. like \left|x-5\right|\leq5. put x-5 between 5 and -5. then solve for x. i don't know how to start this. can someone provide some insight.
  22. S

    How can I prove the statement |a-b| < |a| + |b| using the triangle inequality?

    Homework Statement a and b are real numbers. Show l a-b l < l a l + l b l Homework Equations Well, I know la+bl < lal + lbl by the triangle inequality. The Attempt at a Solution If I can prove that la-bl < la+bl, then I'm done, but that most recent inequality almost seems...
  23. S

    Absolute Value Proof (Need Help)

    Homework Statement Prove that abs(a+b+c) is less than or equal to abs(a)+abs(b)+abs(c) Homework Equations None The Attempt at a Solution This makes sense to me that this would always be true, but i just can't seem to figure out how to write it out
  24. F

    What Does the Solution Set for |x² - 4| < 1 Look Like?

    Hello, abs(x^2 - 4) < 1 implies that: x^2 - 4 < 1 and 4 - x^2 < 1 solving first equation for x gives: -sqrt(5) < x < sqrt(5) solving second equation for x gives: -sqrt(3) < x < sqrt(3) Now, my question is, what does that mean?? How do I give the solution set, without a...
  25. L

    Why Do Solutions Sometimes Differ from Absolute Values?

    In equations dealing with absolute value sometimes the answer to the equation does not coorospond with the absolute value. Why is this?
  26. V

    Absolute Value Equations: Setting Restrictions

    Hello everyone, For the following absolute value equations, I have no trouble solving them and finding the valid x solutions by plugging all the x solutions into the original equation. However, I am just wondering if could someone please show me how to set restrictions for the following...
  27. Q

    How do I solve a quadratic equation with absolute value?

    Homework Statement 25|x| = x^2 + 144 Homework Equations none The Attempt at a Solution okay well, I'm not quite sure what to do, do i try to isolate the |x|? and then break it up into a postive and negative? |x| = (x^2 + 144)/25 ? but from here i become lost...
  28. A

    Absolute Value in a double integral

    [SOLVED] Absolute Value in a double integral Homework Statement If \Omega = [-1,1] x [0,2], evaluate the double integral \int\int_{\Omega} \sqrt{|y-x^{2}|} dA given that it exists. Homework Equations None The Attempt at a Solution I know that in order to integrate with the...
  29. W

    Why is f(x,y) not differentiable at (0,0)?

    Hello, My question is as follows: Show that the function f(x,y) = sqrt(abs(xy)) is not differentiable at (0,0). I was going to go with trying to show that the directional derivatives don't all exist here, but that would require finding the gradient, and I always get confused when trying to...
  30. J

    Absolute value of a complex number

    Homework Statement Find the real/imaginary parts of sinh(x+yi) and its abs value.Homework Equations The Attempt at a Solution I am able to decompose sinh(x+yi) = cosy*sinh(x) + isin(y)cosh(x) - (which is correct according to my book) Now finding the absolute value is kind of causing some...
  31. B

    Absolute Value of a Difference with Heaviside Function

    Homework Statement If |x| = -x + 2x*H(x) what is |x - a|? This isn't the actual question, just something I need to know to solve the question. Homework Equations H(x) is the Heaviside function which is: y = 1 if x >= 0 y = 0 if x < 0 The Attempt at a Solution Well, I'm...
  32. C

    How do I graph the absolute value function with multiple layers?

    Homework Statement The problem is: Draw the graph of the following function: f(x)=|x+|x+|x-1||| Homework Equations |x|=\left\{\begin{array}{cc}x,&\mbox{ if } x \geq 0\\-x,&\mbox{ if }x<0\end{array}\right The Attempt at a Solution If the function were, for instance...
  33. T

    Discrete Mathematics Absolute Value Proof

    Homework Statement Prove the following statement: For all real numbers x and y, |x| times |y| = |xy| Homework Equations I really don't know how to start this as a formal proof. The Attempt at a Solution I was thinking I'd have to break it down into four cases and logically prove...
  34. U

    How Can I Write an Absolute Value Function for the Outline of the Khufu Pyramid?

    Okay, we don't understand this and we need much help the largest pyramid included in the first wonder of the world is Khufu. IT stands 450 feet tall and its base is 755 feet long. Imagine that a coordinate plane is placed over side of the pyramid. In the coordinate plane, each unit...
  35. L

    A weird absolute value problem

    Please help me out here with this problem: 37. Express the function f(x)=l x l + l x-2 l without using absolute value signs. Okay, I've took a sneak peak at the answer key and it is given in kinda of like piece-wise notation where they have 3 equations and different intervals of x for...
  36. D

    How Should Absolute Value Be Handled in Trigonometric Integrals?

    Hi. For an integral like this, for example: \int {\sqrt {1 - \cos ^2 x} dx} The most obvious way of solving would be to make use of the pythagorean identity, to get: \int {\sqrt {\sin ^2 x} dx} Now, I've been taught to simply evaluate it like this: \int {\sqrt {\sin ^2...
  37. M

    Derivative absolute value and also curve help

    Hey i have 2 questions here that i have finished and i have no answers to my solutions so i would like someone to check it over and see where i went wrong , I am not 100% sure on this stuff and need some help , thanks a lot ! Question 1: Using the definition of a derivative, show that ƒ (x) =...
  38. D

    Possible Values of a and b for |2x+a| = |b-x| at x=-4 and x=2/3

    The equation |2x+a|=|b-x| has exactly 2 solutions, at x=-4 and x=2/3. Find the value(s) of a and b. Ok so the questions is asking me to find possible values of a and b which make the equation true for ONLY x=-4 and x=2/3. So for: |a-8|=|b+4| |a+4/3|=b-2/3| I need to find the values...
  39. D

    Absolute Value functions => Piecewise Function

    Given the function: f(x) = \left| {x + 1} \right| + \left| {x + 2} \right| How can i write that as a piecewise function? If i was given something in the form of f(x) = \left| {g(x)} \right|, i know to write it as: f(x) = \left\{ {\begin{array}{*{20}c} {g(x),} & {g(x) \ge 0}...
  40. S

    Max of the absolute value of a polynomial

    What I have is this: Let P_n(x)=(x-x_0)(x-x_1)...(x-x_n), _i are subscripts. Prove that the maximum value of |P_1(x)| for x in [x_0,x_1] is h^2/4, where h =x_1 - x_0. All the x_i terms are evenly spaced. That is, x_(i+1)-x_i is the same for all i. What I noticed is that P_1(x_0)=P_1(x_1)=0...
  41. T

    How do I graph an absolute value inequality in 2 variables?

    I need some help graphing an Absolute value. I know what it look like from a graphing calculator but I don't understand how to get to the answer. http://media.twango.com/m1/original/0042/5fda15431b96400eb32e66afa3627624.jpg Thanks
  42. A

    Why is the sign reversed on -4 in solving 3x^{2}+12x>0?

    I'm confused by why my work is wrong: 3x^{2}+12x>0 3x(x+4)>0 3x>0 , x+4>0 x>0 , x>-4 However, the correct answers are: x>0 and x<-4? Why is the sign reversed on -4? I thought you were only suppose to reverse it if you multiply or divide the equation by a...
  43. R

    Definition of Absolute Value of a Function

    Homework Statement Question straight off the book "In this lesson, you have explored the absolute value of a function. How is it defined?" Homework Equations In this lesson, I did questions like xE[0,9] and was told to write each intervals in absolute value notation. The Attempt at a...
  44. A

    Absolute value and square root

    what is /squared root sign4x+1 / what does this equal because I'm confused when it has the square root sign on it all and it's absolute value
  45. T

    Calculate absolute value of A+B

    Homework Statement Consider two vectors by A=5i-3j and B=-i - 2j Calculate A+B Calculate A-B Calculate absolute value of A+B Calculate absolute value of A-B Homework Equations R= (Ax + Bx)i + (Ay + By)j R= (Ax + Bx)i - (Ay + By)j The Attempt at a Solution I'm not sure...
  46. B

    What is the absolute value of imaginary numbers, why not supernatural numbers?

    what is the absolute value of imaginary numbers, why not "queer" numbers? the square root of -1 is "i". the absolute value of an interger is itself, and of a negative number, it is a positive interger. |-5| = 5 |5| = 5 what is |5i| = ? |-5i| = ? why not invent a queer number...
  47. C

    Computing the range for a rational function involving absolute value

    Hi. I need help computing a range. The question is : Find the domain and range of y=\frac{|x+2|}{x}. The domain is obvious, x can't be 0, (-inf,0,) U (0,inf). But how do I find the range?? Can someone help me out? I have tried messing around with the definition of absolute value... if...
  48. A

    Measuring the absolute value of the AU

    How can they be sure that the value for the AU is 149,597,870 km? What methods are used to measure it?
  49. S

    Limits involving absolute value.

    \lim_{x\rightarrow\infty}\frac{\sin|x|}{x} \lim_{x\rightarrow 0}\frac{|x|-|x-2|}{x-1} thanks!
  50. L

    Meaning of the s-domain absolute value function

    I am trying to understand the meaning of the s-domain absolute function derived from taking the laplace transform of a t-domain function. I know for sure that the real part of the complex frequency in the time domain is the sinusoidal frequency and the imaginary part of the complex frequency in...
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