# 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. ### Trying to understand the property of absolute value inequality

First lets focus on ##|x|## which is defined as distance between ##x##and ##0##. But if we look into it closely $$13=|-11-2|$$ which is distance between -11 and 2 but $$13=|11-(-2)|$$ which means this is distance between 11 and -2. Which is it? In the same way $$x=|x-0|$$ is distance between 0...
2. ### I Absolute value bars in dot product derivation

Dose someone please know why they have the absolute value bars in this derivation? many thanks!
3. ### Solving an absolute value quadratic inequality

Im a having trouble understanding how this exactly works. $$|x^2 - 4| < |x^2+2|$$ So I know the usual thing to do when you have absolute values,here it is even simpler since the right part of the inequality is always positive so I just have these 2 cases. 1. ## x^2-4 >= 0 ## and 2. ## x^2-4...
4. ### Quadratic inequalities with absolute values

I was given a problem to solve that goes like this ##\frac{3}{|x+3|-1}\geq |x+2|## . I got the correct solution for all possible cases and here they are; for ##|x+3|\geq0## and ##|x+2|\geq## i got ##x\epsilon <-2, -2\sqrt{3} ]## and for ##|x+3|\leq0## , ##|x+2|\leq0## I got ##x\epsilon [-5...
5. ### Quadratic equation: Which way is correct? pic1 or pic2?

I am a bit confused, so if anyone can explain to me which way is right I would be very thankful. I think that the way in pic 1 is right because of the properties written next to the procedure but the professor who posts videos on youtube solved it the way as written in pic 2 where he didn't...

35. ### Mathematical Analysis Proof: |x-y|<= |x|+|y|

Homework Statement 1. Show that for all real numbers x and y: a) |x-y| ≤ |x| + |y| Homework Equations Possibly -|x| ≤ x ≤ |x|, and -|y| ≤ y ≤ |y|? The Attempt at a Solution I tried using a direct proof here, but I keep getting stuck, especially since this is my first time ever coming...
36. ### Must we always use absolute value for lens magnification?

My brother's physics teacher says that magnification and height of image are always positive. Is she right?
37. ### B Absolute Value Inequality, |x|>|x-1|....where's my mistake?

Rule: Suppose a>0, then |x|>a if and only if x>a OR x<-a So |x|>|x-1| becomes: x>x-1 which is false (edit: or more accurately doesn't give the whole picture, it implies true for all x) OR x<-x+1 2x<1 x<1/2 which is false
38. ### MHB Why can't real numbers satisfy this absolute value equation?

Explain, in your own words, why there are no real numbers that satisfy the absolute value equation | x^2 + 4x | = - 12. Can we say there is no real number solution here? If so, is the answer then imaginary taught in some advanced math class?
39. ### I Comparing two absolute value equations

Hello, How does one go about algebraically checking if |x+|y+z|| and ||x+y|+z| are equal?
40. ### Find the cdf given a pdf with absolute value

Homework Statement Consider a continuous random variable X with the probability density function fX(x) = |x|/5 , – 1 ≤ x ≤ 3, zero elsewhere. I need to find the cumulative distribution function of X, FX (x). 2. Homework Equations The equation to find the cdf. The Attempt at a Solution FX(x)...
41. S

### B How to interpret the integral of the absolute value?

This is rather basic, and may be a misconception of the notation, however, I can't make the following sum up: The following is given: x_n(t) = 1 -nt , (if 0 <= t <= 1/n) and 0, (if 1/n < t <= 1) However, this part I can't grasp this part in the book: ||x_n||^2 = \int_0^1...
42. ### MHB Why Are There No Real Solutions to the Equation $$|x^2 + 4x| = -12$$?

Explain why there are no real numbers that satisfy the equation $$|x^2 + 4x| = - 12$$ How is this done algebraically?
43. ### MHB Absolute Value Definition....3

Use the algebraic definition of absolute value to rewrite the expression below in a form that does not contain absolute value.
44. ### MHB Absolute Value Definition....2

Use the algebraic definition of absolute value to rewrite the expression below in a form that does not contain absolute value.
45. ### MHB Absolute Value Definition....1

Use the algebraic definition of absolute value to rewrite the expression below in a form that does not contain absolute value.
46. ### Absolute value notation removal

Homework Statement Rewrite |x| < 1 and |x| > 1 by eliminating the absolute value sign Homework Equations |x| < 1 = -1 < x < 1 |x| > 1 = ? The Attempt at a Solution I know that |x| < 1 can be rewritten as -1 < x < 1 but I'm not sure about |x| > 1. Am I right to assume that |x| > 1 = -1 > x > 1?
47. ### MHB 232.q1.2c Double integral with absolute value in integrand

$\displaystyle \int_{-1}^{1} \int_{-2}^{3}(1-|x|) \,dy\,dx$ ok i was ? about the abs
48. ### MHB How Do You Solve the Absolute Value Equation -3|x+5| + 1 = 7|x+5| + 8?

-3|x+5| + 1 = 7|x+5| + 8 Solution: -3|x+5| – 7|x+5| = 7 -10 |x+5| = 7 |x+5| = -7/10 x+5 = ±(-7/10) x = ±(-7/10) – 5 x₁ = -7/10 – 5 x₁ = -57/10 x₂ = 7/10 – 5 x₂ = 2/10 x₂ = 1/5 Correct?
49. B

### B Solving Absolute Value Inequalities: How to Define Cases

Hi there, I'm having trouble understanding this math problem: |x| + |x-2| = 2 The answer says its: 0<=x<=2 I understand you need different "cases" in order to solve this. For example, cases for when x is less than 0, when x-2 is less than 0, etc. Thanks, blueblast
50. ### I Limits and Continuity - Absolute Value Technicality ....

I am reading Manfred Stoll's Book: "Introduction to Real Analysis" ... and am currently focused on Chapteer 4: Limits and Continuity ... I need some help with an inequality involving absolute values in Example 4.1.2 (a) ... Example 4.1.2 (a) ... reads as follows: In the above text we read ...