For questions about or involving the absolute value function.

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2
votes
1answer
15 views

Piecewise linear function and absolute value

While writing a solution to homeworks for my students, I had to write the function $$f(x)=\left\{\begin{array}{ll} \frac{x+2}{2}, & x\leqslant -4\\ \frac{x}{4}, & -4\leqslant x\leqslant 4 \\ ...
2
votes
6answers
58 views

Adding $2$ absolute values together: $|x+2| + |x-3| =5.$

I came across a very basic absolute value question $|x+2| + |x-3| =5.$ Initially, I thought the answer was $x=-2$ and $x=3$ because I let each absolute values be either positive and negative and ...
0
votes
5answers
129 views

Determine all solutions to $|x+12|+|x-5|=15$

Determine all solutions to the following. $$ \lvert x+12\rvert +\lvert x-5\rvert =15.$$ How can I solve problems like this? Should I try graphing the function? Should I somehow consider various ...
11
votes
2answers
69 views

Finding all solutions to the equation $|||||x|-1|-1|-1|-1|=0$

I was presented this question by a student I was tutoring: Suppose $x \in \mathbb{R}$. Find all solutions of the equation $$|||||x|-1|-1|-1|-1|=0.$$ What I explained to the student: Given ...
1
vote
2answers
45 views

How to find roots for $y = ||x^2-x-20|-8|$

$$ y = ||x^2-x-20|-8| $$ After I set $y = 0$, I do not know how to deal with multiple absolute values.
8
votes
9answers
2k views

What's wrong with solving absolute value equations in this way?

Say I have $3x-2 = |x|$. Why can't I just do this: $3x - 2 = -x$ and $3x - 2 = x$ and then get two values for $x$: $1$ and $0.5$? I know the answer $0.5$ doesn't work if you plug this in. However, I ...
1
vote
1answer
52 views

Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which says what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
-3
votes
1answer
460 views

What properties of the absolute value function should one remember? [closed]

When one begins to study real analysis, the absolute value function quickly enters and a large number of exercises involve manipulations with it. What are the basic properties of absolute value that ...
1
vote
1answer
84 views

Inequality: $\left|x^3-y^3\right|<|x|^3+|y|^3$

Could anyone show me why $$\left|x^3-y^3\right|<|x|^3+|y|^3$$ for all real numbers (x,y) except 0? I'm thinking of whether of how to remove the modulus sign on the left hand side of the ...
0
votes
2answers
40 views

Expressing absolute value equations as piecewise functions

I'm not sure how to express this function in piecewise form without using absolute values: $$ f(x) = 3|x-2| - |x+1|$$ I know how to do it when there is just one absolute value, such as: $$g(x) = ...
3
votes
2answers
88 views

Does my proof of $|x+y| \le |x| + |y|$ make sense? How do I conclude a proof?

Thank you for reading it. I know I made a lot of mistakes. This is my first ever proof that I have attempted. Another note is that I only have been studying proofs for about a week. Any advice will be ...
2
votes
2answers
29 views

Taking root from absolute expression

Why is the following true? (Where all terms are positive) $$|x-y| < \epsilon^2 \implies |\sqrt x - \sqrt y| < \epsilon$$
3
votes
1answer
65 views

Geometric idea behind equations of the form $|x-a|\pm|x-b|=c$

So let's say I want to solve $$|x-a|\pm|x-b|=c$$ Using the classic multiple cases approach, one can show that the solutions are given by $$x=\frac{a+b\pm c}2 $$ But how can one make sense of this ...
4
votes
2answers
100 views

Calculating the best match between two sets

I’m a PHP developer and I have a problem calculating the perfect match between two different data sets. I have data sets from companies, where each company defines the requirements for a specific ...
2
votes
3answers
368 views

How to solve inequalities with absolute values on both sides?

If you have an inequality that has two absolute value bars like $|4x+1|<|3x|$, how do you go about doing this? I know that if $4x+1<3x$, then those $x$'s will work but what else do I do? I think ...
-1
votes
3answers
135 views

An inequality with absolute value and a parameter: $|x-4|>a$

Solve : $|x-4|>a$. Case 1: $a>0$; Case 2: $a<0$ Progress I am getting answers which look similar in both cases: Let $a>0$ so $x>4+a$ or $x<4-a$ , Let $a<0$ so ...
3
votes
0answers
36 views

Nested absolute operations

The question is: are the following two functions equivalent? And if yes, what properties of the absolute value should I use to prove it? $f_1(x,y,z)$ = $|\, x + |y+z| \,|$ $f_2(x,y,z)$ = $| \,|x+y| ...
3
votes
1answer
51 views

Why is the value of $\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|\,dx$ independent of integer parameter $n$?

I am not able to find an easy solution for the following formula $$\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|dx=4+\frac{4}{3}\pi\sqrt{3}.$$ Please help me prove it. Why it does not depend on the (positive) ...
-3
votes
1answer
66 views

Someone can solve this limit? [closed]

$$f(x) = \frac{9-2\sqrt{\left\vert\,x\,\right\vert}}{3\sqrt{-x}}$$ $$\lim_{x\to-\infty} f(x) = l$$ I need the method of calculate l and solve this: (proving limit using epsilon-delta definition) ...
3
votes
0answers
75 views

Dealing with absolute values after trigonometric substitution in $\int \frac{\sqrt{1+x^2}}{x} \text{ d}x$.

I was doing this integral and wondered if the signum function would be a viable method for approaching such an integral. I can't seem to find any other way to help integrate the $|\sec \theta|$ term ...
0
votes
2answers
36 views

Graphs for mod functions

Can someone please teach me how to obtain graphs for the following types of functions: $2+3|x-1|$ $|x-1|+|x|+|x+1|$ $|x-1|-|x|-|x+1|$ $|x-1|^2$ Thanks.
0
votes
1answer
19 views

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$, where $aj+b$ is a complex number, and $|f(x)|$ is the modulus function. In the past I've been calculating $|(aj+b)^{-1}|$ by multiplying the numerator and ...
4
votes
3answers
93 views

Is it always true? $\left|A-B\right| \le \left|A\right| + \left|B\right|$

Is it always right to claim that: $$\left|A - B\right| \le \left|A\right| + \left|B\right|$$ where $A, B \in \mathbb{R}$ ?
2
votes
0answers
30 views

Solving $n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt$

I have to solve $$ n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt $$ where $\psi(t)=(2\pi)^{-\frac{1}{2}}e^{-\frac{1}{2}t^2}$ is the density ...
0
votes
3answers
70 views

Absolute value quadratic inequalities not the usual?

$ | -x^2 + 6x | \gt 13 $,for example. I would start off solving $ -x^2 + 6x = \pm 13 $ and either get 4 solutions, 3 solutions or two simply do the the nature of the graph. Without knowing if the two ...
0
votes
1answer
48 views

$\left | -(x+2)^2+6(x+2) \right |>13$

I did $-(x+2)^2+6(x+2)>13$ and $-(x+2)^2+6(x+2)< -13$. The first inequality had complex solutions and therefore can be disregarded but the second one has two real solutions, $x \approx -3.7$ and ...
0
votes
1answer
26 views

Find absolute value inequality describing the result of measurement

This is a problem from my homework where a sample of a quantity is $37.5\pm 1.2$ grams. And if the actual quantity is $x$, write the results as an absolute value inequality and solve for $x$. I ...
2
votes
1answer
50 views

How to draw $|y|=|(x-2)^2-1|$?

This correspondence's domain and codomain is available for all real numbers. So the codomain and domain are the set of all real numbers on? And, this graph is true? (This graph was called ...
0
votes
1answer
316 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 ...
1
vote
4answers
55 views

How to solve equations involving modulus function of the type $|x+1| - |1-x|=2 $ and $ |x-1|=|x|+a$?

I am able to solve equation of the type $ |5x+1|=|11-2x|$. I square both the side and my equation becomes $ (5x+1)^2=(11-2x)^2 $ further simplification gives me $ (5x+1)=\pm (11-2x)$. I get have ...
1
vote
6answers
395 views

How could we solve $x$, in $|x+1|-|1-x|=2$?

How could we solve $x$, in $|x+1|-|1-x|=2$? Please suggest a analytical way that I could use in other problems too like this $ |x+1|+|1-x|=2$ and of this genre. Thank you,
1
vote
1answer
32 views

Don't understand inequality in order to prove Algebraic Limit Theorem

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I'm stuck on Theorem 2.3.3 on page 45, i.e., the Algebraic Limit Theorem. In particular, letting $\lim a_n = a$ and $\lim ...
2
votes
1answer
30 views

If $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ are complex numbers, then $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$

Let $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ be two complex numbers. Ahlfors says that $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$. I don't understand why that is. Any help would be greatly appreciated.
1
vote
1answer
53 views

What is the minimum of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ over all complex numbers?

Find the Least value of $\left|z-2(1+i)\right|+\left|z+1-5i\right|+\left|z-6+2i\right|$ My try:: Let $A(2,2)$ and $B(-1,5)$ and $C(6,-2)$ and $P(x,y)$ be a point Here $A,B$ and $C$ are the point ...
2
votes
4answers
61 views

Evaluate the integral $\int_{-1}^{1}\left\vert\, x^{3} - x\,\right\vert\,{\rm d}x$

I'm trying to solve: $$\int_{-1}^{1}\left\vert\, x^{3} - x\,\right\vert\,{\rm d}x$$ I tried to solve this integral as follows: solving $x^{3} - x = 0$ which implies $x = 0$ , $x = -1$ or $x = 1$. ...
0
votes
0answers
17 views

A system of absolute value equalities

Background: I'm trying to show that the transformation $T:\Bbb R^n\to\Bbb R^n$ defined by $T(x_1,\dots,x_n) := (|x_2-x_1|,|x_3-x_2|,\dots,|x_1-x_n|)$ is (or is not, this is out of curiosity only) ...
0
votes
2answers
63 views

Solving an equation with absolute values: $ | 2x - 5| + | 2x - 3 | = m $

Given that the following equation does not have solutions in $\mathbb{R}$, find the value of $m$: $$| 2x - 5| + | 2x - 3 | = m $$ I try to resolve this equation on cases, when $| 2x - ...
2
votes
2answers
121 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
2
votes
1answer
52 views

Solve $2\sqrt{(x-1)(x+2)}\ge|x+1|-2$

Can you please show me how can I solve this inequality. I would like to see how it can be done without the graph of the functions. Thank you! $$2\sqrt{(x-1)(x+2)}\ge|x+1|-2$$
1
vote
2answers
27 views

Where did I go wrong with this inequality involving absolute value function?

Question: Find all $x \in \mathbb R$ such that the inequality $4<|x+2|+ |x-1|<5$ is satisfied. This is my attempt at solving the problem: Case (i): If $x+2 \geq 0 $ and $ x-1\geq0$, then ...
-5
votes
2answers
54 views

SAT question stuck? [closed]

I am preparing for sat and this question, I have no idea how to solve it. Please provide step wise solution also. If $2|x+3|=4$ and $\frac{|y+1|}{3}=2$, then $|x+y|$ could equal of the following ...
0
votes
2answers
58 views

$|x| - |y| \leq |x-y|?$

Is there a clever way to show that $$|x| - |y| \leq |x-y|$$ I believe I can think of a way to solve this using cases, but the book I'm working out of said that "A very short proof is possible if you ...
0
votes
3answers
33 views

I do not quite understand this difference in limits

According it my study material: $\lim_{x\to 0^-}\frac {x}{|x|}= -1$ and $\lim_{x\to0^-} \frac {1}{|x|}= \infty$ Why does $\lim_{x\to0^-} \frac {1}{|x|}\ne -\infty$ as 1 still devided by a negative ...
1
vote
3answers
58 views

Proving a limit exists - solving for epsilon with absolute values

I have the equation that I want to prove the limit goes to 1: $$\lim_{n \to \infty} \frac {(n+8)(n+1)}{n(n-10)} = 1$$ Using definition of limit, I get this equation: $$ \left | \frac ...
2
votes
3answers
62 views

For which $a$ does $|x+1|+|2-x|=a^2 -1$ have exactly two solutions?

If it is not a problem, I would really appreciate if someone could explain to me how to solve and graph the following equation: For which real numbers $a$ does the equation $|x+1| +|2-x|=a^2 -1$ ...
2
votes
1answer
36 views

Does “Expected Absolute Deviation” or “Expected Absolute Deviation Range” exist in stats and have another name?

So everyone is familiar with Variance and Standard Deviation from high school, but it seems no one has any familiarity with a philosophical justification for such weird, seemingly arbitrary measures. ...
-1
votes
2answers
59 views

Sketching a set of complex numbers and deducing the value of $|z +1 - i|$ for such numbers

The point $P$ represents the complex number $z$. a) Given that $\arg(\frac{z-2i}{z+2}) = \frac{\pi}{2}$ , sketch the locus of $P$. Ok so I've sketched this and this is what it looks like : b) ...
0
votes
4answers
58 views

How to solve Absolute Value Inequality: |x-1| ≥ 3-x

I am learning the topic of solving absolute value inequality question. I had mostly understood the steps in order to solve for an inequality. However, I'm still clueless of a step to solve the ...
-5
votes
3answers
79 views

How to solve this: $|3-x|\ge2$ [closed]

How to solve $|3-x|\ge2$ ? I know that if $|x| < y$, then $-y < x < y$. But in this case what to do? Thanks. Here, $|x|$ is the absolute value of $x$.
1
vote
2answers
33 views

Spivak Absolute Value Problem (Prologue 9-v)

I'm working on the following problem Express the following with at least one less pair of absolute value signs $$|(| \sqrt2 + \sqrt3| - |\sqrt5 - \sqrt7|)|$$ Now I can see that the ...