1
vote
5answers
52 views

For what real number $c$, this equation has exactly three solutions?

For what real number c does the equation $|x^2 + 12x + 34| = c$ has exactly three solutions?
-2
votes
2answers
72 views

What is $\sqrt{x^2}$ when $x<0$? [closed]

$x\in \mathbb{R}$\ $\{0\}$ $$\frac{\sqrt{x^2}}{|x|}+1 =?$$ What is the answer when $x \lt 0$? $2$ or $0$?
1
vote
4answers
26 views

Showing an Absolute Value Inequality Problem Proof

I tried solving this question but it does not works for me. Q.) Show that $\left|x + \frac1{x}\right| \ge 2$ for all $x \ne 0$ There are two ways to do. One is squaring and other is to use absolute ...
1
vote
3answers
36 views

What are the steps to solving |3x + 1| > |2x - 7| with the given answer as $(-∞,-8)\cup(6/5,∞)$?

What are the steps to solving $|3x + 1| > |2x - 7|$ with the given answer as $(-∞,-8)\cup(6/5,∞)$? I am having difficulty with understanding inequalities with absolute value functions on both ...
4
votes
2answers
52 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 \\ ...
12
votes
2answers
99 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 ...
2
votes
6answers
79 views

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

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
2answers
48 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.
1
vote
1answer
56 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 ...
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 ...
0
votes
2answers
73 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
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 ...
2
votes
4answers
908 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 ...
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 ...
0
votes
1answer
49 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 ...
1
vote
4answers
84 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 ...
0
votes
0answers
18 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) ...
2
votes
2answers
122 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
-5
votes
2answers
56 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
4answers
64 views

solving the system

solve the system : $$ y+|x-2|=3 $$, $$ |x+y|= m $$ graphicly when $m$ equals $6$. I can easily (realtively) skecth the first graph , however, how the bloody hell do you sketch $|x+y|= 6$??
2
votes
4answers
247 views

inequality method of solution

Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$ I've tried first determining when the absolute value ...
2
votes
5answers
119 views

Finding the minimum value of a sum [closed]

Let $x,y,z$ be real numbers . Find the real number $a$ so that $S$ has a minimum value , where $$S=|x-a|+|y-a|+|z-a| .$$
1
vote
4answers
96 views

Simple question about the range of possible values for a function

So we have $2 |3-x| + 5 = k$, where $k$ is a constant. Provided this equation has two real solutions for $x$, what is the range of possible values for $k$?
0
votes
1answer
25 views

Absolutet Value Inequality with cases number line

I was wondering if anyone knows how to solve $|ax+b|<cx+d$ type questions by using cases and the number line to finish. I am personally struggling with the number line, I have half-finished a ...
0
votes
2answers
34 views

Need help with this absolute value equation

I need to solve the following equation involving absolute value: $$|x-1| = 1-x$$ Looking at the term $x-1$, I thought I'd divide the interval into parts: $x < 1$ and $x \geq 1$. Now, when ...
2
votes
2answers
112 views

Why can't absolute values be expressed with negative numbers. [closed]

The answer to this question seems obvious. 'An absolute value expresses the quantity of ones between any number and 0'. But does that mean it must be positive? I took a shot at answering my ...
2
votes
2answers
65 views

Basic question about solving modulus equation

It common in the literature to solve the modulus equation like $|x+5|+|x-1|=8$ by dividing into cases when $x<-5$, $-5\leq x<1$ and $x\geq1$. My question is whether dividing into cases is ...
0
votes
3answers
103 views

Absolute values don't work

I don't understand, how absolute valued could possibly be considered well defined. As shown here, $|a| = |-a| , ||a|| = |-|a||$ So lets take $a=-2, |a| = -2 = |-a|,$ but $|-a| = |2| = 2$ But it ...
1
vote
0answers
59 views

Absolute values nested multiple times

Is there any algorithm to quickly determine "zero points" (i.e. points with undefined derivation) of absolute values functions which are nested multiple times? I do know, that any part of this ...
2
votes
7answers
130 views

for $n$ an integer, why is $n^0=1$ ??

This is so going to cost me.... I was wondering why for any integer $n$: $n^0 =1$. Perhaps It's because $n$ is a round number and if $m$ is a non negative integer, also round then: $$n^m = 1 \cdot ...
1
vote
1answer
192 views

Logarithmic inequalities

Full disclosure: This is a homework problem, but my question is regarding a concept that came about during solving the problem, not the actual solution to the problem. Problem: Rewrite as geometric ...
0
votes
2answers
43 views

Why is the following simplification possible?

I have seen the following simplification: $$\left|\frac{1}{(-1-\frac{1}{n})^4 - 1}\right| = \frac{1}{\left|-1-\frac{1}{n}\right|^4 - 1}$$ I really don't have a clue why this is possible... I am ...
2
votes
1answer
52 views

Solving inequation with two absoulte values

I need to solve the following inequation: $$ |x| \cdot |x-1|-1>-x\\ $$ I cant get the correct result. I tried to solve it like this: $$ |x| \cdot |x-1|-1>-x $$ I know that I can write $|x ...
-5
votes
3answers
81 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 $x$.
3
votes
4answers
149 views

Is $\sqrt{x^2}=|x|$ or $=x$? Isn't $(x^2)^\frac12=x?$ [duplicate]

$|x|=\sqrt{x^2}$ as Wolfram|Alpha shows. But, as $(x^2)^\frac12=x$, I can't understand where am I wrong interpreting Square-root.
1
vote
0answers
443 views

Properly Solving Absolute Value Inequality and Quadratic Inequality Problems

How do I solve the following absolute value inequality and inequality problems properly? 1) $\newcommand\abs[1]{|#1|}\abs{2x+9}>x$ Solving this problem algebraically, I get When $x > 0, x ...
1
vote
4answers
76 views

Inequalities and absolute values

My book asks that if $$-5\leq x\leq 1$$ then find the boundaries of absolute value of $x$. Can you please help me in finding that?
0
votes
1answer
57 views

Taking “Absolute Value Operator” as a common factor?

If I have an equation like this and Im trying to solve for X |x| + 4|x| = 40 Can I take the absolute Value (Modulus) as a common factor? ...
1
vote
2answers
835 views

Sum of absolute values and the absolute value of the sum of these values?

I'm working on a proof and I need some help with this: I determined that for some situations ($x$ or $y$ are negative but not both): $|x| + |y| > x + y$ How can I conclude using that statement ...
3
votes
3answers
141 views

Absolute Value inequality help: $|x+1| \geq 3$

Find the solutions to the inequality: $$|x+1| \geq 3$$ I translate this as: which numbers are at least $3$ units from $1$? So, picturing a number line, I would place a filled in circle at the ...
2
votes
1answer
85 views

what is the value of $a+b?$

Can anyone help me to solve this problem: $x$ and$y$ are real numbers which satisfy $x>y$ and $xy<0$. If $\left | x \right | + \left | y \right | + \left | 42y-x \right | + \left | 23x-y \right ...
1
vote
3answers
91 views

To what extent can I square both sides of an absolute equation?

I am working on some absolute equation problems like the following: $$\begin{align} & {|x-4|} \lt 1 \\ & 1 \le |x| \le 4 \\ & |x+3| = |2x+1| \end{align}$$ Now, for both of these ...
2
votes
2answers
114 views

How to prove this max absolute value equation?

How to prove this equation? $$\max(|x_1-x_2|,|y_1-y_2|) = \frac{\left|x_1+y_1-x_2-y_2\right|+\left|x_1-y_1-(x_2-y_2)\right|}{2}$$
4
votes
5answers
121 views

How is it, that $\sqrt{x^2}$ is not $ x$, but $|x|$?

As far as I see, $\sqrt{x^2}$ is not $x$, but $|x|$, meaning the "absolute". I totally get this, because $x^2$ is positive, if $x$ is negative, so $\sqrt{y}$, whether $y = 10^2$ or $y = -10^2$: $y$ is ...
0
votes
2answers
47 views

Is this a correct way to express $\left|f(x)\right| \leq \left|x\right|^9$?

If $\left|f(x)\right| \leq \left|x\right|^9$, then, is it correct to say that $f(x) \leq x^9$ and $f(x) \geq -x^9$ ? If it is not, could someone explain why? Thank you.
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 - ...
0
votes
3answers
612 views

I need help finding the x intercept of an absolute value equation.

$y= |2x-3| + 2x +6$ Find the $x$ intercept. (P.S.: In my Algebra teacher's answer document it says that there is no $x$ intercept for this equation. I'm confused as to why that is. I keep ...
1
vote
3answers
72 views

Value and simplify

I want to find the value and simplify square root 36 ? Square root of 36 is 6 But I would know how to find the value and simplify it .
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 ...
1
vote
1answer
260 views

Finding domain of a rational function

Find the domain and graph: $$f(t)=\frac{-t}{|t|}$$ My book says to define it piecewise. My questions: $\mathbf{1)}$ Do all rational functions have to be defined piecewise, or just this ...