3
votes
4answers
232 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
106 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
85 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
22 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
31 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
86 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
45 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
92 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
50 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
129 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 ...
0
votes
1answer
173 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
46 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 ...
1
vote
0answers
309 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 ...
0
votes
4answers
70 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
48 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
548 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
108 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 ...
1
vote
2answers
77 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 ...
0
votes
3answers
76 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 ...
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
3answers
440 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 .
1
vote
1answer
208 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 ...
0
votes
2answers
101 views

How to solve $|2x +1|< 1/4$?

How do you solve $$|2x +1|< \frac{1}4$$
0
votes
2answers
82 views

Prove $|x+1|\leq 4$ implies that $-4\leq x\leq 2$.

How do I prove that if $x$ is a real number, then $\lvert x+1 \rvert\leq 3$ implies that $-4\leq x\leq 2$. EDIT: $\lvert x+1 \rvert\leq 4$ should be $\lvert x+1 \rvert\leq 3$
0
votes
1answer
96 views

Determine all the values of the parameter $a$ for which the inequality $3-|x-a|>x^2$ is satisfied by at least one negative $x$.

I wanted to know, how can I determine all the values of the parameter $a$ for which the inequality $3 - |x-a| > x^2$ is satisfied by at least one negative $x$. I tried for $x<a, |x-a|=-(x-a)$ ...
1
vote
2answers
56 views

Finding $x$ from inequality: $\left | \frac{3^n + 2}{3^n + 1} - 1 \right | \le \frac{1}{28}$

Find $x$ in $\mathbb{Z}$ satisfying this inequality: $$\left | \frac{3^n + 2}{3^n + 1} - 1 \right | \le \frac{1}{28}.$$ I tried something, but I don't think it's correct. $$-\frac{1}{28} ...
1
vote
2answers
762 views

Proof the maximum function $\max(x,y) = \frac {x +y +|x-y|} {2}$ [duplicate]

I want to prove the maximum function max: $\mathbb{R} \rightarrow \mathbb{R}$, which is defined by $$\max(x,y) = \begin{cases}x, \text { if } x \geq y , \\ y, \text { if } x < y \end{cases}$$ ...
0
votes
1answer
101 views

Find the value of this logarithmic expression involving fifth root of unity.

Let $\alpha$ be the fifth root of unity. We then want to evaluate the expression $$\log |1 + \alpha + \alpha^2 + \alpha^3 - 1/\alpha |$$ Thanks in anticipation for your help in solving this!
10
votes
7answers
921 views

what does $|x-2| < 1$ mean?

I am studying some inequality properties of absolute values and I bumped into some expressions like $|x-2| < 1$ that I just can't get the meaning of them. Lets say I have this expression $$ ...
4
votes
5answers
336 views

Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$

Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$ example1: $|x-1|+|x-2|\geq 1$ my solution:(substitution) $x-1=t,x-2=t-1,|t|+|t-1|\geq 1,|t-1|\geq 1-|t|,$ square, $t^2-2t+1\geq ...
12
votes
6answers
883 views

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
4
votes
1answer
233 views

intersection of two graph

i would like to clarify some questions from GRE,which at first seems a little difficult to understand,suppose that we have some function $f(x)=|2*x|+4$ and graph of this function is given ...
3
votes
1answer
122 views

Question based on Triangle Inequality $\displaystyle |x+y|\leq |x|+|y|$

If $x,y,z\in \mathbb{R}-\left\{0\right\}$. Then prove that $\displaystyle 1\leq \frac{|x+y|}{|x|+|y|}+\frac{| y+z|}{| y |+| z |}+\frac{| z+x|}{| z |+| x |}\leq 3$ My Try:: Using Triangle Inequality ...
2
votes
1answer
63 views

Simplifying $\left|\left|\sqrt{-x^2}-1\right|-2\right|$

How do we simplify the expression $\left|\left|\sqrt{-x^2}-1\right|-2\right|$? This is very confusing. Do they cancel out and become just simply $\sqrt{-x^2}-1-2$?
7
votes
2answers
344 views

How does one DERIVE the formula for the maximum of two numbers

I want to derive (not prove that this is true) the formula $\max (x,y) = \dfrac{x + y + |y-x|}{2}$ I was reading a proof (which they have the result ahead of time already) that we do cases and then ...
2
votes
3answers
95 views

Question about absolute value in inequalities

My book presents the following: $$7 \le x \le 9 $$ so $$ -1 \le x - 8 \le 1 $$ and $$ |x-8| \le 1$$ I usually get confused with the way that taking the absolute value of an expression works. Could ...
1
vote
2answers
98 views

How to evaluate the inequality $|x+1|<-1$?

Okay perhaps the title isn't specific enough, I didn't know how to word it exactly. I'm finding the interval of convergence for a power series and i know the answer to be (-2,0] I end up with the ...
0
votes
1answer
56 views

Is this (or when) does this equality hold?

Let $a,b,c,d \in \mathbb{R}$ and $x,y$ are variables which are also real numbers $$|ax + by|^2 + |cx + dy|^2 + 2|ax + by||cx + dy| = (ax + by)^2 + (cx + dy)^2 + 2(ax + by)(cx + dy)$$ Is this always ...
0
votes
1answer
97 views

Finding The Contour Maps Of A Function Of Two Variables

I am given the function $f(x,y) = \ln|y-x^2|$, and am suppose to find the contour maps. Let $z = c = f(x,y)$. $c = \ln|y-x^2| \rightarrow e^c = e^{\ln|y-x^2|} \rightarrow e^c = |y-x^2|$ I know I ...
1
vote
3answers
73 views

Solving $2|x+1|>|x+4|$

I'm trying to solve the following equations and inequalities for $x\in\mathbb R$: $$2|x+1|>|x+4|$$ I know I'm supposed to consider the intervals $(-\infty,-4), [-4,-1]$ and $(-1,\infty)$ but ...
-1
votes
3answers
115 views

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

Determine all solutions to the following. $$ \lvert x+12\rvert +\lvert x-5\rvert =15.$$
3
votes
3answers
280 views

Exposition On An Integral Of An Absolute Value Function

At the moment, I am trying to work on a simple integral, involving an absolute value function. However, I am not just trying to merely solve it; I am undertaking to write, in detail, of everything I ...
1
vote
2answers
199 views

How to prove this simple statement: $\max\{a,b\}=\frac{1}{2}(a+b+|a-b|)$ [duplicate]

I am trying to prove this statement. for any $a,b \in \mathbb{R}$, $$\max\{a,b\}=\frac{1}{2}\big(a+b+|a-b|\big)$$ and $$\min\{a,b\}=\frac{1}{2}\big(a+b-|a-b|\big)$$ I am eating myself not knowing ...
1
vote
2answers
157 views

When does a absolute value equation have one unique solution?

Find $m \in \mathbb R$ for which the equation $|x-1|+|x+1|=mx+1$ has only one unique solution. When does a absolute value equation have only 1 solution? I solved for $x$ in all 4 cases and got ...
1
vote
2answers
73 views

Confusion solving $\sqrt{4m^2-4m+1}+|1-2m|\leq2$, weird solution.

I am trying to solve $\sqrt{4m^2-4m+1}+|1-2m|\leq2$. Since i know $|1-2m| = \pm(1-2m)$ i tough solving $\sqrt{4m^2-4m+1}+1-2m\leq2$ and $\sqrt{4m^2-4m+1}-1+2m\leq2$. As solutions i get $0\leq2$ and ...
1
vote
2answers
92 views

How to prove this inequality $x,y\in\Bbb R$, $|x|<1,|y|<1$ show that $\bigg|\frac{x-y}{1-xy}\bigg| < 1$ (and similar ones)

I have to show that the inequality below is true, i tried some thing but got stuck, i tried to eliminate the absolute value $-1<\frac{x-y}{1-xy}<1$ and then solve for $x$ and $y$ with no ...
7
votes
2answers
164 views

Maximum of the difference

What is the maximum value of $f(… f(f(f(x_{1} – x_{2}) – x_{3})-x_{4}) … – x_{2012})$ where $x_{1}, x_{2}, … , x_{2012}$ are distinct integers in the set ${1, 2, 3, …, 2012}$ and $f$ is the absolute ...
3
votes
2answers
102 views

Two solutions for $x$

I am trying to solve this. I already got the answer but my doubt is if I am doing it right. There are two solutions for $x$ in the equation, to 2 decimal places. What is the value of the greater ...