The absolute-value tag has no wiki summary.
0
votes
6answers
272 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,
7
votes
2answers
157 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 ...
5
votes
2answers
612 views
How to use triangle inequality to establish the following one
I need to use $|a+b| \leq |a|+|b|$ to show that $||a|-|b|| \leq |a-b|$. I have tried to represent $||a|-|b||$ as $||a|+(-|b|)|$, and then get $||a|+(-|b|)| \leq |a|+|-|b||$, but that isn't leading ...
4
votes
1answer
499 views
Prove variant of triangle inequality containing p-th power for 0 < p < 1
Sorry if this is a trivial question, but I am kind of stuck with proving the following inequality and have been searching for a while:
$\rho \left( \sum\limits_i^n d_i \right) \leq \sum\limits_i^n ...
3
votes
2answers
238 views
Why exactly can you take the absolute value of one side of this inequality and assume it is still true?
Exercise:
Show that if $(b_n) \to b$, then the sequence of absolute values $\left| b_n \right|$ converges to $\left| b \right|$.
Solution (partial):
By the triangle inequality, ...
0
votes
1answer
129 views
Absolute value of a real number
My question is:
Solve: $|x-4|< a$, where $a$ belongs to the real numbers. Solve this by considering various cases depending upon whether $a$ is negative, positive or zero.
What I have tried ...
5
votes
2answers
68 views
Is there a lower-bound version of the triangle inequality for more than two terms?
The triangle inequality $|x+y|\leq|x|+|y|$ can be generalized by induction to $$|x_1+\ldots+ x_n|\leq|x_1|+\ldots+|x_n|.$$
Can we generalize the version $|x+y|\geq||x|-|y||$ to $n$ terms too? I need ...
3
votes
1answer
253 views
Why do definitions of distinct conic sections produce a single equation?
I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
4
votes
3answers
395 views
Proving two integral inequalities
Can anyone help me to prove that these integral inequalities hold?
Here $x$ is a real value:
$$
\left| \int_a^b\ f(x) dx \right| \leq \int_a^b\ |f(x)| dx
$$
Here $z$ is a complex value:
$$
\left| ...
4
votes
3answers
153 views
Is there an alternate definition for $\{ z \in \mathbb{C} \colon \vert z \vert \leq 1 \} $.
Is there a method of constructing a subset of a reasonably arbitrary ring so that when the construction is applied the $\mathbb{C}$ the result is $B = \{ z \in \mathbb{C} \colon |z| \leq 1 \} $?
My ...
2
votes
3answers
138 views
Solve an absolute value equation simultaneously
My question is :
Solve simultaneously
$$\left\{\begin{align*}&|x-1|-|y-2|=1\\&y = 3-|x-1|\end{align*}\right.$$
What I did :
$y=3 - |x-1|$ is given.
Thus $y = 3-(x-1)$ or $y = ...
1
vote
3answers
117 views
Proving square root of a square is the same as absolute value
Lets say I have a function defined as $f(x) = \sqrt {x^2}$. Common knowledge of square roots tells you to simplify to $f(x) = x$ (we'll call that $g(x)$) which may be the same problem, but it isn't ...
1
vote
1answer
2k views
Solving inequality with two absolute values
Hey, !
In my pre-calculus class the teacher showed the solution of the following example:
\begin{align}
\vert x-3 \vert \lt \vert x - 4 \vert + x
\end{align}
He started by stated the domains ...
0
votes
5answers
76 views
Prove That $|a +b| = |a| +|b|$ if $a$ and $b$ Have Same Signs, And $|a +b| < |a| + |b|$ if $a$ and $b$ Have Opposite Signs
My Proof:
$|a +b| = |a| +|b|$ ..... $(i)$
$|a +b| < |a| + |b|$ ..... $(i)$
If $'a'$ and $'b'$ have same signs:
Let $a$ and $b$ be equal to $-x$. Replacing $a$ and $b$ with $-x$ in the equation ...
0
votes
2answers
88 views
solving absolute value equation 2
My question is : Solve simultaneously-
$$\left\{\begin{align*}&|x-1|+|y-2|=1\\&y = 3-|x-1|\end{align*}\right.$$
I tried to solve this question by the method told by Marvis as I had ...
0
votes
2answers
113 views
Solving $|x-2| + |x-5|=3$ [duplicate]
Possible Duplicate:
How could we solve $x$, in $|x+1|-|1-x|=2$?
How should I solve:
$|x-2| + |x-5|=3$
Please suggest a way that I could use in other problems of this genre too
Any help ...
0
votes
3answers
101 views
Absolute value on a number line
Solve : |x-4|>a if case1:a>0 and case2:a<0
I am getting answers which look similar in both cases.
please i wish to know why it is so and how different both answers are when plotted on a number ...
0
votes
5answers
161 views
Solving an equation with absolute values
The equation I am trying to solve is this : $\newcommand\abs[1]{|#1|}\abs{3y+7}=\abs{2y-1}$.
My conventional approach is to split this into three intervals with $1/2$ and $-7/3$ being the two "split" ...
