All Questions
Tagged with systems-of-equations inequality
194 questions
0
votes
1
answer
79
views
How to find a condition on $a,b,c$ such that solutions exist to a system of inequalities
How can I find a condition on $a,b,c \in \mathbb{R_{> 0}}$ such that there exist solutions $x,y,z \in \mathbb{R}$
$$
\begin{align}
ax-y-z&>0 \\
-x+by-z&>0 \\
-x-y+cz&>0
\end{...
1
vote
1
answer
85
views
How do you solve this incline plane forces question?
Question
I would greatly appreciate any assistance solving this question!
Working out so far
- 41
0
votes
1
answer
30
views
Prove that if there is $a\in \mathbb R$, $2x + y = a - 2$ and $4xy = a² + 3a + 1$, then $x\in[-3,0]$ and $y\in[-6,0]$
You have to arrive at $(2x + 3)² \le 9$ and $(y + 3)² \le 9$ after eliminating $a$ but I'm struggling to find out how
Thanks in advance
- 3
0
votes
0
answers
25
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Proving a the distance between Cauchy sequences converges [duplicate]
Assume we have two Cauchy sequences { $x_n$ } and {$y_n$} in the metric space $(X,d)$. Is it true that the sequence {$a_n$}$=d(x_n,y_n)$ is convergent in $\mathbb{R}$? Here is my try: $$$$
Since those ...
- 571
0
votes
0
answers
41
views
Insights for Outcome Function Involving Multiple Interdependent Variables
I am working on a model involving multiple interdependent variables and systems of equations, and I am trying to gain insights into the behavior and properties of a specific outcome function. Despite ...
- 51
1
vote
1
answer
87
views
Count number of possible combinations of $\sum_{i=1}^{n} a_i \leq 10$
If I have $a_1+a_2 \leq 10$, with $a_1, a_2 \in \{0, \, 1, \, 2, \, \cdots, \, 10 \}$:
To count the number of possible combinations for $a_1$ and $a_2$ such that
$$a_1+a_2 \leq 10\quad\mbox{and}\quad
...
- 21
-1
votes
1
answer
73
views
Prove that for $x,y,z$ positive integers the form of $\frac{x^2+y^2+z^2}{xy+yz+zx}$ can't be equal to 3. [duplicate]
Because it is positive integers so I can multiply both side by $(xy+yz+zx)$
I have tried to use completing square, like below
$(3x-3y)^2+(3y-3z)^2+(3z-3x)^2=12x^2+12y^2+12z^2$
But in above form, it ...
- 13
2
votes
2
answers
223
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Prove these equations have only zero solution.
Original problem: consider the function $f = f_{a,b,c}(u,v,w)$:
$$
f_{a,b,c}(u,v,w) = (v + T)^3 + v T (v+ T) - u^2 T - v w^2, \quad u,v,w \in\mathbb{R},
$$
where
$$
T = -a u -b v- c w,
$$
and $a,b,c\...
- 133
2
votes
4
answers
146
views
Solve the equation $\left(\frac{1+\sqrt{1-x^2}}{2}\right)^{\sqrt{1-x}} = (\sqrt{1-x})^{\sqrt{1-x}+\sqrt{1+x}}$
Solve in $\mathbb{R}$:
$
\left(\frac{1+\sqrt{1-x^2}}{2}\right)^{\sqrt{1-x}} = (\sqrt{1-x})^{\sqrt{1-x}+\sqrt{1+x}}
$
My approach:
Let $a = \sqrt{1-x}$ and $b = \sqrt{1+x}$ so $a^2 + b^2 = 2$. The ...
- 679
3
votes
0
answers
65
views
What are the maximum number of solutions to a system of non-linear inequalities?
What are the maximum number of solutions to a system of non-linear inequalities? In particular:
Let $A, B, C$ be real nonzero numbers. Consider the set $S = \{\frac 1 A, \frac 1 B, \frac 1 C, \frac {...
- 5,275
3
votes
3
answers
156
views
Solve $a=\frac{b+c}{1+b^2c^2},b=\frac{a+c}{1+a^2c^2}, c=\frac{a+b}{1+a^2b^2}$ [closed]
Let $a,b,c> 0$ such that $$a=\frac{b+c}{1+b^2c^2},b=\frac{a+c}{1+a^2c^2}, c=\frac{a+b}{1+a^2b^2}$$
Prove that the only solution to this system of equations is $$a=b=c=1$$
I am getting this answer ...
- 169
2
votes
1
answer
183
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How to solve a system of inequality and equality systematically?
How to solve this system of inequalities and equalities systematically:
$\begin{array}{l}
\left\{ {\begin{array}{*{20}{c}}
{\frac{x}{{2x + 3y}} + \frac{y}{{y + z}} + \frac{z}{{x + z}} = \frac{{34}}{{...
-1
votes
2
answers
52
views
Solving a system of inequalities and equalities [closed]
I am considering the following system:
$$0 < x < 1,$$
$$x = y + z - 1,$$
$$x \geq y \geq 0,$$
$$x \geq z \geq 0,$$
which I am pretty sure does not have any solutions. But I'm struggling to prove ...
- 35
2
votes
1
answer
78
views
About Solving the Following System of Inequalities
I would like to solve the following system of inequalities for a range of $\alpha$:
\begin{align*}
p_1 + p_4 + p_3 & = \alpha\tag1 \\
p_1 + p_2 + p_5 & = \alpha\tag2 \\
1 - p_1 - p_4 - p_5 &...
- 52
0
votes
2
answers
86
views
Maximization question using inequalities (Cauchy-Schwarz, AM-GM)
Consider this system of equations:
\begin{equation*}
a^2 + b^2 + c^2 + d^2 = 14 \\
3a + 2b + c + d = 14
\end{equation*}
I want to find the maximum value of $d$ given that $a, b, c, d \in \mathbb{R}$. ...
0
votes
1
answer
36
views
How to prove that C(x,y) coordinates which have parameters satisfies inequation with two absolute values
I have two linear functions:
f(x) = 2x - m + 6
g(x) = -x + 2m + 3
I have to prove that for all ...
- 219
0
votes
0
answers
64
views
Derive property of solutions linear equations from matrices
My question concerns the following systems of linear equations:
$x = 1 + Ax$ and $y = 1 + By$, where 1 denotes the $n$x1 vector of 1's and $A$ and $B$ are $n$x$n$ substochastic matrices that satisfy ...
- 69
0
votes
0
answers
32
views
Solving specific system of equations with inequalities
I am coding something in Ruby and I came across this math problem.
I have a system of equations and inequalities that looks like this:
$$
a = \sum \frac{a_n * x_n}{100}\\
b = \sum \frac{b_n * x_n}{100}...
- 101
0
votes
1
answer
67
views
Textbooks on basic algebra (with exercises)
I am looking for textbooks (ideally with a lot of exercises) on basic algebra.
In particular, I am interested in the following topics:
basic calculations with real and complex numbers
use and ...
- 123
3
votes
1
answer
80
views
Let $f(x)=x^n+p_1x^{n-1}+p_2x^{n-2}+ ...+p_n.$ The positive roots of the equation $f(x)=0$ do not exceed $\sqrt[r]{p}+1$ where $-p$ is the greatest
Let $$f(x)=x^n+p_1x^{n-1}+p_2x^{n-2}+ ...+p_n.$$The positive roots of the equation $f(x)=0$ do not exceed $\sqrt[r]{p}+1$ where $-p$ is the greatest negative coefficient and $p_r$ is the first ...
- 2,630
3
votes
1
answer
47
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How to find $L$ if $L=\frac{c}{(1-L)^a}$
How to find $L$ if $L=\frac{c}{(1-L)^a}$
I was trying to apply log but $\ln L +a\ln (1-L)=\ln c$. How can continued please?
Thank you
- 1,265
1
vote
1
answer
90
views
How $~2.576 \sqrt{{pq \over n}}\leq 0.02 ~$is derived from$~P(|\hat p-p|\leq 0.02)\geq 0.99 ,~P\left(|\hat p-p|<2.576\sqrt{{pq/n }}\right)=0.99$?
We want to estimate the proportion of customers each of whom uses the particluar brand of detergent.
$$\begin{align}
\underbrace{p:=\text{population proportion which satisfies}~~0.8<p<0.9 }_{\...
- 1,539
4
votes
1
answer
254
views
System of inequations
If $p$, $q$, $r$, $s$ and $t$ are real numbers such that $q+r<s+t$, $r+s<t+p$, $s+t<p+q$ and $p+q<r+s$, then find the largest and the smallest term among them.
This is how I solved it:
$$...
4
votes
1
answer
73
views
Proof of positivity of $~ x+\sqrt{x^2+1} ~$ of $~\operatorname{arsinh}(x)=\operatorname{arcsinh}(x)=\sinh^{-1}(x)= \ln \left( x+\sqrt{x^2+1}\right)$
Proof of positivity of $~ x+\sqrt{x^2+1} ~$
I found this formula appears at $~ \operatorname{arsinh}(x)= \ln \left( x+\sqrt{x^2+1}\right)~$
So, of course this argument inside the natural log function ...
- 1,539
1
vote
1
answer
120
views
Generators for the solution set of a system of inequalities
Given a system of linear equations of the form
\begin{align*}
a_{1,1}x_1+a_{1,2}x_2+&\dots+a_{1,n}x_n = b_1 \\
a_{2,1}x_1+a_{2,2}x_2+&\dots+a_{2,n}x_n = b_2 \\
\vdots \\
a_{n,1}x_1+a_{n,2}x_2+&...
- 11
0
votes
2
answers
49
views
Proof of $x+\sqrt{x^2+e} >0~~~~\text{where}~~x\in[-1,1]$
$$e:=\exp(1)\tag{1}$$
I want to prove the following positivity.
$$x+\sqrt{x^2+e} >0~~~~\text{where}~~x\in[-1,1]\tag{2}$$
I've tried to prove it using proof of contradiction.
$$\underbrace{x+\sqrt{x^...
- 1,539
0
votes
1
answer
100
views
Derivation of Inequality of arithmetic and geometric means using a circle
$$a,b:=\text{positive numbers}\tag{1}$$
I want to derive the following inequality.
$$\underbrace{\sqrt{ab}\leq{a+b\over 2}}_{\text{Inequality of arithmetic and geometric mean}}\tag{2}$$
To derive it, ...
- 1,539
1
vote
0
answers
24
views
Reasoning about equation combined with inequalities
The given problem eventually boils down to finding the interval for $x_5$ given these facts
$\begin{equation}\begin{cases}x_2+x_4+x_5=590\\180<x_2<200\\x_4>200\\x_5>x_4\end{cases}\end{...
- 519
0
votes
2
answers
128
views
Solve for $a,b,c,d$ over $a^4+b^4+c^4+d^4=48, abcd=12$
Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that
\begin{align*}
a^4 + b^4 + c^4 + d^4 &= 48, \\
abcd &= 12.
\end{align*}
I think I should apply some inequalities, ...
- 800
1
vote
0
answers
29
views
Checking for a solution in system of inequalities
I have a system modeled by the following:
$S \in [4000000, 4000001, ..., 48000000]$
$M \in [1, 2, ..., 8]$
$N \in [8, 9, ..., 86]$
$R \in [2, 4, 6, 8]$
$4000000 \le \frac S M \le 16000000$
$S * N = ...
- 135
0
votes
1
answer
90
views
solution to $\sum_{i=1}^{n}\frac{1}{a_{i}x+b_{i}} = 0$ [closed]
Is there any general procedure to solve the equation
$$
\sum_{i=1}^{n}\frac{1}{a_{i}x+b_{i}}=0
$$
with respect to $x$ for given $a_{i}$ and $b_{i}$, with $i=1,\dots,n$?
- 99
11
votes
2
answers
397
views
Find the smallest $n$ for which there are real $a_{1}, a_{2}, \ldots,a_{n}$
Find the smallest $n$ for which there are real $a_{1}, a_{2}, \ldots,a_{n}$ such that
$$\left\{\begin{array}{l} a_{1}+a_{2}+\ldots+a_{n}>0 \\a_{1}^{3}+a_{2}^{3}+\ldots+a_{n}^{3}<0 \\a_{1}^{5}+a_{...
- 2,344
1
vote
1
answer
51
views
Word problem with systems of inequations that involves a mixture
I've been struggling to convert this world problem into mathematical expressions:
In an oil mill, they decide to make a mixture from two types of oil: the extra virgin whose
price is \$4 per liter ...
- 25
5
votes
2
answers
194
views
Maximizing $a^2+b^2+c^2+d^2$ with given constraints
The following problem is from a local contest which ended today:
Let $a,b,c,d$ be positive real numbers such that $$(a+b)(c+d)=143\\ (a+c)(b+d)=150\\ (a+d)(b+c)=169$$ Find the maximum value of $a^2+b^...
- 4,026
4
votes
1
answer
174
views
Solving a system of quadratic inequalities
I have the following equation:$$\bigg\lfloor \sqrt {c^2(x^2+y^2)+2c(x+y)+2} +\frac{1}{2} \bigg\rfloor = c$$For a given positive integer $c$, I am trying to define algebraically, in terms of $x$ & $...
- 821
5
votes
1
answer
428
views
Finding the prices of a pen, an eraser and a notebook from the given system of inequalities
The sum of the prices of a pen, an eraser and a notebook is $100$ rupees. The price of a notebook is greater than the price of two pens. The price of three pens is greater than the price of four ...
- 4,026
0
votes
0
answers
242
views
Methods for solving multivariable system of linear inequalities
I'm looking for a way to solve multivariable system of linear inequalities.
The systems I'm trying to solve have more that 6 unknowns and I'm interested in all the intervals that are solutions.
I ...
- 1
2
votes
3
answers
84
views
If $\varepsilon > a + b$, then $\varepsilon^2 > (\sqrt{a}+\sqrt{b})^2$
For $a,b>0$ and $\varepsilon <1$, if $\varepsilon > a + b$, then does it follow: $\varepsilon^2 > (\sqrt{a}+\sqrt{b})^2$?
My attempt:
$(\sqrt{a}+\sqrt{b})^2=a+b+2\sqrt{ab}$ and further we ...
- 1,018
1
vote
0
answers
93
views
Show that a solution doesn't exist for this system of equations
I'm trying to show that there is not a solution for this system of equations: (the unknowns are $P_1$ and $P_2$)
$\left\{ \begin{array}{l}
{P_1} = \left( {\mu + 1} \right) + b{P_2} - \sqrt {\mu \left(...
- 31
1
vote
1
answer
43
views
If $uA=0, u\geq0, u1=1$ has not solution then $Ax<0$ yes has solution.
Prove that given a matrix $m \times n$, the system $A x < 0$ has
solution if and only if $u A = 0, u \geqslant 0, u 1 = 1$ has not solution.
My attempt:
I was able to prove the necessary condition ...
- 175
3
votes
1
answer
65
views
Is this necessary condition sufficient for "geometric realization"?
Let $0<\sigma_1 < \sigma_2$ be fixed positive reals satisfying $\sigma_1 \sigma_2=1$.
Let $0<a \le b$ satisfy $ab \ge 1$. I am looking for necessary and sufficient conditions on $a,b$ that ...
- 25.6k
2
votes
2
answers
179
views
Finding minima from simultaneous equations
We are given that a point $(x,y,z)$ in $\mathbb{R}^3$ satisfies the following equations
$x\cos\alpha-y\sin\alpha+z =1+\cos\beta$
$x\sin\alpha+y\cos\alpha+z =1-\sin\beta$
$x\cos(\alpha+\beta)-y\sin(\...
- 548
0
votes
0
answers
47
views
Solving a system of equations with inequalities ? all solutions needed?
I have a set of variables $\{x_{ij}\} \in \{1,2,3,4\}$ such that $1\leq i\leq 5$ and $1\leq j\leq 4$. And the following set of equations :
$$\forall i : \sum_{j}x_{ij} =4$$
$$\forall i \forall i' \...
- 1,797
2
votes
0
answers
62
views
Probability that a system of linear inequalities of random variables is satisfied
Let $X_i$ (i = 1...n) be a set of independent uniform random variables. Is there a tool/method/concept to find the probability that a system of linear inequalities of $X_i$ is satisfied ?
$$
\left\{
\...
- 21
2
votes
0
answers
72
views
For which solution $(x, y)$ is $x + y$ minimum in this system?
In the system:
\begin{align} 8y − 3x & \leq 16 \\ 3x + 8y &\geq −18\end{align}
for which solution $(x, y)$ is $x + y$ minimum?
I just tried to find the solution by finding the intersection ...
- 95
2
votes
3
answers
51
views
Linear system of inequalities.
Given linear system:
$$4+\delta_1-\delta_2\ge 0 \\ 2-\delta_1+\delta_2 \ge 0 \\ 1+\delta_1-\delta_2+\delta_3 \ge 0$$
How from there it follows that $\delta_2=\delta_3 = 0$ and $\delta_1\neq0$?
- 1,412
0
votes
2
answers
127
views
How to solve this $3\times3$ system of inequalities
Suppose that $x$, $y$, $z$, $\nu$, $N$, $g$ are some postitive parameters and $A$, $B$ and $C$ are variables that belong to $(0,+\infty)$. I want to identify one or all of the feasible points taht ...
- 253
3
votes
0
answers
76
views
Positive solution for a system of linear inequalities
Question
Let $A=[a_{ij}]$ be $n \times n$ real matrix with the positive diagonal. The question is to find (all) the weakest sufficient conditions on the entries of $A$ such that the system $Ax\gg 0$ ...
- 1,091
3
votes
0
answers
33
views
Set of solutions for given inequality
Given the matrix $A\in\mathbb{R}^{n\times n}$ with all eigenvalues inside the unit circle and the symmetric positive definite matrix $P\in\mathbb{R}^{n\times n}$ satisfying $ APA^\top-P+I=0 $, I need ...
- 484
0
votes
0
answers
16
views
Choosing the best electricity fare upon consumption habits
I have a real-life problem that I would like to express in maths, especially to use an algorithm to solve it over time. Both because I want to learn something new, both because, as a wise consumer, I ...
