measgachyx5q9

2022-05-15

Let A and B represent two linear inequalities:

$A:{a}_{1}{x}_{1}+...+{a}_{n}{x}_{n}\ge {k}_{1}$

$B:{b}_{1}{x}_{1}+...+{b}_{n}{x}_{n}\ge {k}_{2}$

If A and B is unsatisfiable (does not have solution), does the following hold in general (the conjunction of two inequalities implies the summation of them )? If so, I am looking for a formal proof?

$A\wedge B\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}A+B$

${\mathit{a}}_{1}{\mathit{x}}_{1}+...+{\mathit{a}}_{n}{\mathit{x}}_{n}\ge \mathit{k}1\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\wedge \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{\mathit{b}}_{1}{\mathit{x}}_{1}+...{\mathit{b}}_{n}{\mathit{x}}_{n}\ge {\mathit{k}}_{2}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{\mathit{a}}_{1}{\mathit{x}}_{1}+...+{\mathit{a}}_{n}{\mathit{x}}_{n}+{\mathit{b}}_{1}{\mathit{x}}_{1}+...{\mathit{b}}_{n}{\mathit{x}}_{n}\ge {\mathit{k}}_{1}+{\mathit{k}}_{2}$

and then I would like to generalize the above theorem to summation of several inequalities.

$A:{a}_{1}{x}_{1}+...+{a}_{n}{x}_{n}\ge {k}_{1}$

$B:{b}_{1}{x}_{1}+...+{b}_{n}{x}_{n}\ge {k}_{2}$

If A and B is unsatisfiable (does not have solution), does the following hold in general (the conjunction of two inequalities implies the summation of them )? If so, I am looking for a formal proof?

$A\wedge B\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}A+B$

${\mathit{a}}_{1}{\mathit{x}}_{1}+...+{\mathit{a}}_{n}{\mathit{x}}_{n}\ge \mathit{k}1\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\wedge \phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{\mathit{b}}_{1}{\mathit{x}}_{1}+...{\mathit{b}}_{n}{\mathit{x}}_{n}\ge {\mathit{k}}_{2}\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}{\mathit{a}}_{1}{\mathit{x}}_{1}+...+{\mathit{a}}_{n}{\mathit{x}}_{n}+{\mathit{b}}_{1}{\mathit{x}}_{1}+...{\mathit{b}}_{n}{\mathit{x}}_{n}\ge {\mathit{k}}_{1}+{\mathit{k}}_{2}$

and then I would like to generalize the above theorem to summation of several inequalities.

Carolyn Farmer

Beginner2022-05-16Added 16 answers

It's still false, even with the unsatisfiability assumption.

Consider the inequalities

$\begin{array}{rl}-2x& >2\\ x& >3\end{array}$

Their sum is

$-x>5$

i.e., $x<-5$. But $x<-5$ does not imply that $x>3$.

Consider the inequalities

$\begin{array}{rl}-2x& >2\\ x& >3\end{array}$

Their sum is

$-x>5$

i.e., $x<-5$. But $x<-5$ does not imply that $x>3$.

quorums15lep

Beginner2022-05-17Added 4 answers

I don't see how the linear combination part is relevant. $A\ge {k}_{1},B\ge {k}_{2}\to A+B\ge {k}_{1}+{k}_{2}$ regardless of where $A$ and $B$come from. This can be seem by

$A\ge {k}_{2}$

$A-{k}_{1}\ge 0$

$B+(A-{k}_{1})\ge B\ge {k}_{2}$

$B+A\ge {k}_{1}+{k}_{2}$

$A\ge {k}_{2}$

$A-{k}_{1}\ge 0$

$B+(A-{k}_{1})\ge B\ge {k}_{2}$

$B+A\ge {k}_{1}+{k}_{2}$

Find the volume V of the described solid S

A cap of a sphere with radius r and height h.

V=??

Whether each of these functions is a bijection from R to R.

a) $f(x)=-3x+4$

b) $f\left(x\right)=-3{x}^{2}+7$

c) $f(x)=\frac{x+1}{x+2}$

?

$d)f\left(x\right)={x}^{5}+1$In how many different orders can five runners finish a race if no ties are allowed???

State which of the following are linear functions?

a.$f(x)=3$

b.$g(x)=5-2x$

c.$h\left(x\right)=\frac{2}{x}+3$

d.$t(x)=5(x-2)$ Three ounces of cinnamon costs $2.40. If there are 16 ounces in 1 pound, how much does cinnamon cost per pound?

A square is also a

A)Rhombus;

B)Parallelogram;

C)Kite;

D)none of theseWhat is the order of the numbers from least to greatest.

$A=1.5\times {10}^{3}$,

$B=1.4\times {10}^{-1}$,

$C=2\times {10}^{3}$,

$D=1.4\times {10}^{-2}$Write the numerical value of $1.75\times {10}^{-3}$

Solve for y. 2y - 3 = 9

A)5;

B)4;

C)6;

D)3How to graph $y=\frac{1}{2}x-1$?

How to graph $y=2x+1$ using a table?

simplify $\sqrt{257}$

How to find the vertex of the parabola by completing the square ${x}^{2}-6x+8=y$?

There are 60 minutes in an hour. How many minutes are there in a day (24 hours)?

Write 18 thousand in scientific notation.