Draw the graph of inequalities {(|x-2|<=5),(|y-4|>2):}

Wribreeminsl 2020-10-21 Answered
Draw the graph of inequalities
{|x2|5|y4|>2
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Answered 2020-10-22 Author has 109 answers

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asked 2022-06-05
Suppose that we are given a subroutine which, given a system of linear inequalities A x b, x R n , either produces a solution or decides that no solution exists.
Construct an algorithm that uses a single call of this subroutine and which returns an optimal solution (if it exists) to the following linear programming problem:
minimize  c T x subject to:  A x = b x 0
Well, I think I should take a look at the dual problem.
The dual is:
maximize  p T b subject to:  A T p   c
Then I would call the subroutine for the matrix A T and to the vector c to find a dual feasible point y.
I tried to use complementary slackness to construct a primal solution and to decide if y is optimal but it didn't work .
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Solve the system of linear inequalities with parameters
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Here x,y are unknown variables and a,b are parameters.
My attempt. By adding the inequalities with some coeficients I separated the variables and get the simple system
(**) { 0 y + 6 a 5 , 0 x + 9 a + 3 b 8.
and I am able to solve it. But the solutions of the last system are not solution of the initial system!
Maple and wolframAlpha cant solve the system.
P.S.1 For a = 63 100 and b = 59 100 Maple gives the solutions
{ x = 1 , 9 50 y , y 11 25 } , { x = 3 / 2 y + 127 100 , 9 50 < y , y < 11 25 } , { 9 50 < y , x < 1 , y < 11 25 , 3 / 2 y + 127 100 < x } , { y = 11 25 , 61 100 x , x < 1 } , { x = 3 / 2 y + 127 100 , 11 25 < y , y < 127 150 } , { 11 25 < y , x < 2 y + 47 25 , y < 127 150 , 3 / 2 y + 127 100 < x } , { x = 2 y + 47 25 , 11 25 < y , y < 127 150 } , { x = 0 , 127 150 y , y 47 50 } , { y = 127 150 , x 14 75 , 0 < x } , { 0 < x , 127 150 < y , x < 2 y + 47 25 , y < 47 50 } , { x = 2 y + 47 25 , 127 150 < y , y < 47 50 }
asked 2022-06-05
The system of inequalities is as follows:
{ a a b b M a a b b m
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In the euclidean space, the points ( x , y , z ) belonging to a regular octahedron are those that satisfy the inequalities
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where a 0. These eight inequalities can be divided into two groups of four according to the number (even or odd) of negative signs they contain. For example, the inequalities
x y + z a x + y + z a x + y z a x y z a
all have one or three negative signs and the points satisfying these form a tetrahedron. The other four inequalities correspond to the dual tetrahedron of the first, which shows that the intersection of two regular dual tetrahedra form a regular octahedron. Moreover, the vertices of the two tetrahedra can be seen as the eight vertices of a cube.
I am wondering if there exists a similar relationship between regular polytopes in four dimensions. As it is another case of a regular cross-polytope, the hexadecachoron (or 16-cell) is defined by the sixteen inequalities
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If one were to take the eight inequalities containing an odd number of negative signs, say
x y z w a x + y z w a x y + z w a x y z + w a x + y + z w a x + y z + w a x y + z + w a x + y + z + w a
which 4-polytope would be obtained ? I doubt it would be a regular 5-cell, since (obviously) the number of cells and the number of hyperplanes don't add up. Besides, the intersection of the two 4-polytopes corresponding to the two sets of eight inequalities should technically correspond to the 16-cell.
The tesseract, having eight cells, could be a candidate, but I have been unable to show that these eight inequalities define one (or any other 4-polytope). Any ideas?
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How to solve this system with equation and inequality?
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How to determine bounds on one variable in a system of inequalities?
I am interested in the point of 'cross-over' between a generalised harmonic number where the denominator of the summand is raised to a power, and a non-exponential harmonic sum operating on some subset of the natural numbers.
For example, take the generalised harmonic number H x ( k ) = n = 1 x 1 n k , and a harmonic number operating only on odd denominators G x = n = 1 x 1 2 n 1 .
Clearly, there exist values of x,k such that G x < H x ( k ) and values such that H x ( k ) < G x . Thus there exists a value c = G x 0 such that
G x 0 = c < H x 0 ( k ) = n = 1 x 1 n k
and
H x 0 + 2 ( k ) < G x 0 + 2 = c + 1 2 x 0 + 1 + 1 2 x 0 + 3
or
H x 0 + 2 ( k ) c < 1 2 x 0 + 1 + 1 2 x 0 + 3
The values of c , x 0 , k are obviously co-dependent. I am searching for a way to solve for x 0 or at least put bounds on it.
I am interested in how to approach this algebraically rather than numerically. This is a single simple example of G and I want to be able to explore how to solve such problems generally, for whatever pattern of G I choose (provided it's formulable!).
Algebraically, how do I put bounds on x 0 in terms of c , k?