The given inequalities are as follows. y<3x y>x−2

York 2021-03-09 Answered
The given inequalities are as follows.
y<3x
y>x−2
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Expert Answer

broliY
Answered 2021-03-10 Author has 97 answers
Graph of the given inequalities on a coordinate plane as shown below
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Proof of addition and subtraction rules for systems of inequalities.
a < b and c < d
Then a + c < b + d is a valid move, but subtraction isn't. And similarly for a system such as e > f and g < h then e g > f h is valid and so is g e < h f, but addition isn't valid.
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Solve the system of linear inequalities with parameters
(*) { 0 2 x + 2 y 3 b + 3 a 2 0 2 x 3 y + 6 b + 3 a 1 0 x 1 0 y 2 0 a 1 0 b 1
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 2020-11-14

If u is an algebraic expression and c is a positive number,
1. The solutions of |u|<c are the numbers that satisfyc<u<c.
2. The solutions of |u|>c are the numbers that satisfyu<c or u<c.
These rules are valid if < is replaced be and > is replaced by

asked 2022-06-16
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 erasers and the price of three erasers is greater than the price of a notebook. If all of the prices are in integers, find each of their prices.
Let the prices of a pen, an eraser and a notebook be p, e and n respectively. Then we have the following:
p + e + n = 100 n > 2 p 3 p > 4 e 3 e > n
We have p > 4 3 e and n > 2 p > 8 3 e.
So,
4 3 e + e + 8 3 e < 100 e < 20
Similarly, we get
p 27         and n 56
Setting the values of e and p and with a bit brute forcing, I got e = 19, p = 26 and n = 55 which I think is the only solution.
Is the solution correct?
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Graph the solution set for the of linear inequalities

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Given a system A x b of linear inequalities, describe a linear programming problem whose optimal solution immediately tells us which inequalities among A x b are always satisfied with equality.
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Trying to solve a system of inequalities
I have this kind of system of inequalities (with one equality):
1. x + y + z = 1
2. a x + b y + c z 2
3. a 2 x + b 2 y + c 2 z 6
4. a 3 x + b 3 y + c 3 z 14
and so on.. (I could continue with 5. 6. 7.... the power always increases by one and I know the value on the right).
x,y,z are my unknowns and a,b,c are known.
Could I get a solution (or an approximation) for x,y and z as the number of inequalities grows, I don't see it?