Let D = x &#x2208; R 3 </msup> : | 2 x 1

Taniyah Estrada

Taniyah Estrada

Answered question

2022-06-05

Let D = x R 3 : | 2 x 1 x 2 + 3 x 3 + 1 | + | x 2 + 2 x 3 2 | + | 5 x 2 3 x 3 | 10 . Express D as the feasible solution set of a linear system of inequalities (meaning, a system of the form A x b).
How is the feasible solution set represented? Is this problem just a matter of removing the absolute signs and setting up two linear equations as such: 2 x 1 + x 2 3 x 3 1 10 and ( 2 x 1 + x 2 3 x 3 1 ) 10?

Answer & Explanation

Sydnee Villegas

Sydnee Villegas

Beginner2022-06-06Added 22 answers

Your original constraint is
| 2 x 1 x 2 + 3 x 3 + 1 | + | x 2 + 2 x 3 2 | + | 5 x 2 3 x 3 | 10
Now consider the case where 5 x 2 3 x 3 0. If we knew that was always true we could write simply
5 x 2 3 x 3 0
If we knew 5 x 2 3 x 3 0 then we could write simply
| 2 x 1 x 2 + 3 x 3 + 1 | + | x 2 + 2 x 3 2 | ( 5 x 2 3 x 3 ) 10
If we have both of these constraints, we capture both possibilities.
We can now extend that reasoning for all the 2 3 possible combinations of terms being negative and non-negative, e.g.
+ ( 2 x 1 x 2 + 3 x 3 + 1 ) + ( x 2 + 2 x 3 2 ) + ( 5 x 2 3 x 3 ) 10
( 2 x 1 x 2 + 3 x 3 + 1 ) + ( x 2 + 2 x 3 2 ) + ( 5 x 2 3 x 3 ) 10
+ ( 2 x 1 x 2 + 3 x 3 + 1 ) ( x 2 + 2 x 3 2 ) + ( 5 x 2 3 x 3 ) 10
( 2 x 1 x 2 + 3 x 3 + 1 ) ( x 2 + 2 x 3 2 ) + ( 5 x 2 3 x 3 ) 10
+ ( 2 x 1 x 2 + 3 x 3 + 1 ) + ( x 2 + 2 x 3 2 ) ( 5 x 2 3 x 3 ) 10
+ ( 2 x 1 x 2 + 3 x 3 + 1 ) ( x 2 + 2 x 3 2 ) ( 5 x 2 3 x 3 ) 10
( 2 x 1 x 2 + 3 x 3 + 1 ) ( x 2 + 2 x 3 2 ) ( 5 x 2 3 x 3 ) 10
( 2 x 1 x 2 + 3 x 3 + 1 ) + ( x 2 + 2 x 3 2 ) ( 5 x 2 3 x 3 ) 10
If thats a bit confusing, consider
| y 1 | + | y 2 | 1
y 1 + y 2 1
y 1 + y 2 1
y 1 y 2 1
y 1 y 2 1
Hopefully you see the connection.

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