$3.$ Consider an LP with variables x$1,$x$2,$x$3,$x$4.$ Suppose that the LP includes constraints x$1,$x$2,$x$3,$x$40.$

$($a$)$ Consider the constraint x$4|$x$32$x$1|.$ Suppose that we want to add to the LP the condition that

this constraint is satisfied. Show how to satisfy this requirement so that the resulting formulation

is an LP$.$

Hint: rewrite the constraint as a pair of linear inequalities.

$($b$)$ Consider the following inequalities:

$6$x$1+2$x$2+3$x$3+5$x$44,$

$2$x$1+4$x$2+2$x$3+6$x$49.$

Suppose that we want to add to an IP the condition that at least one of these two constraints is

satisfied. Show how to satisfy this requirement so that the resulting formulation is an IP$.$

Hint: add a binary variable indicating which of the two constraints must be satisfied.

$($c$)$ Let non$-$negative vectors a$1,...,$a$$ with four entries and numbers $1,...,$$$ be given, and $1$r$.$

Consider the following set of inequalities:

$($ai$)$T x$$i$,$ i$=1,...,.$

We want to add to an IP the condition that at least r of the constraints are satisfied. Show how

to satisfy this requirement so that the resulting formulation is an IP$.$

$($d$)$ Consider the following set of values, S$=\{5,7,11,19,41,74,1919\}.$ Suppose that we want to add

to an IP the condition that the variable x takes only one of the values in S$.$ Show how to satisfy

this requirement so that the resulting formulation is an IP$.$