# Determine the elements of the set S The elements have the property |x-1|=<4 . The elements are in Z set.

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l x-1l =< 4

First let us solve the inequality:

l x-1 l =< 4

Then we have two cases:

1: (x-1) = < 4

==> x = < 5 ==> x= {-inf,....2,3,4,5}........(1)

2: -(x-1) =< 4

==> -x + 1 =< 4

==> -x =< 3

==> x >= -3 ==> x = { -3, -2, -1, .....inf}........(2)

From (1) and (2) , we conclude that:

S = { -inf, ...., 3, 4, 5} Intersect { -3, -2, -1, ....inf}

** S = { -3, -2, -1, 0, 1, 2, 3, 4, 5}**

S = {x: |x-1| + < 4 and the elements are in z(or integers.)

To find the elements of S.

|x-1| =< 4

So,

-4 <= x-1 =< 4. Add 1.

-4+1 < = x =< 4+1

-3 <= x =< 5. Since x is also integer, therefore S = {-3,-2,-1,-0, 1,2,3,4,5}

From enunciation, we find out that the elements of the set S, have to respect the constraint that they are integer numbers which have the property |x-1|=<4.

We'll re-write the constraint |x-1|=<4:

-4=<x-1=<4

We'll solve the left side of the inequality:

-4=<x-1

We'll add 4 both sides:

0=<x-1+4

0=<x+3

We'll subtract 3 both sides:

-3=<x

Now, we'll solve the right side:

x-1=<4

We'll add 1 both sides, to isolate x:

x=<5

So, -3=<x=<5

The integer elements of the set S are:

**S = {-3 ; -2 ; -1 ; 0 ; 1 ; 2 ; 3 ; 4 ; 5}**