Solve the inequality x/2 ≥ 1+4/x and express the solution in terms of intervals.

We have to solve the inequation x/2 >= 1 +
4/x

x/2 >= 1 +
4/x

=> x >= 2 +
8/x

If we assume that x >=0, we can multiply both
the sides of the inequation with x without changing the
sign.

=> x^2 >= 2x +
8

=> x^2 - 2x - 8 >=
0

=> x^2 - 4x + 2x - 8 >=
0

=> x( x - 4) + 2(x - 4)
>=0

=> (x - 4)(x + 2) >=
0

For (x - 4)(x + 2) >= 0 both (x - 4) >=0
and (x + 2)>= 0

=> x >= 4 and x
>= -2

This is satisfied by x >= 4
...(1)

or both (x - 4) <=0 and (x + 2) <=
0

=> x <= 4 and x <= -2, which gives
no results as x >= 0

If we assume that x < 0,
we can multiply both the sides with x but the inequation changes
to

x^2 < 2x +
8

=> x^2 - 2x - 8 <
0

=> x^2 - 4x + 2x - 8 <
0

=> x( x - 4) + 2(x - 4) <
0

=> (x - 4)(x + 2) <
0

This is true if either ( x - 4) < 0 and (x + 2)
> 0

=> x < 4 and x >
-2

As we have assumed x<
0,

0 > x >= -2
...(2)

Or if x - 4 >= 0 and x + 2 <= 0 and
x< 0 which we have assumed

=> x >= 4
and x <= 0, which gives no results.

So from (1) and
(2) we arrive at x >= 4 and 0 > x >=
-2

The required x lies in [-2, 0) and [4,
inf)