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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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)

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