(1)   1/4x+4 > 2/4x-2

(2) x²+5x+3/x+2 < x

 please give me a answer with detalied solution. rates for sure

1)  

Taking reciprocals on both sides (and thus reversing the inequality sign),

8x + 84x - 2

4x-10

x

x²+5x+3/x+2 < x  ......................................[Main inequality]

Now,we must multiply the inequality by x+2.

However 2 cases arise depending on whether x+2 is positive or not.

If it isnt then the sign of the inequality will change.

CASE 1: x+2 is positive

x+2 > 0

x > -2........................................................[condition 1]

Multiplying the Main inequality by x+2 (without changing the inequality sign)

x² + 5x + 3 < x² + 2x

3x + 3 < 0

x < -1.....................................................[condition 2]

Combining conditions 1 and 2 gives   -2 < x < -1 as the solution to case 1)

CASE 2:  x+2 is negative

x + 2 < 0

x < -2...................................................[condition 3]

Multiplying the Main inequality by x + 2 (and changing the sign of the inequality),

x²+5x+3 > x² + 2x

3x + 3 > 0

x > -1.....................................................[condition 4]

Since conditions 3 and 4 are contradictory,we must reject case 2.

Hence the final answer is,

-2 < x < -1