# Help Solve This Inequality?

1/(5x^2 - 2x - 7) < 5/13

Can someone show me the steps to do this please? I only ever get half of the answer right and I have no idea why!!

1..   1/(5x^2 - 2x - 7) < 5/13

multiply each side by  5x^2 -2x -7

1< (5/13 ) * (5x^2 -2x -7)

multiply each side by 13

13 < 5*(5x^2 -2x -7)

13< 25x^2 -10x - 35

subtract 13 from eachside

13 -13<25x^2 -10x - 35- 13

25x^2 - 10x - 48 >0

comparing  the above inequality with ax^2+bx +c>0

a =25 , b =- 10 ,c =-48

the qradratic equation formula (- b± sqrt( b^2 -4ac)/2a

= 10 ± sqrt(10^2 -(4*25*-48)/50

=10 ± sqrt(100 +4800)50

=10± sqrt(4900)/50

=(10 +70)/50 ,(10 -70)/50

=8/5 ,-6/5

x -8/5>0  ,x +6/5>0

x > 8/5,  x< 6/5

the solution is  8/5 <x < 6/5

The rational inequality is

Rewrite as

And simplify left side.

Multiply each side of inequality by negitive one and flip the symbol.

The inequality critical values at x  = 8/5, = -6/5

x  = -1 x  = 7/5

The set of solutions of the inequality is the union of the interval [-6/5, 8/5] and the interval

(-1, 7/5).

Solution x  < -6/5

x  > 8/5

-1 < < 7/5.