Domain inequality problem

Question

If g is decreasing on it domain, solve this inequality - g(x²)>g(5x+6), D_g=[1, ∞)

Answer 1

We need to know where x^2 > 5x+6 on x=[1,infinity],

x^2 - 5x - 6 > 0

( x - 6 )( x + 1 ) > 0

THe solutions are x=6 and x = -1

Sign table:

interval x < -1 -1 < x < 6 x > 6

sample x -2 0 7

result 8 -6 8

so for x=[1,6), x^2 < 5x+6

and for x(6,infinity) x^2 > 5x+6

since the function is decreasing, g(x^2) > g(5x+6) for the first inequality and

g(x^2) < g(5x+6) for the second inequality

So the solution is [1,6)