Given h(x) = 9^x + 9 and g(x) = 10*3^x, find {x | h(x) < g(x)}.

This question is asking to find the values for x, such that h(x) is strictly less than g(x). We can write this as 9^x + 9 < 103^x and solve for x as follows. 9^x + 9 < 103^x => 3^(2x) - 103^x + 9 < 0 We let t = 3^x : => t^2 - 10t + 9 < 0 => (t - 9)(t - 1) < 0 By either sketching the quadratic, or by a sign diagram we find the values of t that satisfy this inequality : 1 < t < 9. By substituting t = 3^x again, we find : 1 < 3^x < 9 => 3^0 < 3^x < 3^2 => 0 < x < 2.