Question

Let f(x)=x^2+7x+7 and g(x)=x+9. Giving your answer in interval notation, find the set of x for which f(x)>g(x).

Answer 1

You want to find the interval in which \(f(x) > g(x) \) which is equivalent to solving the inequality \(f(x) - g(x) > 0 \)

Substituting for \(f(x)\) and \(g(x)\) to get:
\(x^2 + 7x + 7 - (x + 9) > 0\)
\(\implies x^2 + 6x - 2 > 0\)

The two roots of the equation \(x^2 + 6x -2 = 0\) are given by the quadratic formula \(\displaystyle \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) to be \(\displaystyle \frac{-6 \pm \sqrt{44}}{2} = -3 \pm \sqrt{11}\)

So, the inequality becomes \((x - (-3 + \sqrt{11}))(x - (-3 - \sqrt{11})) > 0\)

\(\implies\) your interval for x is \(x \in (-\infty, -3-\sqrt{11}) \text{ U } (-3 + \sqrt{11}, \infty)\)