Solve the inequality $x(x + 6) > 16 - 5x + 33.$ Write your answer in interval notation.

cooIcooIcooI17 Jul 31, 2024

#1**+1 **

Let's rewrite this equation into standard form ax^2+bx+c > 0. We have

\(x^2+6x>16-5x+33\\ x^2+11x-49>0\)

We could probably try out some form of the quadratic formula, but let's take another approach.

Let's first complete the square for x. We get

\(\left(x+\frac{11}{2}\right)^2-\frac{317}{4}>0\)

Now, we can conformtably isolate x. We get

\(\left(x+\frac{11}{2}\right)^2>\frac{317}{4}\)

Since we square root it, we have to split the inequality into two different ones. We have

\(x+\frac{11}{2}<-\sqrt{\frac{317}{4}}\quad \mathrm{or}\quad \:x+\frac{11}{2}>\sqrt{\frac{317}{4}}\)

\(x<\frac{-\sqrt{317}-11}{2}\quad \mathrm{or}\quad \:x>\frac{\sqrt{317}-11}{2}\)

This is not the prettiest case, but it does work. Now, let's put this in interval notation. We have

\((-\infty, \frac{-\sqrt{317}-11}{2} )U(\frac{\sqrt{317}-11}{2}, \infty)\)

Thanks! :)

NotThatSmart Jul 31, 2024