solve the rational inequality. express the solution using interval notation

First of all this inequality does not exist in x=(-5), because we cannot divide by zero.

(x^{2} - 45)/(x+5) ≤ x .

1. Let's assume that (x+5) > 0 , we can multiply both sides of inequality and keep the same sign "≤"

x^{2} - 45 ≤ x(x+5) ----> x^{2} - 45 ≤ x^{2} +5x

- x^{2} -x^{2}

----> -45 ≤ 5x ----> 5x / 5 ≥ -45 / 5 ------> x ≥ -9 and x > -5 ---->** x > -5**

2. Now let's assume (x+5) < 0 , x < -5 ----> when we will divide by (x+5), we have switch sign of inequality to opposite.

x^{2} - 45 ≥ x(x+5) -------------------> x ≤ -9 and x < -5 ----> **
x < -5**

**The answer: (-∞ , -5) U (-5 , ∞) , all real numbers except (-5)**

## Comments

I read it backwards, duh. Nice job! :)

Thank you, Michael :-)