It is to be noted that where the above is given, the correct answer is
x < (-9/2) + (-1/2) √(69)
(x-5)/(x² + 9x + 3) ≤0
Let's find the critical points of the inequality.
(x-5)/(x² + 9x + 3) = 0
x - 5 = 0 (Multiply both sides by x² + 9x + 3)
x - 5 + 5 = 0 +5 (Add 5 to both sides)
x = 5
Check possible critical points
x = 5 (Works in Original Equation)
Critical Points:
x = 5 (Make both sides equal)
x = -9/2 + 1/2√(69) or x = -9/2 + -1/2 √(69) (Makes left denominator equal to 0)
Check intervals in between critical points. Test values in the intervals to see if they work.
x < -9/2 + -1/2√(69) (works in original inequality)
-9/2 + -1/2√(69) < x < -9/2 + 1/2√(69) (doesn't work in original inequality)
-9/2 + 1/2√(69) < x ≤ 5 (Works in original inequality)
x ≥ 5 (Doesn't work in original inequality)