Inequality math problem.

Question

1/|x²-9x+36|<3/5

(| indicates modulus not brackets btw)

Find the range of values of x.

Answer 1

If you consider the equation y = x² - 9x + 36, it is a parabola with a vertex that can be calculated as follows:

x_vertex = -b/(2a) = -(-9)/(2•1) = 4.5

y_vertex = (4.5)² - 9(4.5) + 36 = 15.75

So the vertex is (4.5, 15.75) and since this parabola opens up, the minimum value is 15.75. Since the inequality in question is 1/(x² - 9x + 36) then the MAXIMUM value of this will occur at the minimum value of x² - 9x + 36 or 15.75. So 1/15.75 = 4/63 or approx 0.063 (occurring at x = 4.5). All other values of x are less than this.

So 1/(x² - 9x + 36) < 3/5 for all values of x.

The same holds true for the negative value of (x² - 9x + 36) which could also occur due to the absolute value which would be -(x² - 9x + 36) in the denominator, This parabola is always negative (opening downwards with a max of -15.75) meaning 1/(x² - 9x + 36) is always less than 3/5.

So the values of x that make this true are (-∞, ∞) or all real numbers