Solve x-9/x^2-4 < 0 and write interval notation
Note:
x-9/x^2-4 < 0 ≠ (x-9)/(x^2-4) < 0
I presume it is the latter, and that you have omitted to insert parentheses when interpreting a fraction.
The best way to visualize the problem is to sketch a graph of the function (of the left-hand-side).
The numerator is positive for x>9, and negative for x<9.
The denominator is >0 for |x|>2, and <0 for x<2.
Note that there is a vertical asymptote at x=2.
So make a table:
interval numerator denominator expression
(-∞,-2) <0 >0 <0
(-2,2) <0 <0 >0
(2,9) <0 >0 <0
(9,+∞) >0 >0 >0
From the table, it should be relatively easy to find the solution to f(x)<0.
For reference:
http://img811.imageshack.us/img811/4262/1291593942.png
To solve the inequality (x - 9)/(x^2 - 4) < 0, we need to find the values of x that make the expression less than zero.
Step 1: Factor the denominator.
The denominator x^2 - 4 can be factored as (x - 2)(x + 2).
Step 2: Determine the critical points.
To find the critical points, we need to set both the numerator (x - 9) and the denominator (x - 2)(x + 2) equal to zero and solve for x.
Setting x - 9 = 0, we find x = 9.
Setting (x - 2)(x + 2) = 0, we find x = -2 and x = 2.
The critical points are x = -2, x = 2, and x = 9.
Step 3: Create a sign chart.
On a number line, mark the critical points -2, 2, and 9.
Choose test points in the intervals between these critical points to determine the sign of the expression (x - 9)/(x^2 - 4).
For example, choose x = -3, which is less than -2, and substitute it into the inequality:
(-3 - 9)/((-3)^2 - 4) = (-12)/(9 - 4) = -12/5 < 0
Therefore, the expression is negative in the interval (-∞, -2).
Choose x = 0, which is between -2 and 2, and substitute it into the inequality:
(0 - 9)/(0^2 - 4) = (-9)/(-4) > 0
Therefore, the expression is positive in the interval (-2, 2).
Choose x = 5, which is greater than 2, and substitute it into the inequality:
(5 - 9)/(5^2 - 4) = (-4)/(21) < 0
Therefore, the expression is negative in the interval (2, 9).
Choose x = 10, which is greater than 9, and substitute it into the inequality:
(10 - 9)/(10^2 - 4) = (1)/(96) > 0
Therefore, the expression is positive in the interval (9, ∞).
Step 4: Write the solution in interval notation.
Based on the sign chart, the solution to the inequality (x - 9)/(x^2 - 4) < 0 is:
(-∞, -2) U (2, 9)