solve the inequality express your solution in interval notation

(x - 4)^2/ (x -6) (x + 6) > 0 Help me!

critical values of x are

x = 4, -6 and 6

Pick any x in the sections of the number line marked by these values
We don not have to do the actual calculation , just consider what happens to the signs

x < -6, say x=-10
+/-- > 0 good
x between -6 and 4, say x=0
+/- + < 0 NO
x between 4 and 6 , say x=5
+/- + <0 , NO
x > 6, say x=10
+/+ + > 0, good

so x < -6 or x > 6

Wolfram shows the graph and illustrates my solution is correct
http://www.wolframalpha.com/input/?i=%28x+-+4%29%5E2%2F+%28x+-6%29+%28x+%2B+6%29+%3Dy

Thanks I can do it that way but she counts it wrong. That is the answer I got. She wants it in interval notation.

Oh my!

That would be expressing it in a different form, I think it would be something like
(-∞ , -6) or (6,∞)

x < -6 or x > 6 says the same thing in a different way, but it certainly is not "wrong".

exactly it is what I thought but it does not match her answer sheet so she counts it wrong.

To solve the given inequality (x - 4)^2/ (x -6) (x + 6) > 0, we need to follow a step-by-step process:

Step 1: Find the critical points by setting the numerator and denominator equal to zero separately and solving for x.

Setting the numerator (x - 4)^2 equal to zero:
(x - 4)^2 = 0
x - 4 = 0
x = 4

Setting the denominator (x - 6)(x + 6) equal to zero:
x - 6 = 0 or x + 6 = 0
x = 6 or x = -6

So, the critical points are x = 4, x = 6, and x = -6.

Step 2: Create a number line by locating the critical points and marking them accordingly.

-∞ |______________|____(-6)_____|_____(4)_____|____(6)______|______________| +∞

Step 3: Test each interval using a test point in that interval.

Test point: x = -7

For the interval (-∞, -6), substitute x = -7 into the inequality:
((-7 - 4)^2) / ((-7 - 6) * (-7 + 6)) > 0
(11^2) / (-13) > 0
121 / (-13) > 0
-9.3 > 0

Since -9.3 is less than zero, the inequality is not satisfied in the interval (-∞, -6).

Test point: x = 0

For the interval (-6, 4), substitute x = 0 into the inequality:
((0 - 4)^2) / ((0 - 6) * (0 + 6)) > 0
(-4^2) / (-6 * 6) > 0
16 / (-36) > 0
-0.44 > 0

Since -0.44 is less than zero, the inequality is not satisfied in the interval (-6, 4).

Test point: x = 5

For the interval (4, 6), substitute x = 5 into the inequality:
((5 - 4)^2) / ((5 - 6) * (5 + 6)) > 0
(1^2) / (-1 * 11) > 0
1 / (-11) > 0
-0.09 > 0

Since -0.09 is less than zero, the inequality is not satisfied in the interval (4, 6).

Test point: x = 7

For the interval (6, ∞), substitute x = 7 into the inequality:
((7 - 4)^2) / ((7 - 6) * (7 + 6)) > 0
(3^2) / (1 * 13) > 0
9 / 13 > 0

Since 9/13 is greater than zero, the inequality is satisfied in the interval (6, ∞).

Step 4: Determine the solution based on the sign changes of the intervals.

Based on the test results, the inequality is satisfied in the interval (6, ∞). Therefore, the solution in interval notation is:

Solution: x ∈ (6, ∞)