please help me with thisone thanks

solve the inequality algebraically. Express the solution in interval notation. (x-8)^2/x^2-36 > 0

Consider the equation:

(x-8)^2/(x^-36) = 0
(x-8)^2/((x+6)(x-6)) = 0

"critical values are -6,6,8

There are 4 segments on the number line of x which we should look at
1. x < -6
2. x between -6 and 6
3. x between 6 and 8
4. x > 8

I usually take an arbitrary value in each segment, and test it in the original. We don't actually have to work out the value, just consider the signs.
In this case, since the top is squared, we only have to look at the denominator's sign.

1. let x = -10 ----+/+ > 0 , good
2. let x = 0, --- +/- < 0 , no good
3. let x = 7, --- +/+> 0 , good
4. let x = 10 --- +/+ > 0 good

so x < -6 OR x > 6, x is a real number.

(The above is a rather general solution. In this case we could have just looked at the denominator.
clearly we cannot have it become negative. It was easy to see that would happen between -6 and 6 for values of x)

i don't get it so whick would be the solution or answer

"so x < -6 OR x > 6, x is a real number"

is the answer.
As Reiny explained, there are two segments to the answer, hence the "OR".

In interval notation, it would be:
(-∞,-6)∪(6,∞).
The (round) parentheses mean that the boundary values are excluded from the interval.

To solve the inequality algebraically, we need to find the values of x that satisfy the given inequality. Here's how you can do it step by step:

1. Start by factoring both the numerator and denominator of the fraction.

(x - 8)^2 / (x^2 - 36) > 0

Factoring the numerator gives us:

(x - 8)(x - 8) / (x^2 - 36) > 0

Factoring the denominator gives us the difference of squares:

(x - 8)(x + 8) / ((x - 6)(x + 6)) > 0

2. Determine the critical points by setting both the numerator and the denominator equal to zero and solving for x.

Numerator: (x - 8) = 0 --> x = 8
Denominator: (x - 6) = 0 --> x = 6 (also, x = -6 but we will address this later)

So, we have two critical points: x = 8 and x = 6.

3. Create a number line and mark the critical points on it.

-∞ | 6 | 8 | +∞

4. Choose a test value from each of the three intervals created on the number line:

a) Interval (-∞, 6): Pick x = 0
b) Interval (6, 8): Pick x = 7
c) Interval (8, +∞): Pick x = 9

5. Substitute each test value into the original inequality and determine the sign:

a) (0 - 8)*(0 + 8) / ((0 - 6)*(0 + 6)) > 0
-8*8 / (-6*6) > 0
64 / 36 > 0
1.77 > 0 (positive)

b) (7 - 8)*(7 + 8) / ((7 - 6)*(7 + 6)) > 0
-1*15 / (1*13) > 0
-15 / 13 > 0
-1.15 > 0 (negative)

c) (9 - 8)*(9 + 8) / ((9 - 6)*(9 + 6)) > 0
1*17 / (3*15) > 0
17 / 45 > 0
0.38 > 0 (positive)

6. Analyzing the signs, we can determine the solution to the inequality.

- From step 5, we see that the inequality is satisfied when either both factors are positive or both factors are negative. This means that the solution is either (-∞, 6) or (8, +∞).

- However, we also need to consider the excluded values. In this case, x = -6 and x = 6 cannot be included in the solution since they would make the denominator equal to zero, which is not allowed.

- Therefore, the solution to the inequality in interval notation is:
(-∞, 6) U (8, +∞)