Solve. Write the solution set using interval notation.

x2 + 16x + 64 > 0

Solve. Write the solution set using interval notation.

x − 4/
(x + 1)(x − 1) ≤ 0

(x+8)^2 > 0

This is true for all x ≠ -8, so
(-∞,-8)U(-8,∞)

(x-4) / (x-1)(x+1) <= 0
(-∞,-1)U(1,4]

http://www.wolframalpha.com/input/?i=%28x-4%29+%2F+%28%28x-1%29%28x%2B1%29%29+for+-2+%3C%3D+x+%3C%3D+5

To solve the inequality x^2 + 16x + 64 > 0, we can use the method of factoring.

First, we need to factor the quadratic expression x^2 + 16x + 64. It can be factored as (x + 8)(x + 8) = (x + 8)^2.

Now, we can rewrite the inequality as (x + 8)^2 > 0.

To find the solution set, we can analyze the sign of (x + 8)^2.

Since (x + 8)^2 is a perfect square, it is always non-negative (or positive), except when it equals zero.

When (x + 8)^2 = 0, we have x + 8 = 0, which gives x = -8.

So, x = -8 is the only value where (x + 8)^2 = 0.

Therefore, the solution set for the inequality (x + 8)^2 > 0 is x ≠ -8.

To write this solution set in interval notation, we use parentheses to indicate that -8 is not included in the solution. So, the interval notation is (-∞, -8) U (-8, ∞).

Moving on to the next equation x - (4/(x + 1)(x - 1)) ≤ 0, we can solve it as follows:

First, we need to determine the values of x that make the expression (x + 1)(x - 1) equal to zero.

Setting (x + 1)(x - 1) = 0, we find x = -1 and x = 1.

These values divide the number line into three intervals: (-∞, -1), (-1, 1), and (1, ∞).

Next, we choose a test point from each interval and evaluate the inequality to determine whether it is true or false.

For the interval (-∞, -1), we can choose x = -2 as a test point. Evaluating the inequality, we get:

(-2 - 4/(-2 + 1)(-2 - 1)) ≤ 0 => (-2 + 4/(-3)) ≤ 0 => (-2 - 4/3) ≤ 0 => (-6/3 - 4/3) ≤ 0 => (-10/3) ≤ 0

Since (-10/3) is negative, this interval satisfies the inequality.

For the interval (-1, 1), we can choose x = 0 as a test point. Evaluating the inequality, we get:

(0 - 4/(0 + 1)(0 - 1)) ≤ 0 => -4/(0 + 1)(0 - 1) ≤ 0 => -4/(1)(-1) ≤ 0

Since -4/(-1)(1) = -4/(-1) = 4, which is positive, this interval does not satisfy the inequality.

For the interval (1, ∞), we can choose x = 2 as a test point. Evaluating the inequality, we get:

(2 - 4/(2 + 1)(2 - 1)) ≤ 0 => (2 - 4/(3)(1)) ≤ 0 => (2 - 4/3) ≤ 0 => (6/3 - 4/3) ≤ 0 => (2/3) ≤ 0

Since 2/3 is positive, this interval also does not satisfy the inequality.

So, the solution set for the inequality x - (4/(x + 1)(x - 1)) ≤ 0 is (-∞, -1] U [1, ∞).