Answer:

To find the complete set of x which satisfy the given inequality, we can break it down into two separate cases based on the two inequalities involved.

For the inequality (x-12)(x-4)(x² +16)(x+4)(x²-16) > 0, we can consider the sign of each factor individually.

1.

(x-12) > 0 implies x > 12

2.

(x-4) > 0 implies x > 4

3.

(x² +16) > 0 is always positive for all real values of x.

4.

(x+4) > 0 implies x > -4

5.

(x²-16) > 0 implies x < -4 or x > 4

Combining all the conditions, we get x > 12.

For the inequality (x-4)(x+5) ≤ 0, we can find the values of x that make either factor negative or zero.

1.

(x-4) ≤ 0 implies x ≤ 4

2.

(x+5) ≤ 0 implies x ≤ -5

Combining the conditions, we get x ≤ -5.

Therefore, the complete set of x which satisfy either (x-12)(x-4)(x² +16)(x+4)(x²-16) > 0 or (x-4)(x+5) ≤ 0 is x ≤ -5 or x > 12.

Verified answer

To find the complete set of values for x that satisfy the given inequality, we will solve it step by step. Let's break it down:

Solve the inequality (x - 12)(x - 4)(x² + 16)(x + 4)(x² - 16) > 0:

Step 1: Find the critical points by setting each factor equal to zero and solving for x:

(x - 12) = 0 => x = 12

(x - 4) = 0 => x = 4

(x² + 16) = 0 has no real solutions

(x + 4) = 0 => x = -4

(x² - 16) = 0 => x = -4 or x = 4

Step 2: Plot these critical points on a number line:

  -∞      -4        4       12       +∞

   |--------|--------|--------|--------|

Step 3: Test the intervals between the critical points:

For (x - 12)(x - 4)(x² + 16)(x + 4)(x² - 16) > 0, we need to determine whether each interval is positive or negative by testing a value from each interval.

Test a value in the interval (-∞, -4), e.g., x = -5:

(-5 - 12)(-5 - 4)(-5² + 16)(-5 + 4)(-5² - 16) = (-17)(-9)(1)(-1)(9) = 1377

Since the product is positive (+), this interval satisfies the inequality.

Test a value in the interval (-4, 4), e.g., x = 0:

(0 - 12)(0 - 4)(0² + 16)(0 + 4)(0² - 16) = (-12)(-4)(16)(4)(-16) = -196608

Since the product is negative (-), this interval does not satisfy the inequality.

Test a value in the interval (4, 12), e.g., x = 10:

(10 - 12)(10 - 4)(10² + 16)(10 + 4)(10² - 16) = (-2)(6)(116)(14)(84) = -8171712

Since the product is negative (-), this interval does not satisfy the inequality.

Test a value in the interval (12, +∞), e.g., x = 13:

(13 - 12)(13 - 4)(13² + 16)(13 + 4)(13² - 16) = (1)(9)(185)(17)(165) = 1127805

Since the product is positive (+), this interval satisfies the inequality.

Solve the inequality (x - 4)(x + 5) ≤ 0:

Step 1: Find the critical points by setting each factor equal to zero and solving for x:

(x - 4) = 0 => x = 4

(x + 5) = 0 => x = -5

Step 2: Plot these critical points on a number line:

  -∞      -5        4       +∞

   |--------|--------|--------|

Step 3: Test the intervals between the critical points:

For (x - 4)(x + 5) ≤ 0, we need to determine whether each interval is positive or negative by testing a value from each interval.

Test a value in the interval (-∞, -5), e.g., x = -6:

(-6 - 4)(-6 + 5) = (-10)(-1) = 10

Since the product is positive (+), this interval does not satisfy the inequality.

Test a value in the interval (-5, 4), e.g., x = 0:

(0 - 4)(0 + 5) = (-4)(5) = -20

Since the product is negative (-), this interval satisfies the inequality.

Test a value in the interval (4, +∞), e.g., x = 5:

(5 - 4)(5 + 5) = (1)(10) = 10

Since the product is positive (+), this interval does not satisfy the inequality.

Combine the results from both inequalities:

From the first inequality, we found that the intervals (-∞, -4) and (12, +∞) satisfy the inequality.

From the second inequality, we found that the interval (-5, 4) satisfies the inequality.

Therefore, the complete set of x values that satisfy either (x - 12)(x - 4)(x² + 16)(x + 4)(x² - 16) > 0 or (x - 4)(x + 5) ≤ 0 is:

(-∞, -4) ∪ (-5, 4) ∪ (12, +∞)

This represents the union of the intervals that satisfy either one of the inequalities.