Question

How close to -6 do we have to take x so that the following inequality is satisfied? (Give the largest possible value.) 1/(x-6)⁶ > 1,000,000

Answer 1

To satisfy the inequality 1/(x-6)⁵ > 1,000,000, x must be less than 6 plus the sixth root of 10^-6 which means x must approach just less than 6 from the left.

Step-by-step explanation:

How Close to -6 Must We Take x So That the Inequality is Satisfied?

To find how close to -6 we must take x so that the inequality 1/(x-6)⁵ > 1,000,000 is satisfied, we start by setting up the inequality properly. First, we isolate the variable on one side of the equation:

1. 1/(x-6)⁵ > 1,000,000
2. (x-6)⁵ < 1/1,000,000 since inverting both sides reverses the inequality.
3. Let's convert 1/1,000,000 into a power of 10, which is 10⁻⁹.
4. Now we have (x-6) < (10⁻⁹)¹ⁱ. Since we're dealing with a positive sixth root, we can take the sixth root of both sides without altering the inequality:
5. x-6 < 10⁻⁹.
6. Add 6 to both sides to solve for x:
7. x < 6 + (10⁻⁹).

We are looking for the largest possible value that x can approach from the left to satisfy the inequality x < 6 + (10⁻⁹). Therefore, we have found that as x approaches just less than 6 from the left, the inequality will be satisfied.