Question

Solve the inequality.
|d 2| ≥ 6d ≥ −8 or d ≥ 4 d ≤ −8 or d ≥ −4 d ≤ −8 or d ≥ 4 d ≤ −4 or d ≥ 4

Answer 1

The inequality |d + 2| ≥ 6 results in two solution sets, d ≥ 4 and d ≤ -8, which means the values of d must be either greater than or equal to 4 or less than or equal to -8.

To solve the inequality |d + 2| ≥ 6, we split the equation into two cases based on the absolute value:

1. Case 1: If d + 2 is positive or zero, we have d + 2 ≥ 6, which simplifies to d ≥ 4.
2. Case 2: If d + 2 is negative, we take the negative of the inside of the absolute value: -(d + 2) ≥ 6, which simplifies to d ≤ -8.

Hence, the final answer comprises two solution sets where d is either greater than or equal to 4 or less than or equal to -8. The inequality does not apply for d values between these two solution sets.

To check the reasonableness of our answer, we consider values in and out of the solution sets and see if they satisfy the original inequality. Testing a value like 5 (greater than 4) and -9 (less than -8) should confirm the correctness of the solution. Therefore, the final answer captures all possible values of d that satisfy the inequality.