Question

Solve the inequality |w| + 8 >= 23 for all real numbers 'w'. Determine whether there is a solution or if there are no solutions. If there is a solution, provide the solution. If there are no solutions, clarify that there are no solutions and explain why.

Answer 1

The inequality |w| + 8 ≥ 23 has solutions. The values of 'w' must be greater than or equal to 15 or less than or equal to -15. Thus, there are infinitely many solutions to this inequality.

Step-by-step explanation:

To solve the inequality |w| + 8 ≥ 23 for all real numbers 'w', we first isolate the absolute value on one side by subtracting 8 from both sides of the inequality:

|w| ≥ 23 - 8

|w| ≥ 15

This means that the value of 'w', regardless of its sign, must be greater than or equal to 15. We then consider two cases since the absolute value of 'w' can account for both positive and negative values of 'w':

1. If w is positive (or zero), w ≥ 15.
2. If w is negative, -w ≥ 15, which means w ≤ -15.

Therefore, the solution to the inequality is w ≥ 15 or w ≤ -15. There is a solution, and in fact, there are infinitely many solutions as any real number greater than or equal to 15 or less than or equal to -15 will satisfy the inequality.