Verified answer

|n| ≥ n

By the definition of absolute values, |n| = n for all n ≥ 0, and |n| is positive whenever n<0. All positive numbers are greater than all negative numbers, so for all n, |n| ≥ n

|n| ≤ n

This is only true for n ≥ 0 (which is when |n|=n). For all n<0, |n|>n

|3k| < 0

Never true, because absolute values are always greater than or equal to 0

Definition of absolute value: For all real positive numbers and for 0, |n|=n. For all negative real numbers, |n|= -(n). In either case, the important thing to remember is that the absolution value of real numbers is always greater than or equal to zero.

|n| ≥ n, Consider the three subcases:

n is positive: then |n|=n and n≥n, so all positive real numbers satisfy the inequality.

n is zero: then |n|=n and n≥n, so the number zero satisfoes the inequality.

n is negative: then |n| is positive, a positive number is always greater than a negative number, so all negative real numbers satisfy the inequality.

Therefor, all real numbers satisfy the inequality.

|n| ≥ n, Consider the three subcases:

n is positive: then |n|=n and n≤n, so all positive real numbers satisfy the inequality.

n is zero: then |n|=n and n≤n, so the number zero satisfoes the inequality.

n is negative: then |n| is positive, a positive number is always greater than a negative number, so |n|≥n. This isn't right (we want the absolute value to be smaller), so no negative real numbers satisfy the inequality.

Therefor, all real positive numbers greater than or equal to zero satisfy the inequality.

|3k| < 0

The definition of absolute value is that the absolute value of any real number is zero or positive. No real number satisfies this inequality.

Technically, the answer to |3k| < 0 is most decidedly not "No Solution", it's "No Real Solution". In the Complex plane, there are several solutions to this inequality. For example, the square root of negative one (i) is not a real number, but satisfied the inequality.