If n is an integer such that |3 - 2n| \le 9 and |n - 1| \ge 2, what is the sum of all poss… — GMAT

If $n$ is an integer such that $|3 - 2n| \le 9$ and $|n - 1| \ge 2$ , what is the sum of all possible values of $n$ ?

Answer: 12

The sum of all possible integer values of

n

is 12.

Solving

|3 - 2n| \le 9

gives

-3 \le n \le 6

. Solving

|n - 1| \ge 2

restricts

n

n \le -1

n \ge 3

. Taking the intersection yields the integer set

\{-3, -2, -1, 3, 4, 5, 6\}

, whose sum equals 12.

Unpack and solve

|3 - 2n| \le 9

-3 \le n \le 6

Removing absolute value yields

-9 \le 3 - 2n \le 9

. Dividing by

-2

requires reversing the inequality direction.

Unpack and solve

|n - 1| \ge 2

n \le -1

n \ge 3

An absolute value greater than or equal to 2 implies distance from 1 is at least 2 units in either direction.

Determine the intersection set of integers

\{-3, -2, -1, 3, 4, 5, 6\}

Filters out integers 0, 1, and 2 from the continuous range

[-3, 6]

Calculate the sum of the valid integers

Summing

(-3) + (-2) + (-1) + 3 + 4 + 5 + 6 = 12

Combining system of absolute value inequalities and handling inequality sign flips when dividing by negative quantities.

Question