Solve the inequality: |x - 1| + |x - 2| + |x - 3| >= 6.
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Step-by-Step Solution
Step 1: Identify critical points and intervals
The critical points are the values of x where the expressions inside the absolute value signs become zero. These points divide the number line into intervals, which we will analyze separately.
Step 2: Case 1: $x < 1$
For x < 1$, all three terms inside the absolute value signs are negative. Therefore, we remove the absolute value signs and multiply each term by -1$. Solving the resulting inequality gives x \le 0$. Combining this with the condition x < 1$, the solution for this case is x \le 0$.
Step 3: Case 2: $1 \le x < 2$
For 1 \le x < 2$, (x - 1) is non-negative, while (x - 2) and (x - 3) are negative. Solving the inequality yields x \le -2$. However, this contradicts our initial condition 1 \le x < 2$. Thus, there is no solution in this interval.
Step 4: Case 3: $2 \le x < 3$
For 2 \le x < 3$, (x - 1) and (x - 2) are non-negative, while (x - 3) is negative. Solving the inequality gives x \ge 6$. This also contradicts our initial condition 2 \le x < 3$. Therefore, there is no solution in this interval.
Step 5: Case 4: $x \ge 3$
For x \ge 3$, all three terms inside the absolute value signs are non-negative. We simply remove the absolute value signs. Solving the inequality gives x \ge 4$. Combining this with the condition x \ge 3$, the solution for this case is x \ge 4$.
Step 6: Combine solutions
The solutions from the valid cases are x \le 0 and x \ge 4$. Combining these, the final solution set for the inequality is x \in (-\infty, 0] \cup [4, \infty)$.