Quadratic Equations
Inequalities with absolute values
Grade 11
Question:
<p>If \(x\) satisfies \(|x - 1| + |x - 2| + |x - 3| > 6\), then</p>
<p>(a) \(x \in (-\infty, 1)\)</p>
<p>(b) \(x \in (-\infty, 0)\)</p>
<p>(c) \(x \in (4, \infty)\)</p>
<p>(d) \(x \in (2, \infty)\)</p>
Step-by-Step Solution
Key Concept: Split the domain into intervals based on critical points where expressions inside absolute values change sign.
<p>Analyze the sum of absolute values by cases. For $x \leq 1$: sum = $-(x-1) - (x-2) - (x-3) = -3x + 6 > 6 \Rightarrow x < 0$. For $1 < x < 2$: sum = $(x-1) - (x-2) - (x-3) = -x + 4 > 6 \Rightarrow x < -2$ (no solution). For $2 \leq x \leq 3$: sum = $(x-1) + (x-2) - (x-3) = x > 6$ (no solution). For $x > 3$: sum = $(x-1) + (x-2) + (x-3) = 3x - 6 > 6 \Rightarrow x > 4$.</p>
Correct Answer: B, C