Sets, Relations & Functions
Grade 11
Question:
<p>If \(\{x\}=\frac{2}{3}\) and the nested floor/fractional-part expression equals 5, which are correct?<br>(A) \(x=\frac{14}{3}\) (B) \([x]=5\) (C) \(x=\frac{17}{3}\) (D) \([x]=4\)</p>
A)x=14/3
B)[x]=5
C)x=17/3
D)[x]=4
Step-by-Step Solution
<div class="solution"><p>Let x=n+2/3. The nested expression reduces to [x+2/3]=5, giving n+4/3+2/3=[n+2]=n+2... wait: [x+2/3]=[n+4/3]=n+1=5 \to n=4.</p><p>So x=4+2/3=14/3. [x]=4.</p><p><strong>Answer: (A),(D)</strong></p><div class="trap-box"><strong>Trap:</strong> Fractional part stays 2/3 after adding integers. Don't lose track.<div class="key-concept"><strong>Key Concept:</strong> Nested [ ] and { } -- reduce to x=n+r with 0\ler<1
Correct Answer: A,D