If a ∈ R and |2a – 1| ≤ 3[a] + 2{a}, Here [x] represents greatest integer value of x and {x} represents fractional part value of x.

Question

If a \(\in\) R and |2a – 1| ≤ 3[a] + 2{a}, Here [x] represents greatest integer value of x and {x} represents fractional part value of x.

Then the value of 72a_min equals to _____.

Answer 1

Correct answer: 18

|2a – 1| ≤ 3[a] + 2{a}

|2a – 1| ≤ [a] + 2a

For 2a – 1 < 0 i.e., a < \(\frac 12\) – 2a + 1 ≤ [a] + 2a

4a + [a] –1 ≥ 0 ⇒ [a] ≥ 1 – 4a ⇒ a ≥ 0.25 and for a > \(\frac 12\)

2a – 1 ≤ [a] + 2a

⇒ [a] ≥ –1

⇒ a ≥ –1

So a_min = 0.25

⇒ 72.(0.25) = \(\frac{72} 4\) = 18