Question

If [x + [2x]] < 3, where [.] denotes the greatest integer function, then

• A. x ∈ [0, 1)
• B. x ∈ (-∞, 2/3]
• C. x ∈ [0, 3/2)
• D. x ∈ (-∞, 1)

Answer 1

Answer: (D)

x ∈ (-∞, 1)

Answer 2

[x + [2x]] < 3 ⇒ [x] + [2x] ≤ 2.

Any non-positive real number will satisfy this inequality.

Now if x ∈ $\left[ 0 , \frac { 1 } { 2 } \right)$ ⇒ [x] = 0, [2x] = 0

⇒ inequality is again satisfied.

For x ∈ $\left[ \frac { 1 } { 2 } , 1 \right)$ ⇒ [x] = 0, [2x] = 1

⇒ inequality is still satisfied.

For x ∈ $\left[ 1 , \frac { 3 } { 2 } \right)$ ⇒ [x] = 1, [2x] = 2

⇒ inequality doesn't hold true. Thus x ∈ (-∞, 1).