Question

If -2 is smaller than (x^2-5)≤3 then the range of x is, where ( ) represents greatest integer function​

Answer 1

Given info : -2 is smaller than [x² - 5]≤ 3 then the range of x is ... Where [.] represents greatest integer function.

Solution : here, -2 < [x² - 5] ≤ 3

Case 1 : [x² - 5] ≤ 3

we know, [x ± k] = [x] ± k, for any integer k

⇒[x²] - 5 ≤ 3

⇒[x²] ≤ 8

we know, if [x] ≤ k ⇒x < k + 1 , where k ∈ Z

so, [x²] ≤ 8 ⇒x² < 8 + 1 = 9

⇒x² < 9

⇒-3 < x < 3

Case 2 : [x² - 5] > -2

⇒[x²] - 5 > -2

⇒[x²] > 3

We know, [x] > k ⇒x ≥ k + 1, where k ∈ Z

so, [x²] > 3 ⇒x² ≥ 3 + 1 = 4

⇒x² ≥ 4

⇒x ≥ 2 or x ≤ -2

Taking common values of case 1 and case 2 we get, 2 ≤ x < 3 or -3 < x ≤ -2

therefore the value of x ∈ [2, 3) U (-3, -2]