Question

`\lfloor(x^2+1)/(10)\rfloor+\lfloor(10)/(x^2+1)\rfloor=1`


`\small\textrm{$\bullet\ \lfloor ()\rfloor$ denotes the greatest integer that does not exceed the number}`​

Answer 1

• See below

Explanation:

This equation is solved if one of the following 2 conditions is met

1.

• (x² + 1) / 10 is between 1 and 2

and

• 10 / (x² + 1) is between 0 and 1

Solve this:

• 1 < (x² + 1) / 10 < 2
• 10 < x² + 1 < 20
• 9 < x² < 19
• 3 < |x| < √19
• x ∈ (- √19, - 3) ∪ (3, √19)

• 0 < 10 / (x² + 1) < 1
• x² + 1 > 10
• x² > 9
• |x| > 3

Solution for this case is x ∈ (- √19, - 3) ∪ (3, √19)

2.

• (x² + 1) / 10 is between 0 and 1

and

• 10 / (x² + 1) is between 1 and 2

Solve this:

• 0 < (x² + 1) / 10 < 1
• 0 < x² + 1 < 10
• - 1 < x² < 9
• 0 ≤ |x| < 3
• x ∈ ( - 3, 3)

• 1 < 10 / (x² + 1) < 2
• x² + 1 < 10 ⇒ x² < 9 ⇒ |x| < 3
• x² + 1 > 5 ⇒ x² > 5 ⇒ |x| > √5

Solution for this case is x ∈ (- 3, - √5) ∪ (√5, 3)