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[tex]\lfloor\dfrac{x^2+1}{10}\rfloor+\lfloor\dfrac{10}{x^2+1}\rfloor=1[/tex]
[tex]\small\textrm{$\bullet\ \lfloor ()\rfloor$ denotes the greatest integer that does not exceed the number}[/tex]​

Sagot :

mhanifa

Answer:

  • See below

Step-by-step 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)

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