Solve each of the following system of equations in R. 1/(|x|-3) < 1/2

Question

Solve each of the following system of equations in R.

\(\frac{1}{|x|-3}\) < \(\frac{1}{2}\) 

1/(|x|-3) < 1/2

Answer 1

Given,

1/(|x|-3) < 1/2

We know that, 

If we take reciprocal of any inequality we need to change the inequality as well. 

Also, 

|x|–3≠0 

⇒ |x|>3 or |x|<3 

For |x|<3 

⇒ –3<x<3

⇒ x∈ (–3, 3) ….(1)

∴ The equation can be re–written as–

|x| - 3 > 2

Adding 2 both the sides, we get– 

|x|–3+3> 2+3 

⇒ |x|>5 

We know that, 

|x |>a ⟺ x<-a or x>a

Here, 

a = 5 

⇒ x< -5 ….(2) 

⇒ x ∈ (–∞,–5 ) or x ∈(5, ∞) 

⇒ x ∈ (–∞,–5 ) ⋃ (–3, 3) ⋃ (5, ∞) 

(from 1 and 2) 

We can verify the answers using graph as well.