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+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 or x ≥ 5 

⇒ x ∈ (–∞,–5 ] or x ∈ [5, ∞) …(2)

⇒ x ∈ (–∞,–5 ] ⋃ (–3, 3) ⋃ [5, ∞) 

(From 1 and 2)

We can verify the answers using graph as well.