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

1.	Solve the system of equations in R.1/(\|x\| – 3) < 1/2
Answer» As 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) Now, the equation can be re–written as \|x\| – 3 > 2 Then, let us add 3 on both the sides, we get \|x\| – 3 + 3 > 2 + 3 \|x\| > 5 Suppose ‘a’ be a fixed real number. Now, \|x\| > a ⟺ x < –a or x > a Here, a = 5 x < –5 or x > 5 …. (2) From (1) and (2) x ∈ (–∞,–5 ) or x ∈ (5, ∞) ∴ x ∈ (–∞,–5 ) ⋃ (–3, 3) ⋃ (5, ∞)

Answer»

As 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)

Now, the equation can be re–written as

|x| – 3 > 2

Then, let us add 3 on both the sides, we get

|x| – 3 + 3 > 2 + 3

|x| > 5

Suppose ‘a’ be a fixed real number. Now,

|x| > a ⟺ x < –a or x > a

Here, a = 5

x < –5 or x > 5 …. (2)

From (1) and (2)

x ∈ (–∞,–5 ) or x ∈ (5, ∞)

∴ x ∈ (–∞,–5 ) ⋃ (–3, 3) ⋃ (5, ∞)

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