Given:
\(\frac{1}{|X|-3} \le \frac{1}{2}, x \in R\)
Intervals of |x|:
|x| = -x, x<0
|x| = x, x ≥ 0
Domain of \(\frac{1}{|X|-3} \le \frac{1}{2}\)
x| + 3 = 0
X = -3 or x = 3
Therefore,
-3 < x < 3
Now, combining intervals with domain:
x < -3, -3 < x < 0, 0 ≤ x <3, x >3
For x < -3
Now, subtracting \(\frac{1}{2}\)from both the sides
Signs of x + 5:
x + 5 = 0 → x = -5 (Subtracting 5 from both the sides)
x + 5 > 0 → x > -5 (Subtracting 5 from both the sides)
x + 5 < 0 → x < -5 (Subtracting 5 from both the sides)
Signs of -2x - 6:
-2x - 6 = 0 → x = -3
(Adding 6 on both the sides, then multiplying both the sides by -1 and then dividing both the sides by 2)
-2x - 6 > 0 → x < -3
(Adding 6 on both the sides, then multiplying both the sides by -1 and then dividing both the sides by 2)
-2x - 6 < 0 → x > -3
(Adding 6 on both the sides, then multiplying both the sides by -1 and then dividing both the sides by 2)
Intervals satisfying the required condition: ≤ 0
x < -5, x= -5, x >-3
Or
x ≤ -5 or x >-3
Similarly, for -3 < x < 0:
x ≤ -5 or x >-3
Merging overlapping intervals:
-3 < x < 0
For, 0 ≤ x < 3:
Subtracting \(\frac{1}{2}\) from both the sides
Multiplying both the sides by 2
Signs of 5 – x:
5 – x = 0 → x = 5
(Subtracting 5 from both the sides and then dividing both sides by -1)
5 – x > 0 → x < 5
(Subtracting 5 from both the sides and then multiplying both sides by -1)
5 – x < 0 → x > 5
(Subtracting 5 from both the sides and then multiplying both sides by -1)
Signs of x – 3:
x – 3 = 0 → x = 3 (Adding 3 to both the sides)
5 – x > 0 → x > 3 (Adding 3 to both the sides)
5 – x < 0 → x < 3 (Adding 3 to both the sides)
Intervals satisfying the condition: x ≤ 0
x < 3 or x = 5 or x > 5
Or
x <3 and x ≥ 5
Similarly, for 0 ≤ x < 3:
x <3 and x ≥ 5
Merging overlapping intervals:
0 ≤ x < 3
Now, combining all the intervals satisfying condition: ≤ 0
x ≤ -5 or -3 < x < 0 or 0 ≤ x < 3 or x ≥ 5
Therefore
x є (-∞, -5] υ (-3, 3) υ [5, ∞)
