Given:
\(\frac{1}{2 - |X|} \ge\) 1, xϵ R – {–2, 2}
Intervals of |x|:
x ≥ 0, |x| = x and x < 0, |x| = -x
Domain of \(\frac{1}{2 - |X|} \ge\)1
\(\frac{1}{2 - |X|} \ge\)1 is undefined at x = -2 and x = 2
Therefore, Domain: x < -2 or x > 2
Combining intervals with domain:
x < -2
For x < - 2
\(\frac{1}{2 - (-{\text{x}} )} \ge\)1
Subtracting 1 from both the sides
Signs of -1 -x:
-1 – x = 0 → x = -1
(Adding 1 to both the sides and then dividing by -1 on both the sides)
-1 – x > 0 → x < -1
(Adding 1 to both the sides and then multiplying by -1 on both the sides)
-1 – x < 0 → x > -1
(Adding 1 to both the sides and then multiplying by -1 on both the sides)
Signs of 2 + x:
2 + x = 0 → x = -2 (Subtracting 2 from both the sides)
2 + x > 0 →x > -2 (Subtracting 2 from both the sides)
2 + x < 0 →x < -2 (Subtracting 2 from both the sides)
Intervals satisfying the required condition: ≥ 0
-2 < x < 1 or x = -1
Merging overlapping intervals:
-2 < x ≤ 1
Combining the intervals:
-2 < x ≤ 1 and x < -2
Merging the overlapping intervals:
No solution.
Similarly, for -2 < x <0
\(\frac{1}{2 - (-{\text{x}} )} \ge\)1
Therefore, Intervals satisfying the required condition: ≥ 0
-2 < x ≤ 1 and x < -2
Merging overlapping intervals:
-2 < x ≤ 1
Combining the intervals:
-2 < x ≤ 1 and -2 < x < 0
Merging the overlapping intervals:
-2 < x ≤ 1
For 0 ≤ x < 2
\(\frac{1}{2 -{\text{x}}} \ge\)1
Subtracting 1 from both the sides
Signs of x -1:
x - 1 = 0 → x = 1(Adding 1 to both the sides)
x - 1 > 0 → x > 1(Adding 1 to both the sides)
x - 1 < 0 → x < 1(Adding 1 to both the sides)
Signs of 2 + x:
2 + x = 0 → x = -2 (Subtracting 2 from both the sides)
2 + x > 0 →x > -2 (Subtracting 2 from both the sides)
2 + x < 0 →x < -2 (Subtracting 2 from both the sides)
Intervals satisfying the required condition: ≥ 0
1 < x < 2 or x = 1
Merging overlapping intervals:
1 ≤ x < 2
Combining the intervals:
1 ≤ x < 2 and 0 ≤ x < 2
Merging the overlapping intervals:
1 ≤ x < 2 Similarly, for x >2:
\(\frac{1}{2 -{\text{x}}} \ge\)1
Therefore,
Intervals satisfying the required condition: ≥ 0
1 < x < 2 or x = 1
Merging overlapping intervals:
1 ≤ x < 2
Combining the intervals:
1 ≤ x < 2 and x > 2
Merging the overlapping intervals:
No solution.
Now, combining all the intervals:
No solution or -2 < x ≤ 1 or 1 ≤ x < 2
Merging the overlapping intervals:
-2 < x ≤ 1 or 1 ≤ x < 2
Thus, x є (-2, -1] υ [1,2)
