Q. 295.0( 1 Vote )

# Solve each of the

Answer :

Given:

1, x ϵ R. – {–2, 2}

Intervals of |x|:

x ≥ 0, |x| = x and x < 0, |x| = -x

Domain of 1

1 is undefined at x = -2 and x = 2

Therefore, Domain: x < -2 or x > 2

Combining intervals with domain:

x<-2, -2<x<0, 0 ≤ x<2, x>2

For x < - 2

1

Subtracting 1 from both the sides

1 -1

0

0

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:

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

1

Subtracting 1 from both the sides

1 -1

0

0

0

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:

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)

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