Solve each of the following in equations and represent the solution set on

Question:

Solve each of the following in equations and represent the solution set on the number line.

$\frac{1}{2-|x|} \geq$ $1, x \in R-\{-2,2\}$

 

Solution:

Given:

$\frac{1}{2-|x|} \geq 1, x \in \mathrm{R} .-\{-2,2\}$

Intervals of $|x|:$

$x \geq 0,|x|=x$ and $x<0,|x|=-x$

Domain of $\frac{1}{2-|x|} \geq_{1}$

$\frac{1}{2-|x|} \geq 1$ is undefined at $x=-2$ and $x=2$

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

Combining intervals with domain:

$x<-2,-22$

For $x<-2$

$\frac{1}{2-(-x)} \geq 1$

Subtracting 1 from both the sides

$\frac{1}{2+x}-1 \geq_{1-1}$

$\frac{1-(2+x)}{2+x} \geq 0$

$\frac{1-2-x}{2+x} \geq 0$

$\frac{-1-x}{2+x} \geq 0$

Signs of $-1-x$ :

$-1-x=0 \rightarrow x=-1$

(Adding 1 to both the sides and then dividing by -1 on both the sides)

$-1-x>0 \rightarrow x<-1$

(Adding 1 to both the sides and then multiplying by -1 on both the sides)

$-1-x<0 \rightarrow x>-1$

(Adding 1 to both the sides and then multiplying by -1 on both the sides)

Signs of 2 + x:

$2+x=0 \rightarrow x=-2$ (Subtracting 2 from both the sides)

$2+x>0 \rightarrow x>-2$ (Subtracting 2 from both the sides)

$2+x<0 \rightarrow x<-2$ (Subtracting 2 from both the sides)

Intervals satisfying the required condition: ≥ 0

$-2

Merging overlapping intervals:

$-2

Combining the intervals:

$-2

Merging the overlapping intervals:

No solution

Similarly, for $-2

$\frac{1}{2-(-x)} \geq 1$

Therefore,

Intervals satisfying the required condition: ≥ 0

$-2

Merging overlapping intervals:

$-2

Combining the intervals:

$-2

Merging the overlapping intervals:

$-2

For $0 \leq x<2$

$\frac{1}{2-x} \geq 1$

Subtracting 1 from both the sides

$\frac{1}{2-x}-1 \geq_{1-1}$

$\frac{1-(2-x)}{2-x} \geq 0$

$\frac{1-2+x}{2+x} \geq 0$

$\frac{x-1}{2+x} \geq 0$

Signs of x -1:

$x-1=0 \rightarrow x=1$ (Adding 1 to both the sides)

$x-1>0 \rightarrow x>1$ (Adding 1 to both the sides)

$x-1<0 \rightarrow x<1$ (Adding 1 to both the sides)

Signs of 2 + x:

$2+x=0 \rightarrow x=-2$ (Subtracting 2 from both the sides)

$2+x>0 \rightarrow x>-2$ (Subtracting 2 from both the sides)

$2+x<0 \rightarrow x<-2$ (Subtracting 2 from both the sides)

Intervals satisfying the required condition: ≥ 0

$1

Merging overlapping intervals:

1 ≤ x < 2

Combining the intervals:

$1 \leq x<2$ and $0 \leq x<2$

Merging the overlapping intervals:

$1 \leq x<2$

Similarly, for $x>2$ :

$\frac{1}{2-x} \geq 1$

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

Merging the overlapping intervals:

-2 < x ≤ 1 or 1 ≤ x < 2

Thus, $x \in(-2,-1] \cup[1,2)$ 

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