, The solutions of the inequality IIxI-1I <=1/2 are ?
∣∣x∣−1∣≤12\lvert \lvert x \rvert - 1\rvert \leq \cfrac {1}{2}∣∣x∣−1∣≤21
There are 3 cases:
1) ∣x∣−1=0\lvert x \rvert - 1 = 0∣x∣−1=0
2) ∣x∣−1>0\lvert x \rvert - 1 > 0∣x∣−1>0
3) ∣x∣−1<0\lvert x \rvert - 1 < 0∣x∣−1<0
∣x∣=1x∈{−1;1}\lvert x \rvert = 1 \\ x \in \lbrace -1; 1\rbrace∣x∣=1x∈{−1;1} - satisfies the inequality.
2) ∣x∣−1>0:\lvert x \rvert - 1 > 0 :∣x∣−1>0:
0<∣x∣−1≤121<∣x∣≤320 < \lvert x \rvert - 1 \leq \cfrac{1}{2} \\ 1 < \lvert x \rvert \leq \cfrac{3}{2}0<∣x∣−1≤211<∣x∣≤23
Answer: x∈[−32;−1}⋃{1;32]x \in \Big [ - \cfrac {3}{2} ; -1 \Big \} \bigcup \Big \{ 1; \cfrac{3}{2} \Big ]x∈[−23;−1}⋃{1;23]
3) ∣x∣−1<0:\lvert x \rvert - 1 < 0 :∣x∣−1<0:
−12≤∣x∣−1<012≤∣x∣<1-\cfrac{1}{2} \leq \lvert x \rvert - 1 < 0 \\ \cfrac{1}{2} \leq \lvert x \rvert < 1−21≤∣x∣−1<021≤∣x∣<1
Answer: x∈{−1;−12]⋃[12;1}x \in \Big \{ - 1 ; - \cfrac{1}{2} \Big ] \bigcup \Big [ \cfrac{1}{2} ; 1 \Big \}x∈{−1;−21]⋃[21;1}
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