Answer to Question #228620 in Algebra for Jackson

Question

, The solutions of the inequality IIxI-1I <=1/2 are ?

Answer 1

$\lvert \lvert x \rvert - 1\rvert \leq \cfrac {1}{2}$

There are 3 cases:

1) $\lvert x \rvert - 1 = 0$

2) $\lvert x \rvert - 1 > 0$

3) $\lvert x \rvert - 1 < 0$

1) $\lvert x \rvert - 1 = 0$

$\lvert x \rvert = 1 \\ x \in \lbrace -1; 1\rbrace$ - satisfies the inequality.

2) $\lvert x \rvert - 1 > 0 :$

$0 < \lvert x \rvert - 1 \leq \cfrac{1}{2} \\ 1 < \lvert x \rvert \leq \cfrac{3}{2}$

Answer: $x \in \Big [ - \cfrac {3}{2} ; -1 \Big \} \bigcup \Big \{ 1; \cfrac{3}{2} \Big ]$

3) $\lvert x \rvert - 1 < 0 :$

$-\cfrac{1}{2} \leq \lvert x \rvert - 1 < 0 \\ \cfrac{1}{2} \leq \lvert x \rvert < 1$

Answer: $x \in \Big \{ - 1 ; - \cfrac{1}{2} \Big ] \bigcup \Big [ \cfrac{1}{2} ; 1 \Big \}$

Final answer: $x \in \Big [ - \cfrac{3}{2}; - \cfrac{1}{2} \Big ] \bigcup \Big [ \cfrac{1}{2} ; \cfrac {3}{2} \Big ]$