InterviewSolution
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InterviewSolution
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Solve: `(1)/(|x|-3)le(1)/(2),x in R` |
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Answer» Correct Answer - `*(-oo,-5]uu(-3,3)uu[5,oo)` Case I When `x ge 0` Then, `|x|=x` `therefore (1)/(|x|-3)le(1)/(2)rArr(1)/(x-3)-(1)/(2)le0rArr(2-x+3)/(2(x-3))le0rArr(5-x)/(x-3)le0`. `therefore (5-xle0 and x-3gt0) or(5-x ge0 and x-3lt0)` `rArr (xge5 and x gt3) or(x le5 and x lt3)` `rArr(xge5) or (xlt3)` `rArr (0lex lt3) or(xge5)` `rArr x in [0,3) uu[5,oo)" "[therefore x ge 0]` Case II When `x lt 0`. Then, `|x| = -x` `therefore(1)/(|x|-3)le(1)/(2) rArr(1)/(x-3)le(1)/(2)rArr(-1)/(x+3)le(1)/(2)rArr (1)/(2)+(1)/(x+3)le0` `rArr (x+3+2)/(x+3)ge0 rArr (x+5)/(X+3)ge0` `therefore (x+5 ge0and x+3gt0) or(x+5le0 and x+3 lt0)` `rArr(xgt-5 and xgt-3) or (xle-5 and x lt-3)` `rArr (xgt-3) or (xle-5)`. Bu `xlt0`. `therefore(-3ltxlt0)or(xle-5)rArr x in(-3,0)uu(-oo,-5]`. Hence, ` x in(-oo,-5]uu(-3,0)uu[0,3)uu[5,oo)` `x in(-oo,-5] uu(-3,3)uu[5,oo)`. |
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