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Find the values of 1/x for the following...

Find the values of 1/x for the following values fo x:
(a) `(2,5) " (b) "[-5,-1) " (c ) "(3,oo)`
(d) `(-oo,-2] " (e ) "[-3,4]`

Text Solution

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The correct Answer is:
(a) `((1)/(5),(1)/(2)) " (b) "(-1,-(1)/(5)]" (c ) " (0,(1)/(3)) " (d) " [-(1)/(2), 0) " (e ) "(- oo, -(1)/(3)] cup [(1)/(4),oo)`
(a) `2 lt x lt 5`
`implies (1)/(2) gt (1)/(x) gt (1)/(5)`
`implies (1)/(x) in ((1)/(5),(1)/(2))`
(b) `-5 le x lt -1`
`implies -(1)/(5) ge (1)/(x) gt -1`
`implies (1)/(x) in (-1,-(1)/(5)]`
(c ) ` x gt 3 or 3 lt x lt oo`
`implies(1)/(3) gt (1)/(x) gt 0`
`(1)/(x) in (0,(1)/(3))`
(d) `x le -2 or -oo lt x le -2`
`implies 0 gt (1)/(x) ge -(1)/(2)`
`implies (1)/(x) in [-(1)/(2),0)`
(e) ` x in [-3,4] or -3 le x le 4`
Now for `(1)/(x)` to get defined, we must have `-3 le x lt 0 or 0 lt x le 4`
`implies -(1)/(3) ge (1)/(x) gt -oo or (1)/(4) le (1)/(x) lt oo`
`implies (1)/(x) in (-oo,-(1)/(3)] cup [(1)/(4),oo)`
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