Question 328818
Break up the number line into 3 regions.
Region 1: ({{{-infinity}}},{{{0}}})
Region 2: ({{{0}}},{{{5/3}}})
Region 3: ({{{5/3}}},{{{infinity}}})
For each region, pick any point in the region (not an endpoint).
Test the inequality.
If the inequality is satisifed, that region is part of the solution.
THen move to the next region.
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Region 1: Let {{{x=-1}}},
 {{{ (3x-5) / x > 0 }}}
{{{ (3(-1)-5) / (-1) > 0 }}}
{{{-8/-1>0}}}
{{{8>0}}}
True, this region is part of the solution.
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Region 2: Let {{{x=1}}},
 {{{ (3x-5) / x > 0 }}}
{{{ (3(1)-5) / (1) > 0 }}}
{{{-2/1>0}}}
{{{-2>0}}}
False, this region is not part of the solution.
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Region 3: Let {{{x=2}}},
 {{{ (3x-5) / x > 0 }}}
{{{ (3(2)-5) / (2) > 0 }}}
{{{1/2>0}}}
True, this region is part of the solution.
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The solution is then ({{{-infinity}}},{{{0}}}) U ({{{5/3}}},{{{infinity}}})
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Verify graphically,
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{{{drawing(300,300,-5,5,-10,10,blue(line(5/3,-10,5/3,10)),graph(300,300,-5,5,-10,10,(3x-5)/x))}}}