Question

Solve the following inequality for real x:

$fraction numerator 3 space left parenthesis straight x minus 2 right parenthesis over denominator 5 end fraction less or equal than fraction numerator 5 space left parenthesis 2 straight x minus 3 right parenthesis over denominator 3 end fraction$

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Solution

fraction numerator 3 space left parenthesis straight x minus 2 right parenthesis over denominator 5 end fraction less or equal than fraction numerator 5 space left parenthesis 2 minus straight x right parenthesis over denominator 3 end fraction

Mulitply by 15, we have

9 space left parenthesis straight x minus 2 right parenthesis space less or equal than space 25 space left parenthesis 2 minus straight x right parenthesis

rightwards double arrow

9 straight x minus 18 space less or equal than space 50 minus 25 straight x space rightwards double arrow space 9 straight x plus 25 straight x less or equal than 50 plus 18 space rightwards double arrow space 34 straight x less or equal than 68 space rightwards double arrow space straight x less or equal than 2

rightwards double arrow

straight x space element of space left parenthesis negative infinity comma space 2 right square bracket

Hence, the solution set is

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