Solve the following inequality for real x:

$fraction numerator left parenthesis 2 straight x minus 1 right parenthesis over denominator 3 end fraction greater or equal than fraction numerator left parenthesis 3 straight x minus 2 right parenthesis over denominator 4 end fraction minus fraction numerator left parenthesis 2 minus straight x right parenthesis over denominator 5 end fraction$

Solution

fraction numerator left parenthesis 2 straight x minus 1 right parenthesis over denominator 3 end fraction greater or equal than fraction numerator left parenthesis 3 straight x minus 2 right parenthesis over denominator 4 end fraction minus fraction numerator left parenthesis 2 minus straight x right parenthesis over denominator 5 end fraction

Multiply by L.C.M. 60 of 3, 4, and 5, we have

rightwards double arrow

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rightwards double arrow

40 straight x minus 45 straight x minus 12 straight x greater or equal than negative 30 minus 24 plus 20 space rightwards double arrow space minus 17 straight x greater or equal than negative 34 space rightwards double arrow space fraction numerator negative 17 straight x over denominator negative 17 end fraction less or equal than fraction numerator negative 34 over denominator negative 17 end fraction space rightwards double arrow space straight x less or equal than 2

[∵ division by negative number reverses the inequality]

rightwards double arrow

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Hence, the solution set is

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