Question

Solve the following inequality for real x:

$1 half open parentheses fraction numerator 3 straight x over denominator 5 end fraction plus 4 close parentheses greater or equal than 1 third left parenthesis straight x minus 6 right parenthesis$

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Solution

space space 1 half open parentheses fraction numerator 3 straight x over denominator 5 end fraction plus 4 close parentheses space greater or equal than space 1 third left parenthesis straight x minus 6 right parenthesis space rightwards double arrow space fraction numerator 3 straight x over denominator 10 end fraction plus 2 greater or equal than straight x over 3 minus 2

Multiply by 30, we have

9 straight x plus 60 greater or equal than 10 straight x minus 60

rightwards double arrow

9 straight x minus 10 straight x greater or equal than negative 60 minus 60 space rightwards double arrow space minus straight x greater or equal than negative 120 space rightwards double arrow space straight x less or equal than 120

(Divide by -1)

rightwards double arrow

straight x element of space left parenthesis negative infinity comma space 120 right square bracket

Hence, the solution set is

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