Solve the following inequality and write your answer using interval notation. x/4x+1 < 5x/6

Expert Answers
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You need to solve the inequality `x/(4x+1) lt 5x/6` , hence you need to move all terms containing x to the left side such that:

`x/(4x+1)- 5x/6 lt 0`

You need to bring the fractions to a common denominator such that:

`(6x - 5x(4x+1))/(6(4x+1)) lt 0`

Opening the brackets yields:

`(6x - 20x^2 - 5x)/(6(4x+1)) lt 0`

`(x-20x^2)/(6(4x+1)) lt 0`

You need to remember that the value of  fraction is less than zero if numerator is less than denominator, hence:

`x-20x^2 lt 6(4x+1)`

You need to bring all terms to the left side:

`x - 20x^2 - 24x - 6 lt 0 =gt - 20x^2 - 23x - 6 lt 0`

Multiplying by -1 the sense of inequality reverses such that:

`20x^2+ 23x+ 6 gt 0`

You need to find the zeroes of the quadratic `20x^2+ 23x+ 6= 0,`  hence you need to use quadratic formula such that:

`x_(1,2) = (-23+-sqrt(529 - 480))/(40)`  

`x_1 = (-23+7)/40 =gt x_1 = -16/40 =gt x_1 = -4/10 =gt x_1 = -2/5`

`x_2 = -30/40 =gt x_2 = -3/4`

Hence, the values of `x in (-oo,-3/4)U(-2/5,oo)`  verify the inequality.