Solve the inequality 2x-1/ 4x+1 <1

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cosinusix | College Teacher | (Level 3) Assistant Educator

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We won't get in the trap of multiplying the inequality by `4x+1` because we don't know the sign of `4x+1` therefore we don't know whether it is going to be < or >.

 

Let's try something different!

`(2x-1)/(4x+1)lt1`

iff `(2x-1)/(4x+1)-1lt0`

iff` (2x-1)/(4x+1)-(4x+1)/(4x+1)lt0`

iff `(2x-1-4x-1)/(4x+1)lt0`

iff `(-2x-2)/(4x+1)lt0`

iff `(x+1)/(4x+1)gt0 ` (divide by -2 that is negative and change the orientation of the inequality)

 

the  sign of the quotient is going change for `x+1=0` and for`4x+1=0`

ie for `x=-1 ` and `x=-1/4.`

 

if `xlt-1, x+1lt0 4x+1lt0 ` therefore (`x+1)/(4x+1)gt0 `

if `-1ltxlt-1/4, x+1gt0 4x+1lt0 ` therefore `(x+1)/(4x+1)lt0`

if `xgt-1/4, x+1gt0 4x+1gt0` therefore

`(x+1)/(4x+1)gt0 `

 

Conclusion Solution `S=(-oo,-1)uu(-1/4,oo)`

 

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