solve the inequality `{6x-2}/{x+2}<=5` 

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lfryerda | High School Teacher | (Level 2) Educator

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To solve the inequality, we need to put the inequality to the left side and then write in factored form.  Then we find points of interest of the inequality that we can then use.

`{6x-2}/{x+2}<=5`   move to left side

`{6x-2}/{x+2}-5<=0`   get common denominator

`{6x-2-5(x+2)}/{x+2}<=0`    simplify numerator

`{6x-2-5x-10}/{x+2}<=0`   simplify numerator further

`{x-12}/{x+2}<=0`

Points of interest of the inequality are the zeros of the numerator and the denominator, which are x=12 and x=-2.

For `x>12` , we see that the numerator is positive and the denominator is positive, so the function is also positive.  This is not a solution.

For `-2<x<=12` , the numerator is negative and the denominator is positive, so the fuunction is negative.  This is a solution.

For `x<-2` , the numerator is negative and the denominator is negative, so the function is positive.  This is not a solution.

The solution is the interval `-2<x<=12` .

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