Question

How do you solve `7/(x+3)<-5`?

Answer 1

`(-22/5,-3)`

Explanation:

`"add 5 to both sides"`

`rArr7/(x+3)+5 < 0`

`"express as a single rational function"`

`(7+5(x+3))/(x+3) < 0`

`rArr(5x+22)/(x+3) < 0`

`"the zeros of the numerator/denominator are"`

`"numerator "x=-22/5," denominator "x=-3`

These indicate where the rational function may change sign.

`"the intervals for consideration are"`

`x < -22/5,color(white)(x)-22/5 < x < -3,color(white)(x)x > -3`

`"consider a "color(blue)"test point "" in each interval"`

`"we want to find where the function is negative"`

`"substitute each test point into the function and consider"`
`"it's sign"`

`color(magenta)"x = - 5"to(-)/(-)tocolor(red)" positive"`

`color(magenta)"x = - 4 "to(+)/(-)tocolor(blue)" negative"`

`color(magenta)"x = 4"to(+)/(+)tocolor(red)" positive"`

`rArr-22/5 < x < -3" or " (-22/5,-3)`