Question

What are the boundaries of x if `(2x-1)/(x+5) >= (x+2)/(x+3)?`

Answer 1

`x = -5, x = -3, x = 1-sqrt(14), x = 1+sqrt(14)`
`>=" occurs for "x <-5" and "x>=1+sqrt(14)" and"`
`-3 < x <= 1-sqrt(14)"."`

Explanation:

`=> (2x-1)/(x+5) - (x+2)/(x+3) >= 0`

`=> ((2x-1)(x+3) - (x+2)(x+5))/((x+5)(x+3)) >= 0`

`=> (2x^2+5x-3-x^2-7x-10)/((x+5)(x+3)) >= 0`

`=> (x^2 -2x-13)/((x+5)(x+3)) >= 0`

`=> ((x - 1 - sqrt(14))(x - 1 + sqrt(14)))/((x+5)(x+3)) >= 0`

`"We have following zeros in order of magnitude : "`

`.... -5 .... -3 .... 1-sqrt(14)....1+sqrt(14).....`
`-----------0+++`
`-------0+++++++`
`-----0+++++++++`
`--0++++++++++++`
`"========================="`
`++0---0++0---0+++`

`"We see ">=0" occurs for "x <-5" and "x>=1+sqrt(14)" and"`
`-3 < x <= 1-sqrt(14)"."`