# Given points A(2,-m),B(m-1,2m-1),B(3m-1,2-3m), what is m if area of ABC is < or equal 9/2?

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You may use the determinant formula to evaluate the area of triangle ABC, such that:

`A_(Delta ABC) = (1/2)|[(2,-m,1),(m-1,2m-1,1),(3m-1,2-3m,1)]|`

`A_(Delta ABC) = (1/2)|(2(2m-1) + (m-1)(2-3m) - m(3m-1) - (3m-1)(2m-1) - 2(2-3m) + m(m-1))|`

`A_(Delta ABC) = (1/2)|(4m - 2 + 2m - 3m^2 - 2 + 3m - 3m^2 + m - 6m^2 + 5m - 1 - 4 + 6m + m^2 - m)|`

`A_(Delta ABC) = (1/2)|(-11m^2 + 20m - 9)|`

The problem requests for you to find m if `A_(Delta ABC) <= 9/2` , hence, you need to solve the following inequality, such that:

`(1/2)|(-11m^2 + 20m - 9)| <= 9/2 => |(-11m^2 + 20m - 9)|<=9`

`-9 <= -11m^2 + 20m - 9 <= 9`

`{(11m^2 - 20m + 9 <= 9),(-11m^2 + 20m - 9 <= 9):} => {(11m^2 - 20m <= 0),(-11m^2 + 20m - 18 <= 0):} => {(m(11m - 20) <= 0),(11m^2 - 20m + 18 >= 0):} `

`m(11m - 20) <= 0 => m in [0,20/11]`

Since `11m^2 - 20m + 18 > 0` for all` m in R` , hence, the solution to the system of inequalities is the following, such that:

`m in [0,20/11] nn R => m in [0,20/11] `

Hence, evaluating m, under the given conditions, yields `m in [0,20/11] .`