Solve: `(x-2)/(x-3)+(x-4)/(x-5)=3(1)/(3),xne3,5.`

Options

A. `6, 7/2`
B. `5, 5/2`
C. `7, 3/2`
D. `4, 5/2`

Answer

A

Explanation

We have
`(x-2)/(x-3)+(x-4)/(x-5)=(10)/(3)`
`implies" "((x-2)(x-5)+(x-4)(x-3))/((x-3)(x-5))=(10)/(3)`
`implies" "((x^(2)-7x+10)+(x^(2)-7x+12))/((x^(2)-8x+15))=(10)/(3)`
`implies" "((2x^(2)-14x+22))/((x^(2)-8x+15))=(10)/(3)`
`implies" "3(2x^(2)-14x+22)=10(x^(2)-8x+15)" "["by cross multiplication"]`
`implies" "6x^(2)-42x+66=10x^(2)-80x+150`
`implies" "4x^(2)-38x+84=0implies2x^(2)-19x+42=0`
`implies" "2x^(2)-12x-7x+42=0implies2x(x-6)-7(x-6)=0`
`implies" "(x-6)(2x-7)=0impliesx-6=0" or "2x-7=0`
`x=6" or "x=(7)/(2).`
Hence, `6` and `(7)/(2)` are the roots of the given equation.