Question

Дробово раціональне рівняння

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

Answer 1

`x=(1)/(2)` or 0.5 in decimal form

Explanation:

The expression is
`(x+1)/(x-2)-(x-3)/(x+2)=0`

Find Least Common Multiplier (LCM) of the denominators
`x-2,\:x+2`

This is
`(x-2)(x+2)` the product of the two denominators

Multiply entire equation by this LCM:
`(x-2)(x+2)`

We get

`(x+1)/(x-2)\left(x-2\right)\left(x+2\right)-(x-3)/(x+2)\left(x-2\right)\left(x+2\right)=0\cdot \left(x-2\right)\left(x+2\right)`

Simplify the first term

`(x+1)/(x-2)\left(x-2\right)\left(x+2\right)`
`=\left(x+1\right)\left(x+2\right)`

Simplify the second term

`-(x-3)/(x+2)\left(x-2\right)\left(x+2\right) =\quad -\left(x-3\right)\left(x-2\right)`

RHS is
`0`

Therefore we get

`\left(x+1\right)\left(x+2\right)-\left(x-3\right)\left(x-2\right)=0`

Expand the first term using the FOIL method
`(x+a) (x + b) = x^2 + ax + bx + ab`

Here a = 1, b = 2. So
`(x+1)(x+2) = x^2 + 2x + 1x + (1)(2) = x^2 + 3x + 2`

`\left(x+1\right)\left(x+2\right)`
`=x^2+3x+2`

Expanding
`-\left(x-3\right)\left(x-2\right)` gives us
`-x^2 + 5x - 6`

(Use the FOIL method:
`(x+b) (x + a) = x^2 + ax + bx + ab` Here a = -2, b = -3. Note there is a negative sign before the entire expression)

So the original expression is

`x^2+3x+2-x^2+5x-6 = 0`

Collecting like terms

`x^2-x^2 + 3x +5x + 2 - 6`
`= 8x -4` =
`0`

Add 4 to both sides

`8x-4+4=0+4`

`8x=4`

Divide both sides by 8:

`(8x)/(8)=(4)/(8)`

`x=(1)/(2)` or
`0.5` in decimal