Simplify the expressions and solve the equations.

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embizze | High School Teacher | (Level 1) Educator Emeritus

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(1) Simplify `(v^2+v-12)/(v^2+6v+8) -:(2v-6)/(v+2)` Factor each term:

`((v+4)(v-3))/((v+4)(v+2))-: (2(v-3))/(v+2)` Divide is the same as multiply by reciprocal

`((v+4)(v-3))/((v+4)(v+2))*(v+2)/(2(v-3))` Use the multiplicative identity to reduce common factors in the numerator and denominator (i.e. cancel common factors):

`1/2`

So the expression simplifies to `1/2`

(2) Solve `2/(b-2)=b/(b^2-3b+2)+b/(2b-2)`

`2/(b-2)=b/((b-2)(b-1))+b/(2(b-1))`

The least common denominator is `2(b-1)(b-2)` -- multiply everything by the LCD to get:

`2(2)(b-1)=b(2)+b(b-2)`

`4b-4=2b+b^2-2b`

`b^2-4b+4=0`

`(b-2)^2=0`

`==> b=2`

However, if b=2 then the left side of the original equation is undefined, so b=2 will not work.

There is no solution to the equation.

Here is the graph of the left side in red, and the right side in blue. Any solution would be where the graphs intersect: