✓ Got It?

1. What is the LCM of the expressions?
   1. 2 x + 4 2 x plus 4  and x 2 − x − 6 x squared , minus x minus 6
   2. x 2 + 3 x − 4 , x 2 + 2 x − 8 , x squared , plus 3 x minus 4 comma , x squared , plus 2 x minus 8 comma  and x 2 − 4 x + 4 x squared , minus 4 x plus 4

Problem 2 Adding Rational Expressions

What is the sum of the two rational expressions in simplest form? State any restrictions on the variable. x x − 1 + 2 x − 1 x 2 − 3 x + 2 fraction x , over x minus 1 end fraction . plus . fraction 2 x minus 1 , over x squared , minus 3 x plus 2 end fraction

x x − 1 + 2 x − 1 x 2 − 3 x + 2 = x x − 1 + 2 x − 1 ( x − 1 ) ( x − 2 ) Factor the denominators .   = x x − 1 ⋅ x − 2 x − 2 + 2 x − 1 ( x − 1 ) ( x − 2 ) Rewrite each expression with the LCD .   = x 2 − 2 x ( x − 1 ) ( x − 2 ) + 2 x − 1 ( x − 1 ) ( x − 2 )     = x 2 − 2 x + 2 x − 1 ( x − 1 ) ( x − 2 ) Add the numerators . Combine like terms .   = x 2 − 1 ( x − 1 ) ( x − 2 )     = ( x − 1 ) ( x + 1 ) ( x − 1 ) ( x − 2 ) Factor the numerator and divide out the common factors .   = x + 1 x − 2 , x ≠ 1   table with 7 rows and 4 columns , row1 column 1 , fraction x , over x minus 1 end fraction . plus . fraction 2 x minus 1 , over x squared , minus 3 x plus 2 end fraction , column 2 equals , column 3 fraction x , over x minus 1 end fraction . plus . fraction 2 x minus 1 , over open x minus 1 close open x minus 2 close end fraction , column 4 cap factorthedenominators . . , row2 column 1 , , column 2 equals , column 3 fraction x , over x minus 1 end fraction . dot . fraction x minus 2 , over x minus 2 end fraction . plus . fraction 2 x minus 1 , over open x minus 1 close open x minus 2 close end fraction , column 4 cap rewriteeachexpressionwiththecap lcap ccap d . . , row3 column 1 , , column 2 equals , column 3 fraction x squared , minus 2 x , over open x minus 1 close open x minus 2 close end fraction . plus . fraction 2 x minus 1 , over open x minus 1 close open x minus 2 close end fraction , column 4 , row4 column 1 , , column 2 equals , column 3 fraction x squared , minus 2 x plus 2 x minus 1 , over open x minus 1 close open x minus 2 close end fraction , column 4 cap addthenumerators . .cap combineliketerms . . , row5 column 1 , , column 2 equals , column 3 fraction x squared , minus 1 , over open x minus 1 close open x minus 2 close end fraction , column 4 , row6 column 1 , , column 2 equals , column 3 fraction open , cross out enclosing , x minus 1 , end crossout , close open x plus 1 close , over open , cross out enclosing , x minus 1 , end crossout , close open x minus 2 close end fraction , column 4 table with 2 rows and 1 column , row1 column 1 , cap factorthenumeratoranddivideoutthe , row2 column 1 , commonfactors . . , end table , row7 column 1 , , column 2 equals , column 3 fraction x plus 1 , over x minus 2 end fraction . comma x not equal to 1 , column 4 , end table

Th e sum of the expressions is x + 1 x − 2 fraction x plus 1 , over x minus 2 end fraction  for x ≠ 1 x not equal to 1  and x ≠ 2 . x not equal to 2 .

✓ Got It?

2. What is the sum of the two rational expressions in simplest form? State any restrictions on the variable.
   1. x + 1 x − 1 + − 2 x 2 − x fraction x plus 1 , over x minus 1 end fraction . plus . fraction negative 2 , over x squared , minus x end fraction
   2. x x 2 − 4 + 1 x + 2 fraction x , over x squared , minus 4 end fraction . plus . fraction 1 , over x plus 2 end fraction
   3. Reasoning Is it possible to add the rational expressions in Problem 2 by finding a common denominator, but not the least common denominator? Explain.