Problem 3 Subtracting Rational Expressions

What is the difference of the two rational expressions in simplest form? State any restrictions on the variable. x + 2 x 2 − 2 x − x + 2 2 x − 4 fraction x plus 2 , over x squared , minus 2 x end fraction . minus . fraction x plus 2 , over 2 x minus 4 end fraction

The LCD is 2 x ( x − 2 ) . 2 x open x minus 2 close .

= x + 2 x ( x − 2 ) ⋅ 2 2 − x + 2 2 ( x − 2 ) ⋅ x x Rewrite each expression with the LCD . = 2 ( x + 2 ) 2 x ( x − 2 ) − x ( x + 2 ) 2 x ( x − 2 )   = 2 x + 4 2 x ( x − 2 ) − x 2 + 2 x 2 x ( x − 2 ) Simplify the numerators . = 2 x + 4 − ( x 2 + 2 x ) 2 x ( x − 2 ) Subtract the numerators . = − x 2 + 4 2 x ( x − 2 ) Combine like terms . = − ( x 2 − 4 ) 2 x ( x − 2 ) Factor  − 1  from the numerator . = − ( x − 2 ) ( x + 2 ) 2 x ( x − 2 ) Factor  x 2 − 4  and divide out the common factors . = − ( x + 2 ) 2 x   table with 8 rows and 3 columns , row1 column 1 , equals , column 2 fraction x plus 2 , over x open x minus 2 close end fraction . dot , 2 halves , minus . fraction x plus 2 , over 2 open x minus 2 close end fraction . dot , x over x , column 3 table with 2 rows and 1 column , row1 column 1 , cap rewriteeachexpression , row2 column 1 , withthecap lcap ccap d . . , end table , row2 column 1 , equals , column 2 fraction 2 open x plus 2 close , over 2 x open x minus 2 close end fraction . minus . fraction x open x plus 2 close , over 2 x open x minus 2 close end fraction , column 3 , row3 column 1 , equals , column 2 fraction 2 x plus 4 , over 2 x open x minus 2 close end fraction . minus . fraction x squared , plus 2 x , over 2 x open x minus 2 close end fraction , column 3 cap simplifythenumerators . . , row4 column 1 , equals , column 2 fraction 2 x plus 4 minus open , x squared , plus 2 x close , over 2 x open x minus 2 close end fraction , column 3 cap subtractthenumerators . . , row5 column 1 , equals , column 2 fraction negative , x squared , plus 4 , over 2 x open x minus 2 close end fraction , column 3 cap combineliketerms . . , row6 column 1 , equals , column 2 fraction negative open , x squared , minus 4 close , over 2 x open x minus 2 close end fraction , column 3 cap factor , minus 1 . fromthenumerator . . , row7 column 1 , equals , column 2 fraction negative open , cross out enclosing , x minus 2 , end crossout , close open x plus 2 close , over 2 x open , cross out enclosing , x minus 2 , end crossout , close end fraction , column 3 cap factor . x squared , minus 4 . anddivideoutthecommonfactors . . , row8 column 1 , equals , column 2 fraction negative open x plus 2 close , over 2 x end fraction , column 3 , end table

The difference is − ( x + 2 ) 2 x fraction negative open x plus 2 close , over 2 x end fraction  for x ≠ 2 x not equal to 2  and x ≠ 0 . x not equal to 0 .

✓ Got It?

3. What is the difference of the two rational expressions in simplest form? State any restrictions on the variable.
   1. x + 3 x − 2 − 6 x − 7 x 2 − 3 x + 2 fraction x plus 3 , over x minus 2 end fraction . minus . fraction 6 x minus 7 , over x squared , minus 3 x plus 2 end fraction
   2. x − 1 x + 5 − x + 3 x 2 + 6 x + 5 fraction x minus 1 , over x plus 5 end fraction . minus . fraction x plus 3 , over x squared , plus 6 x plus 5 end fraction