11-4 Adding and Subtracting Rational Expressions

Quick Review

You can add and subtract rational expressions. To add or subtract expressions with like denominators, add or subtract the numerators and write the result over the common denominator. To add or subtract expressions with different denominators, write the expressions with the LCD and then add or subtract the numerators.

Example

What is 1 a + 7 + a a − 5 ? fraction 1 , over eh plus 7 end fraction . plus . fraction eh , over eh minus 5 end fraction . question mark

1 a + 7 + a a − 5 = 1 ( a − 5 ) ( a + 7 ) ( a − 5 ) + a ( a + 7 ) ( a + 7 ) ( a − 5 )   = a − 5 ( a + 7 ) ( a − 5 ) + a 2 + 7 a ( a + 7 ) ( a − 5 )   = a − 5 + a 2 + 7 a ( a + 7 ) ( a − 5 )   = a 2 + 8 a − 5 ( a + 7 ) ( a − 5 ) table with 4 rows and 2 columns , row1 column 1 , fraction 1 , over eh plus 7 end fraction . plus . fraction eh , over eh minus 5 end fraction , column 2 equals . fraction 1 . open , eh minus 5 , close , over open , eh plus 7 , close . open , eh minus 5 , close end fraction . plus . fraction eh . open , eh plus 7 , close , over open , eh plus 7 , close . open , eh minus 5 , close end fraction , row2 column 1 , , column 2 equals . fraction eh minus 5 , over open , eh plus 7 , close . open , eh minus 5 , close end fraction . plus . fraction eh squared , plus 7 eh , over open , eh plus 7 , close . open , eh minus 5 , close end fraction , row3 column 1 , , column 2 equals . fraction eh minus 5 plus , eh squared , plus 7 eh , over open , eh plus 7 , close . open , eh minus 5 , close end fraction , row4 column 1 , , column 2 equals . fraction eh squared , plus 8 eh minus 5 , over open , eh plus 7 , close . open , eh minus 5 , close end fraction , end table

Exercises

Add or subtract.

17. 8 x x + 1 − 3 x + 1 fraction 8 x , over x plus 1 end fraction . minus . fraction 3 , over x plus 1 end fraction
18. 6 7 x + 1 4 fraction 6 , over 7 x end fraction , plus , 1 fourth
19. 5 2 + x + x x − 4 fraction 5 , over 2 plus x end fraction . plus . fraction x , over x minus 4 end fraction
20. 9 3 x − 1 − 5 x 2 x + 3 fraction 9 , over 3 x minus 1 end fraction . minus . fraction 5 x , over 2 x plus 3 end fraction
21. Air Travel The distance between Atlanta, Georgia, and Albuquerque, New Mexico, is about 1270 mi. The groundspeed for jet traffic from Albuquerque to Atlanta can be about 18% faster than the groundspeed from Atlanta to Albuquerque. Let r be the speed from Atlanta to Albuquerque in miles per hour. What is a simplified expression for the round-trip flying time?

11-5 Solving Rational Equations

Quick Review

You can solve a rational equation by multiplying each side by the LCD. Check possible solutions to make sure each satisfies the original equation.

Example

What is the solution of 3 8 + 4 x = 7 x ? 3 eighths , plus , 4 over x , equals , 7 over x . question mark

3 8 + 4 x = 7 x 8 x ( 3 8 + 4 x ) = 8 x ( 7 x ) 8 1 x ( 3 8 1 ) + 8 x 1 ( 4 x 1 ) = 8 x 1 ( 7 x 1 ) 3 x + 32 = 56 3 x = 24 x = 8 table with 6 rows and 2 columns , row1 column 1 , 3 eighths , plus , 4 over x , column 2 equals , 7 over x , row2 column 1 , 8 x . open . 3 eighths , plus , 4 over x . close , column 2 equals 8 x . open , 7 over x , close , row3 column 1 , cross out enclosing , 8 , end crossout to the first , x . open . fraction 3 , over prescript , . lower , 1 with base cross out enclosing , 8 , end crossout . end fraction . close . plus 8 , cross out enclosing , x , end crossout to the first . open , fraction 4 , over

Check 3 8 + 4 8 = ? 7 8 7 8 = 7 8 ✓ table with 2 rows and 2 columns , row1 column 1 , 3 eighths , plus , 4 eighths , column 2 modified equals with question mark above , 7 eighths , row2 column 1 , 7 eighths , column 2 equals , 7 eighths , check mark , end table

Exercises

Solve each equation. Check your solutions.

22. 1 2 + 3 t = 5 8 1 half , plus , 3 over t , equals , 5 eighths
23. 3 m − 4 + 1 3 ( m − 4 ) = 6 m fraction 3 , over m minus 4 end fraction . plus . fraction 1 , over 3 . open , m minus 4 , close end fraction . equals , 6 over m
24. 2 c c − 4 − 2 = 4 c + 5 fraction 2 c , over c minus 4 end fraction . minus 2 equals . fraction 4 , over c plus 5 end fraction
25. 5 2 x − 3 = 7 3 x fraction 5 , over 2 x minus 3 end fraction . equals , fraction 7 , over 3 x end fraction
26. Business A new photocopier can make 72 copies in 2 min. When an older photocopier is operational, the two photocopiers together can make 72 copies in 1.5 min. How long would it take the older photocopier to make 72 copies working alone?

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